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Annals of Mathematics The Annals of Mathematics is a mathematical journal published every two months by Princeton University and the Institute for Advanced Study. Not to be confused with Mathematische Annalen. Annals of Mathematics DisciplineMathematics LanguageEnglish Edited byNicholas M. Katz, Sergiu Klainerman, Fernando C. Marques, Assaf Naor, Peter Sarnak, Zoltán Szabó Publication details Former name(s) The Analyst History1874–present Publisher Princeton University and the Institute for Advanced Study (United States) FrequencyEvery two months Open access Delayed, after 5 years Impact factor 5.24 (2019) Standard abbreviations ISO 4 (alt) · Bluebook (alt1 · alt2) NLM (alt) · MathSciNet (alt ) ISO 4Ann. Math. MathSciNetAnn. of Math. Indexing CODEN (alt · alt2) · JSTOR (alt) · LCCN (alt) MIAR · NLM (alt) · Scopus CODENANMAAH ISSN0003-486X LCCN49006640 JSTOR0003486X OCLC no.01481391 Links • Journal homepage History The journal was established as The Analyst in 1874[1] and with Joel E. Hendricks as the founding editor-in-chief. It was "intended to afford a medium for the presentation and analysis of any and all questions of interest or importance in pure and applied Mathematics, embracing especially all new and interesting discoveries in theoretical and practical astronomy, mechanical philosophy, and engineering".[2] It was published in Des Moines, Iowa, and was the earliest American mathematics journal to be published continuously for more than a year or two.[3] This incarnation of the journal ceased publication after its tenth year, in 1883, giving as an explanation Hendricks' declining health,[4] but Hendricks made arrangements to have it taken over by new management,[5] and it was continued from March 1884 as the Annals of Mathematics.[6] The new incarnation of the journal was edited by Ormond Stone (University of Virginia). It moved to Harvard in 1899 before reaching its current home in Princeton in 1911. An important period for the journal was 1928–1958 with Solomon Lefschetz as editor.[7] During this time, it became an increasingly well-known and respected journal. Its rise, in turn, stimulated American mathematics. Norman Steenrod characterized Lefschetz' impact as editor as follows: "The importance to American mathematicians of a first-class journal is that it sets high standards for them to aim at. In this somewhat indirect manner, Lefschetz profoundly affected the development of mathematics in the United States."[7] Princeton University continued to publish the Annals on its own until 1933, when the Institute for Advanced Study took joint editorial control. Since 1998 it has been available in an electronic edition, alongside its regular print edition. The electronic edition was available without charge, as an open access journal, but since 2008 this is no longer the case. Issues from before 2003 were transferred to the non-free JSTOR archive, and articles are not freely available until 5 years after publication. Editors The current (as of October 2022) editors of the Annals of Mathematics are Helmut Hofer, Nick Katz, Sergiu Klainerman, Fernando Codá Marques, Assaf Naor, Peter Sarnak and Zoltán Szabó (all but Helmut Hofer from Princeton University, with Hofer being a professor at the Institute for Advanced Study and Peter Sarnak also being a professor there as a second affiliation).[8] Abstracting and indexing The journal is abstracted and indexed in the Science Citation Index, Current Contents/Physical, Chemical & Earth Sciences, [9] and Scopus.[10] According to the Journal Citation Reports, the journal has a 2020 impact factor of 5.246, ranking it third out of 330 journals in the category "Mathematics".[11] References 1. Diana F. Liang, Mathematical journals: an annotated guide. Scarecrow Press, 1992, ISBN 0-8108-2585-6; p. 15 2. Hendricks, Joel E. (1874). "Introductory remarks". The Analyst. 1 (1): 1–2. Bibcode:1876Ana.....1....1.. doi:10.1039/an8760100001. 3. Fiske, Thomas S. (1905). "Mathematical progress in America" (PDF). Bulletin of the American Mathematical Society. 11 (5): 238–246. doi:10.1090/S0002-9904-1905-01210-6. Archived (PDF) from the original on 2022-10-09. Reprinted in Bulletin of the American Mathematical Society, New Series, 37 (1), 3–8, 1999. 4. Hendricks, Joel E. (1883). "Announcement". The Analyst. 10 (5): 159–160. JSTOR 2635801. 5. Hendricks, Joel E. (1883). "Announcement". The Analyst. 10 (6): 166. JSTOR 2635728. 6. Raymond Garver (1932). "The Analyst, 1874-1883". Scripta Mathematica. 1 (1): 247–251. 7. J. J. O'Connor and E. F. Robertson. Solomon Lefschetz. MacTutor History of Mathematics archive. Accessed February 2, 2010 8. Editorial Board. Annals of Mathematics, Princeton University 9. "Master Journal List". Intellectual Property & Science. Thomson Reuters. Archived from the original on 2017-09-26. Retrieved 2014-04-29. 10. "Scopus title list". Elsevier. Archived from the original (Microsoft Excel) on 2013-12-02. Retrieved 2014-04-29. 11. "Journals Ranked by Impact: Mathematics". 2020 Journal Citation Reports. Web of Science (Science ed.). Clarivate. 2021. External links • Official website	Wikipedia
The Archimedeans The Archimedeans are the mathematical society of the University of Cambridge, founded in 1935.[1] It currently has over 2000 active members,[2] many of them alumni, making it one of the largest student societies in Cambridge. The society hosts regular talks at the Centre for Mathematical Sciences, including in the past by many well-known speakers in the field of mathematics. It publishes two magazines, Eureka and QARCH.[1] The Archimedeans Named afterArchimedes Formation1935 TypeStudent Society Location • Cambridge, England President Yaël Dillies[1] Parent organization University of Cambridge Websitearchim.org.uk One of several aims of the society, as laid down in its constitution, is to encourage co-operation between the existing mathematical societies of individual Cambridge colleges, which at present are just the Adam's society of St John's College and the Trinity Mathematical Society, but in the past have included many more. The society is mentioned in G. H. Hardy's essay A Mathematician's Apology. Past presidents of The Archimedeans include Michael Atiyah and Richard Taylor. Activity The main focus of the society's activities are the regular talks, which generally concern topics from mathematics or theoretical physics, and are accessible to students on an undergraduate level. Among the list of recent speakers are Fields medalists Michael Atiyah, Wendelin Werner and Alain Connes, as well as authors Ian Stewart and Simon Singh. Many of the speakers are international, and are hosted by The Archimedeans during their visit. After exams and University-wide project deadlines, the society is also known to organise social events. Publications Eureka is a mathematical journal that is published annually by The Archimedeans. It includes articles on a variety of topics in mathematics, written by students and academics from all over the world, as well as a short summary of the activities of the society, problem sets, puzzles, artwork and book reviews. The magazine has been published 65 times since 1939,[3] and authors include many famous mathematicians and scientists such as Paul Erdős, Martin Gardner, Douglas Hofstadter, Godfrey Hardy, Béla Bollobás, John Conway, Stephen Hawking, Roger Penrose, Ian Stewart, Chris Budd, Fields Medallist Timothy Gowers and Nobel laureate Paul Dirac. It can also be read on Mathigon.[4] The journal is distributed free of charge to all current members of the Archimedeans. In addition, there are many subscriptions by other students, alumni and libraries. Subscriptions to Eureka are the society's main source of income. The Archimedeans also publish QARCH, a magazine containing problem sets and solutions or partial solutions submitted by readers. It is published on an irregular basis and distributed free of charge. References 1. "The Archimedeans". www.archim.org.uk. Retrieved 5 August 2020. 2. "The Archimedeans: Shaken or stirred? Interview with Yanni Du". Faculty of Mathematics, University of Cambridge. Retrieved 20 May 2018. 3. "Eureka Archive". www.archim.org.uk. Retrieved 12 December 2021.{{cite web}}: CS1 maint: url-status (link) 4. "Eureka Magazine". Mathigon. Retrieved 12 December 2021. Authority control • ISNI • VIAF	Wikipedia
Arnold Mathematical Journal The Arnold Mathematical Journal is a quarterly peer-reviewed mathematics journal established in 2014. It is organized jointly by the Institute for Mathematical Sciences at Stony Brook University, USA, and Springer Science+Business Media. The editor-in-chief is Askold Khovanskii.[1] The journal is abstracted and indexed in ZbMATH and Scopus.[2] Arnold Mathematical Journal DisciplineMathematics LanguageEnglish Edited byAskold Khovanskii, Vladlen Timorin Publication details History2015–present Publisher Springer Science+Business Media on behalf of the Institute for Mathematical Sciences (Stony Brook University) FrequencyQuarterly Standard abbreviations ISO 4 (alt) · Bluebook (alt1 · alt2) NLM (alt) · MathSciNet (alt ) ISO 4Arnold Math. J. Indexing CODEN (alt · alt2) · JSTOR (alt) · LCCN (alt) MIAR · NLM (alt) · Scopus ISSN2199-6792 (print) 2199-6806 (web) OCLC no.904797046 Links • Journal homepage • @Springer External links • Official website References 1. "Editorial Board of Arnold Mathematical Journal". 2. "Arnold Mathematical Journal". MIAR: Information Matrix for the Analysis of Journals. University of Barcelona. Retrieved 2019-09-04.	Wikipedia
The Art of Mathematics The Art of Mathematics (Korean: 수학의 정석, romanized: suhakui jeongseok), written by Hong Sung-Dae (Korean: 홍성대), is a series of mathematics textbooks for high school students in South Korea. First published in 1966, it is the best-selling book series in South Korea, with about 46 million copies sold as of 2016.[1] In Jeongeup, North Jeolla Province, the hometown of Hong Sung-Dae, a street is named Suhakjeongseok-gil (Korean: 수학정석길) in honor of the author.'[2] Controversy The similarities with the Japanese Textbook series Chart-Style Math (Japanese: チャート式数学) have caused the author to receive accusations of plagiarism.[3] The chapter division, style of explanation, and formatting are visibly similar between the books. For instance, in the Japanese books, the order of questions are in "Example Questions, Practice Questions, Exercise Questions," while in The Art of Mathematics it is "Example Questions, Similar Questions, Practice Questions". The author Hong has denied all accusations, although he has admitted that the questions in the books were selected from 20 reference books around the world. Major topics in the 11th edition[4] Changes in the 11th edition, published 2013-2015, reflect the 2009 revision of South Korea's National Curriculum. Each of the six volumes consist of two versions, one for average students (Korean: 기본편, romanized: Gibon-pyeon) and one for higher-ability students (Korean: 실력편, romanized: silyeok-pyeon). Mathematics I Korean: 수학 I suhak I • Polynomials (다항식 da-hang-sik) • Equations and Inequalities (방정식과 부등식 bang-jeong-sik-gwa boo-deung-sik) • Graphs of Equations (방정식의 그래프 bang-jeong-sik-eui geu-re-pu) Mathematics II Korean: 수학 II suhak II • Sets and Propositions (집합과 명제 jib-hab-gwa myung-jeh) • Functions (함수 ham-soo) • Sequences (수열 soo-yeul) • Exponents and Logarithms (지수와 로그함수 ji-soo-wa lo-geu-ham-soo) Probability and Statistics Korean: 확률과 통계 hwanglyulgwa tonggye • Permutations and Combinations (순열과 조합 soon-yeul-gwa jo-hab) • Probability (확률 hwang-lyul) • Statistics (통계 tong-gye) Calculus I Korean: 미적분 I mijuckboon I • Limits of Sequences (수열의 극한 soo-yeul-eui geuk-han) • Limits and Continuity (극한과 연속성 geuk-han-gwa yeon-sok-sung) • Differentiation of Polynomial Functions (다항식의 미분 da-hang-sik-eui mi-boon) • Integration of Polynomial Functions (다항식의 적분 da-hang-sik-eui juck-boon) Calculus II Korean: 미적분 II mijuckboon II • Exponential and Logarithmic Functions (지수와 로그 함수 ji-soo-wa lo-geu-ham-soo) • Trigonometric Functions (삼각함수 sam-gak-ham-soo) • Differentiation (미분 mi-boon) • Integration (적분 juck-boon) Geometry and Vectors Korean: 기하와 벡터 Gihawa Begteo • Plane Curves (평면곡선 pyung-myun-gog-seon) • Vectors in the Plane (평면 벡터 pyung-myun beg-teo) • Graphs and Vectors in Space (공간의 그래프와 벡터 gong-gan-eui geu-re-pu-wa beg-teo) References 1. Kim (23 June 2019). "'수학의 정석' / 김영배". The Hankyoreh. Retrieved 2019-08-17. 2. 35°39′10″N 126°56′34″E 3. "[金成東의 인간탐험] 수학의 定石 저자 洪性大 상산학원 이사장". monthly.chosun.com (in Korean). 2019-06-20. Retrieved 2019-08-17. 4. "2014년도 적용 새 교육 과정 안내". Sung Ji Publications. Retrieved 2015-07-16. External links • (in Korean) publisher's website	Wikipedia
Numerical Recipes Numerical Recipes is the generic title of a series of books on algorithms and numerical analysis by William H. Press, Saul A. Teukolsky, William T. Vetterling and Brian P. Flannery. In various editions, the books have been in print since 1986. The most recent edition was published in 2007. Numerical Recipes: The Art of Scientific Computing Cover of the third (C++) edition AuthorWilliam H. Press, Saul A. Teukolsky, William T. Vetterling and Brian P. Flannery LanguageEnglish DisciplineNumerical analysis PublisherCambridge University Press Websitenumerical.recipes Overview The Numerical Recipes books cover a range of topics that include both classical numerical analysis (interpolation, integration, linear algebra, differential equations, and so on), signal processing (Fourier methods, filtering), statistical treatment of data, and a few topics in machine learning (hidden Markov model, support vector machines). The writing style is accessible and has an informal tone. The emphasis is on understanding the underlying basics of techniques, not on the refinements that may, in practice, be needed to achieve optimal performance and reliability. Few results are proved with any degree of rigor, although the ideas behind proofs are often sketched, and references are given. Importantly, virtually all methods that are discussed are also implemented in a programming language, with the code printed in the book. Each version is keyed to a specific language. According to the publisher, Cambridge University Press, the Numerical Recipes books are historically the all-time best-selling books on scientific programming methods. In recent years, Numerical Recipes books have been cited in the scientific literature more than 3000 times per year according to ISI Web of Knowledge (e.g., 3962 times in the year 2008).[1] And as of the end of 2017, the book had over 44000 citations on Google Scholar.[2] History The first publication was in 1986 with the title,”Numerical Recipes, The Art of Scientific Computing”, containing code in both Fortran and Pascal; an accompanying book, “Numerical Recipes Example Book (Pascal)” was first published in 1985. (A preface note in “Examples" mentions that the main book was also published in 1985, but the official note in that book says 1986.) Supplemental editions followed with code in Pascal, BASIC, and C. Numerical Recipes took, from the start, an opinionated editorial position at odds with the conventional wisdom of the numerical analysis community: If there is a single dominant theme in this book, it is that practical methods of numerical computation can be simultaneously efficient, clever, and — important — clear. The alternative viewpoint, that efficient computational methods must necessarily be so arcane and complex as to be useful only in "black box" form, we firmly reject.[3] However, as it turned out, the 1980s were fertile years for the "black box" side, yielding important libraries such as BLAS and LAPACK, and integrated environments like MATLAB and Mathematica. By the early 1990s, when Second Edition versions of Numerical Recipes (with code in C, Fortran-77, and Fortran-90) were published, it was clear that the constituency for Numerical Recipes was by no means the majority of scientists doing computation, but only that slice that lived between the more mathematical numerical analysts and the larger community using integrated environments. The Second Edition versions occupied a stable role in this niche environment.[4] By the mid-2000s, the practice of scientific computing had been radically altered by the mature Internet and Web. Recognizing that their Numerical Recipes books were increasingly valued more for their explanatory text than for their code examples, the authors significantly expanded the scope of the book, and significantly rewrote a large part of the text. They continued to include code, still printed in the book, now in C++, for every method discussed.[5] The Third Edition was also released as an electronic book,[6] eventually made available on the Web for free (with nags) or by paid or institutional subscription (with faster, full access and no nags). In 2015 Numerical Recipes sold its historic two-letter domain name nr.com[7] and became numerical.recipes instead. Reception Content Numerical Recipes is a single volume that covers very broad range of algorithms. Unfortunately that format skewed the choice of algorithms towards simpler and shorter early algorithms which were not as accurate, efficient or stable as later more complex algorithms.[8][9] The first edition had also some minor bugs, which were fixed in later editions; however according to the authors for years they were encountering on the internet rumors that Numerical Recipes is "full of bugs". They attributed this to people using outdated versions of the code, bugs in other parts of the code and misuse of routines which require some understanding to use correctly.[10] The rebuttal does not, however, cover criticisms regarding lack of mentions to code limitations, boundary conditions, and more modern algorithms, another theme in Snyder's comment compilation.[9] A precision issue in Bessel functions has persisted to the third edition according to Pavel Holoborodko.[8] Despite criticism by numerical analysts, engineers and scientists generally find the book conveniently broad in scope.[9] Norman Gray concurs in the following quote:[11] Numerical Recipes [nr] does not claim to be a numerical analysis textbook, and it makes a point of noting that its authors are (astro-)physicists and engineers rather than analysts, and so share the motivations and impatience of the book's intended audience. The declared premise of the NR authors is that you will come to grief one way or the other if you use numerical routines you do not understand. They attempt to give you enough mathematical detail that you understand the routines they present, in enough depth that you can diagnose problems when they occur, and make more sophisticated choices about replacements when the NR routines run out of steam. Problems will occur because [...] License The code listings are copyrighted and commercially licensed by the Numerical Recipes authors.[12] A license to use the code is given with the purchase of a book, but the terms of use are highly restrictive.[13] For example, programmers need to make sure NR code cannot be extracted from their finished programs and used – a difficult requirement with dubious enforceability.[14] However, Numerical Recipes does include the following statement regarding copyrights on computer programs: Copyright does not protect ideas, but only the expression of those ideas in a particular form. In the case of a computer program, the ideas consist of the program's methodology and algorithm, including the necessary sequence of steps adopted by the programmer. The expression of those ideas is the program source code ... If you analyze the ideas contained in a program, and then express those ideas in your own completely different implementation, then that new program implementation belongs to you.[6] One early motivation for the GNU Scientific Library was that a free library was needed as a substitute for Numerical Recipes.[15] Style Another line of criticism centers on the coding style of the books, which strike some modern readers as "Fortran-ish", though written in contemporary, object-oriented C++.[15] The authors have defended their very terse coding style as necessary to the format of the book because of space limitations and for readability.[4] Titles in the series (partial list) The books differ by edition (1st, 2nd, and 3rd) and by the computer language in which the code is given. • Numerical Recipes. The Art of Scientific Computing, 1st Edition, 1986, ISBN 0-521-30811-9. (Fortran and Pascal) • Numerical Recipes in C. The Art of Scientific Computing, 1st Edition, 1988, ISBN 0-521-35465-X. • Numerical Recipes in Pascal. The Art of Scientific Computing, 1st Edition, 1989, ISBN 0-521-37516-9. • Numerical Recipes in Fortran. The Art of Scientific Computing, 1st Edition, 1989, ISBN 0-521-38330-7. • Numerical Recipes in BASIC. The Art of Scientific Computing, 1st Edition, 1991, ISBN 0-521-40689-7. (supplemental edition) • Numerical Recipes in Fortran 77. The Art of Scientific Computing, 2nd Edition, 1992, ISBN 0-521-43064-X. • Numerical Recipes in C. The Art of Scientific Computing, 2nd Edition, 1992, ISBN 0-521-43108-5. • Numerical Recipes in Fortran 90. The Art of Parallel Scientific Computing, 2nd Edition, 1996, ISBN 0-521-57439-0. • Numerical Recipes in C++. The Art of Scientific Computing, 2nd Edition, 2002, ISBN 0-521-75033-4. • Numerical Recipes. The Art of Scientific Computing, 3rd Edition, 2007, ISBN 0-521-88068-8. (C++ code) The books are published by Cambridge University Press. References 1. Thomson Reuters, Web of Knowledge, Cited Reference Search. 2. , Google Scholar 3. Press, William H.; Teukolsky, Saul A.; Vetterling, William T.; Flannery, Brian P. (1986). "Preface". Numerical Recipes: The Art of Scientific Computing. New York: Cambridge University Press. p. xi. ISBN 0-521-30811-9. 4. Press, William H.; and Teukolsky, Saul A.; "Numerical Recipes: Does This Paradigm Have a Future?," Computers in Physics, 11, 416 (1997). Preprint. 5. Press, William H.; Teukolsky, Saul A.; Vetterling, William T.; Flannery, Brian P. (2007). "Preface to the Third Edition". Numerical Recipes: The Art of Scientific Computing (3rd ed.). New York: Cambridge University Press. p. xi. ISBN 978-0-521-88068-8. 6. Press, William H.; Teukolsky, Saul A.; Vetterling, William T.; Flannery, Brian P. (2007). Numerical Recipes: The Art of Scientific Computing (3rd ed.). New York: Cambridge University Press. ISBN 978-0-521-88068-8. 7. "Two letter domain NR.com sold : Rebrands to Numerical.Recipes". 14 October 2015. 8. "Reviews: Numerical Recipes". www.quut.com. Retrieved 28 January 2019. (updated for the third edition; clone URL) 9. Van Snyder, W. (March 1991). "Why not use Numerical Recipes?". stat.uchicago.edu. Retrieved 28 January 2019. (Date estimated by Editor's remark. Last update circa September 1999; older clone) 10. "Numerical Recipes Distressing Rumors". numerical.recipes. February 1999. Retrieved 28 January 2019. (Date given is first archive.org date for the page on the old nr.com domain.) 11. Gray, Norman. "Numerical Recipes". Theory and Modelling Resources Cookbook, www.astro.gla.ac.uk. 12. Numerical Recipes Web site, Numerical Recipes Code 13. Weiner, Benjamin. "Boycott Numerical Recipes". Buy the book if you feel like it, learn from it, but use a library like the GNU Scientific Library instead. Especially if you ever want other people to use your work. The NR license is the RIAA of the scientific community. 14. Hornbeck, Haysn (January 28, 2020). Fast Cubic Spline Interpolation (Technical report). University of Calgary. arXiv:2001.09253. 15. Galassi, Mark; Theiler, James; Gough, Brian. "GNU Scientific Library -- Design document". GNU Operating System. GNU.org. Retrieved January 5, 2019. External links • Official website • Current electronic edition of Numerical Recipes (limited free page views). • Numerical Recipes at Google Books • Older versions of Numerical Recipes available electronically (links to C, Fortran 77, and Fortran 90 versions in various formats, plus other hosted books) • W. Van Snyder, Why not use Numerical Recipes? , full four-page mirror by Lek-Heng Lim (includes discussion of alternatives)	Wikipedia
Banach–Tarski paradox The Banach–Tarski paradox is a theorem in set-theoretic geometry, which states the following: Given a solid ball in three-dimensional space, there exists a decomposition of the ball into a finite number of disjoint subsets, which can then be put back together in a different way to yield two identical copies of the original ball. Indeed, the reassembly process involves only moving the pieces around and rotating them without changing their shape. However, the pieces themselves are not "solids" in the usual sense, but infinite scatterings of points. The reconstruction can work with as few as five pieces.[1] An alternative form of the theorem states that given any two "reasonable" solid objects (such as a small ball and a huge ball), the cut pieces of either one can be reassembled into the other. This is often stated informally as "a pea can be chopped up and reassembled into the Sun" and called the "pea and the Sun paradox". The theorem is called a paradox because it contradicts basic geometric intuition. "Doubling the ball" by dividing it into parts and moving them around by rotations and translations, without any stretching, bending, or adding new points, seems to be impossible, since all these operations ought, intuitively speaking, to preserve the volume. The intuition that such operations preserve volumes is not mathematically absurd and it is even included in the formal definition of volumes. However, this is not applicable here because in this case it is impossible to define the volumes of the considered subsets. Reassembling them reproduces a set that has a volume, which happens to be different from the volume at the start. Unlike most theorems in geometry, the mathematical proof of this result depends on the choice of axioms for set theory in a critical way. It can be proven using the axiom of choice, which allows for the construction of non-measurable sets, i.e., collections of points that do not have a volume in the ordinary sense, and whose construction requires an uncountable number of choices.[2] It was shown in 2005 that the pieces in the decomposition can be chosen in such a way that they can be moved continuously into place without running into one another.[3] As proved independently by Leroy[4] and Simpson,[5] the Banach–Tarski paradox does not violate volumes if one works with locales rather than topological spaces. In this abstract setting, it is possible to have subspaces without point but still nonempty. The parts of the paradoxical decomposition do intersect a lot in the sense of locales, so much that some of these intersections should be given a positive mass. Allowing for this hidden mass to be taken into account, the theory of locales permits all subsets (and even all sublocales) of the Euclidean space to be satisfactorily measured. Banach and Tarski publication In a paper published in 1924,[6] Stefan Banach and Alfred Tarski gave a construction of such a paradoxical decomposition, based on earlier work by Giuseppe Vitali concerning the unit interval and on the paradoxical decompositions of the sphere by Felix Hausdorff, and discussed a number of related questions concerning decompositions of subsets of Euclidean spaces in various dimensions. They proved the following more general statement, the strong form of the Banach–Tarski paradox: Given any two bounded subsets A and B of a Euclidean space in at least three dimensions, both of which have a nonempty interior, there are partitions of A and B into a finite number of disjoint subsets, $A=A_{1}\cup \cdots \cup A_{k}$, $B=B_{1}\cup \cdots \cup B_{k}$ (for some integer k), such that for each (integer) i between 1 and k, the sets Ai and Bi are congruent. Now let A be the original ball and B be the union of two translated copies of the original ball. Then the proposition means that you can divide the original ball A into a certain number of pieces and then rotate and translate these pieces in such a way that the result is the whole set B, which contains two copies of A. The strong form of the Banach–Tarski paradox is false in dimensions one and two, but Banach and Tarski showed that an analogous statement remains true if countably many subsets are allowed. The difference between dimensions 1 and 2 on the one hand, and 3 and higher on the other hand, is due to the richer structure of the group E(n) of Euclidean motions in 3 dimensions. For n = 1, 2 the group is solvable, but for n ≥ 3 it contains a free group with two generators. John von Neumann studied the properties of the group of equivalences that make a paradoxical decomposition possible, and introduced the notion of amenable groups. He also found a form of the paradox in the plane which uses area-preserving affine transformations in place of the usual congruences. Tarski proved that amenable groups are precisely those for which no paradoxical decompositions exist. Since only free subgroups are needed in the Banach–Tarski paradox, this led to the long-standing von Neumann conjecture, which was disproved in 1980. Formal treatment The Banach–Tarski paradox states that a ball in the ordinary Euclidean space can be doubled using only the operations of partitioning into subsets, replacing a set with a congruent set, and reassembling. Its mathematical structure is greatly elucidated by emphasizing the role played by the group of Euclidean motions and introducing the notions of equidecomposable sets and a paradoxical set. Suppose that G is a group acting on a set X. In the most important special case, X is an n-dimensional Euclidean space (for integral n), and G consists of all isometries of X, i.e. the transformations of X into itself that preserve the distances, usually denoted E(n). Two geometric figures that can be transformed into each other are called congruent, and this terminology will be extended to the general G-action. Two subsets A and B of X are called G-equidecomposable, or equidecomposable with respect to G, if A and B can be partitioned into the same finite number of respectively G-congruent pieces. This defines an equivalence relation among all subsets of X. Formally, if there exist non-empty sets $A_{1},\dots ,A_{k}$, $B_{1},\dots ,B_{k}$ such that $A=\bigcup _{i=1}^{k}A_{i},\quad B=\bigcup _{i=1}^{k}B_{i},$ $\quad A_{i}\cap A_{j}=B_{i}\cap B_{j}=\emptyset \quad {\text{for all }}1\leq i<j\leq k,$ and there exist elements $g_{i}\in G$ such that $g_{i}(A_{i})=B_{i}{\text{ for all }}1\leq i\leq k,$ then it can be said that A and B are G-equidecomposable using k pieces. If a set E has two disjoint subsets A and B such that A and E, as well as B and E, are G-equidecomposable, then E is called paradoxical. Using this terminology, the Banach–Tarski paradox can be reformulated as follows: A three-dimensional Euclidean ball is equidecomposable with two copies of itself. In fact, there is a sharp result in this case, due to Raphael M. Robinson:[7] doubling the ball can be accomplished with five pieces, and fewer than five pieces will not suffice. The strong version of the paradox claims: Any two bounded subsets of 3-dimensional Euclidean space with non-empty interiors are equidecomposable. While apparently more general, this statement is derived in a simple way from the doubling of a ball by using a generalization of the Bernstein–Schroeder theorem due to Banach that implies that if A is equidecomposable with a subset of B and B is equidecomposable with a subset of A, then A and B are equidecomposable. The Banach–Tarski paradox can be put in context by pointing out that for two sets in the strong form of the paradox, there is always a bijective function that can map the points in one shape into the other in a one-to-one fashion. In the language of Georg Cantor's set theory, these two sets have equal cardinality. Thus, if one enlarges the group to allow arbitrary bijections of X, then all sets with non-empty interior become congruent. Likewise, one ball can be made into a larger or smaller ball by stretching, or in other words, by applying similarity transformations. Hence, if the group G is large enough, G-equidecomposable sets may be found whose "size"s vary. Moreover, since a countable set can be made into two copies of itself, one might expect that using countably many pieces could somehow do the trick. On the other hand, in the Banach–Tarski paradox, the number of pieces is finite and the allowed equivalences are Euclidean congruences, which preserve the volumes. Yet, somehow, they end up doubling the volume of the ball! While this is certainly surprising, some of the pieces used in the paradoxical decomposition are non-measurable sets, so the notion of volume (more precisely, Lebesgue measure) is not defined for them, and the partitioning cannot be accomplished in a practical way. In fact, the Banach–Tarski paradox demonstrates that it is impossible to find a finitely-additive measure (or a Banach measure) defined on all subsets of a Euclidean space of three (and greater) dimensions that is invariant with respect to Euclidean motions and takes the value one on a unit cube. In his later work, Tarski showed that, conversely, non-existence of paradoxical decompositions of this type implies the existence of a finitely-additive invariant measure. The heart of the proof of the "doubling the ball" form of the paradox presented below is the remarkable fact that by a Euclidean isometry (and renaming of elements), one can divide a certain set (essentially, the surface of a unit sphere) into four parts, then rotate one of them to become itself plus two of the other parts. This follows rather easily from a F2-paradoxical decomposition of F2, the free group with two generators. Banach and Tarski's proof relied on an analogous fact discovered by Hausdorff some years earlier: the surface of a unit sphere in space is a disjoint union of three sets B, C, D and a countable set E such that, on the one hand, B, C, D are pairwise congruent, and on the other hand, B is congruent with the union of C and D. This is often called the Hausdorff paradox. Connection with earlier work and the role of the axiom of choice Banach and Tarski explicitly acknowledge Giuseppe Vitali's 1905 construction of the set bearing his name, Hausdorff's paradox (1914), and an earlier (1923) paper of Banach as the precursors to their work. Vitali's and Hausdorff's constructions depend on Zermelo's axiom of choice ("AC"), which is also crucial to the Banach–Tarski paper, both for proving their paradox and for the proof of another result: Two Euclidean polygons, one of which strictly contains the other, are not equidecomposable. They remark: Le rôle que joue cet axiome dans nos raisonnements nous semble mériter l'attention (The role this axiom plays in our reasoning seems to us to deserve attention) They point out that while the second result fully agrees with geometric intuition, its proof uses AC in an even more substantial way than the proof of the paradox. Thus Banach and Tarski imply that AC should not be rejected solely because it produces a paradoxical decomposition, for such an argument also undermines proofs of geometrically intuitive statements. However, in 1949, A. P. Morse showed that the statement about Euclidean polygons can be proved in ZF set theory and thus does not require the axiom of choice. In 1964, Paul Cohen proved that the axiom of choice is independent from ZF – that is, it cannot be proved from ZF. A weaker version of an axiom of choice is the axiom of dependent choice, DC, and it has been shown that DC is not sufficient for proving the Banach–Tarski paradox, that is, The Banach–Tarski paradox is not a theorem of ZF, nor of ZF+DC.[8] Large amounts of mathematics use AC. As Stan Wagon points out at the end of his monograph, the Banach–Tarski paradox has been more significant for its role in pure mathematics than for foundational questions: it motivated a fruitful new direction for research, the amenability of groups, which has nothing to do with the foundational questions. In 1991, using then-recent results by Matthew Foreman and Friedrich Wehrung,[9] Janusz Pawlikowski proved that the Banach–Tarski paradox follows from ZF plus the Hahn–Banach theorem.[10] The Hahn–Banach theorem does not rely on the full axiom of choice but can be proved using a weaker version of AC called the ultrafilter lemma. So Pawlikowski proved that the set theory needed to prove the Banach–Tarski paradox, while stronger than ZF, is weaker than full ZFC. A sketch of the proof Here a proof is sketched which is similar but not identical to that given by Banach and Tarski. Essentially, the paradoxical decomposition of the ball is achieved in four steps: 1. Find a paradoxical decomposition of the free group in two generators. 2. Find a group of rotations in 3-d space isomorphic to the free group in two generators. 3. Use the paradoxical decomposition of that group and the axiom of choice to produce a paradoxical decomposition of the hollow unit sphere. 4. Extend this decomposition of the sphere to a decomposition of the solid unit ball. These steps are discussed in more detail below. Step 1 The free group with two generators a and b consists of all finite strings that can be formed from the four symbols a, a−1, b and b−1 such that no a appears directly next to an a−1 and no b appears directly next to a b−1. Two such strings can be concatenated and converted into a string of this type by repeatedly replacing the "forbidden" substrings with the empty string. For instance: abab−1a−1 concatenated with abab−1a yields abab−1a−1abab−1a, which contains the substring a−1a, and so gets reduced to abab−1bab−1a, which contains the substring b−1b, which gets reduced to abaab−1a. One can check that the set of those strings with this operation forms a group with identity element the empty string e. This group may be called F2. The group $F_{2}$ can be "paradoxically decomposed" as follows: Let S(a) be the subset of $F_{2}$ consisting of all strings that start with a, and define S(a−1), S(b) and S(b−1) similarly. Clearly, $F_{2}=\{e\}\cup S(a)\cup S(a^{-1})\cup S(b)\cup S(b^{-1})$ but also $F_{2}=aS(a^{-1})\cup S(a),$ and $F_{2}=bS(b^{-1})\cup S(b),$ where the notation aS(a−1) means take all the strings in S(a−1) and concatenate them on the left with a. This is at the core of the proof. For example, there may be a string $aa^{-1}b$ in the set $aS(a^{-1})$ which, because of the rule that $a$ must not appear next to $a^{-1}$, reduces to the string $b$. Similarly, $aS(a^{-1})$ contains all the strings that start with $a^{-1}$ (for example, the string $aa^{-1}a^{-1}$ which reduces to $a^{-1}$). In this way, $aS(a^{-1})$ contains all the strings that start with $b$, $b^{-1}$ and $a^{-1}$, as well as the empty string $e$. Group F2 has been cut into four pieces (plus the singleton {e}), then two of them "shifted" by multiplying with a or b, then "reassembled" as two pieces to make one copy of $F_{2}$ and the other two to make another copy of $F_{2}$. That is exactly what is intended to do to the ball. Step 2 In order to find a free group of rotations of 3D space, i.e. that behaves just like (or "is isomorphic to") the free group F2, two orthogonal axes are taken (e.g. the x and z axes). Then, A is taken to be a rotation of $ \theta =\arccos \left({\frac {1}{3}}\right)$ about the x axis, and B to be a rotation of $\theta $ about the z axis (there are many other suitable pairs of irrational multiples of π that could be used here as well).[11] The group of rotations generated by A and B will be called H. Let $\omega $ be an element of H that starts with a positive rotation about the z axis, that is, an element of the form $\omega =\ldots b^{k_{3}}a^{k_{2}}b^{k_{1}}$ with $k_{1}>0,\ k_{2},k_{3},\ldots ,k_{n}\neq 0,\ n\geq 1$. It can be shown by induction that $\omega $ maps the point $(1,0,0)$ to $ \left({\frac {k}{3^{N}}},{\frac {l{\sqrt {2}}}{3^{N}}},{\frac {m}{3^{N}}}\right)$, for some $k,l,m\in \mathbb {Z} ,N\in \mathbb {N} $. Analyzing $k,l$ and $m$ modulo 3, one can show that $l\neq 0$. The same argument repeated (by symmetry of the problem) is valid when $\omega $ starts with a negative rotation about the z axis, or a rotation about the x axis. This shows that if $\omega $ is given by a non-trivial word in A and B, then $\omega \neq e$. Therefore, the group H is a free group, isomorphic to F2. The two rotations behave just like the elements a and b in the group F2: there is now a paradoxical decomposition of H. This step cannot be performed in two dimensions since it involves rotations in three dimensions. If two rotations are taken about the same axis, the resulting group is the abelian circle group and does not have the property required in step 1. An alternate arithmetic proof of the existence of free groups in some special orthogonal groups using integral quaternions leads to paradoxical decompositions of the rotation group.[12] Step 3 The unit sphere S2 is partitioned into orbits by the action of our group H: two points belong to the same orbit if and only if there is a rotation in H which moves the first point into the second. (Note that the orbit of a point is a dense set in S2.) The axiom of choice can be used to pick exactly one point from every orbit; collect these points into a set M. The action of H on a given orbit is free and transitive and so each orbit can be identified with H. In other words, every point in S2 can be reached in exactly one way by applying the proper rotation from H to the proper element from M. Because of this, the paradoxical decomposition of H yields a paradoxical decomposition of S2 into four pieces A1, A2, A3, A4 as follows: $A_{1}=S(a)M\cup M\cup B$ $A_{2}=S(a^{-1})M\setminus B$ $\displaystyle A_{3}=S(b)M$ $\displaystyle A_{4}=S(b^{-1})M$ where we define $S(a)M=\{s(x)\|s\in S(a),x\in M\}$ and likewise for the other sets, and where we define $B=a^{-1}M\cup a^{-2}M\cup \dots $ (The five "paradoxical" parts of F2 were not used directly, as they would leave M as an extra piece after doubling, owing to the presence of the singleton {e}!) The (majority of the) sphere has now been divided into four sets (each one dense on the sphere), and when two of these are rotated, the result is double of what was had before: $aA_{2}=A_{2}\cup A_{3}\cup A_{4}$ $bA_{4}=A_{1}\cup A_{2}\cup A_{4}$ Step 4 Finally, connect every point on S2 with a half-open segment to the origin; the paradoxical decomposition of S2 then yields a paradoxical decomposition of the solid unit ball minus the point at the ball's center. (This center point needs a bit more care; see below.) N.B. This sketch glosses over some details. One has to be careful about the set of points on the sphere which happen to lie on the axis of some rotation in H. However, there are only countably many such points, and like the case of the point at the center of the ball, it is possible to patch the proof to account for them all. (See below.) Some details, fleshed out In Step 3, the sphere was partitioned into orbits of our group H. To streamline the proof, the discussion of points that are fixed by some rotation was omitted; since the paradoxical decomposition of F2 relies on shifting certain subsets, the fact that some points are fixed might cause some trouble. Since any rotation of S2 (other than the null rotation) has exactly two fixed points, and since H, which is isomorphic to F2, is countable, there are countably many points of S2 that are fixed by some rotation in H. Denote this set of fixed points as D. Step 3 proves that S2 − D admits a paradoxical decomposition. What remains to be shown is the Claim: S2 − D is equidecomposable with S2. Proof. Let λ be some line through the origin that does not intersect any point in D. This is possible since D is countable. Let J be the set of angles, α, such that for some natural number n, and some P in D, r(nα)P is also in D, where r(nα) is a rotation about λ of nα. Then J is countable. So there exists an angle θ not in J. Let ρ be the rotation about λ by θ. Then ρ acts on S2 with no fixed points in D, i.e., ρn(D) is disjoint from D, and for natural m<n, ρn(D) is disjoint from ρm(D). Let E be the disjoint union of ρn(D) over n = 0, 1, 2, ... . Then S2 = E ∪ (S2 − E) ~ ρ(E) ∪ (S2 − E) = (E − D) ∪ (S2 − E) = S2 − D, where ~ denotes "is equidecomposable to". For step 4, it has already been shown that the ball minus a point admits a paradoxical decomposition; it remains to be shown that the ball minus a point is equidecomposable with the ball. Consider a circle within the ball, containing the point at the center of the ball. Using an argument like that used to prove the Claim, one can see that the full circle is equidecomposable with the circle minus the point at the ball's center. (Basically, a countable set of points on the circle can be rotated to give itself plus one more point.) Note that this involves the rotation about a point other than the origin, so the Banach–Tarski paradox involves isometries of Euclidean 3-space rather than just SO(3). Use is made of the fact that if A ~ B and B ~ C, then A ~ C. The decomposition of A into C can be done using number of pieces equal to the product of the numbers needed for taking A into B and for taking B into C. The proof sketched above requires 2 × 4 × 2 + 8 = 24 pieces - a factor of 2 to remove fixed points, a factor 4 from step 1, a factor 2 to recreate fixed points, and 8 for the center point of the second ball. But in step 1 when moving {e} and all strings of the form an into S(a−1), do this to all orbits except one. Move {e} of this last orbit to the center point of the second ball. This brings the total down to 16 + 1 pieces. With more algebra, one can also decompose fixed orbits into 4 sets as in step 1. This gives 5 pieces and is the best possible. Obtaining infinitely many balls from one Using the Banach–Tarski paradox, it is possible to obtain k copies of a ball in the Euclidean n-space from one, for any integers n ≥ 3 and k ≥ 1, i.e. a ball can be cut into k pieces so that each of them is equidecomposable to a ball of the same size as the original. Using the fact that the free group F2 of rank 2 admits a free subgroup of countably infinite rank, a similar proof yields that the unit sphere Sn−1 can be partitioned into countably infinitely many pieces, each of which is equidecomposable (with two pieces) to the Sn−1 using rotations. By using analytic properties of the rotation group SO(n), which is a connected analytic Lie group, one can further prove that the sphere Sn−1 can be partitioned into as many pieces as there are real numbers (that is, $2^{\aleph _{0}}$ pieces), so that each piece is equidecomposable with two pieces to Sn−1 using rotations. These results then extend to the unit ball deprived of the origin. A 2010 article by Valeriy Churkin gives a new proof of the continuous version of the Banach–Tarski paradox.[13] Von Neumann paradox in the Euclidean plane Main article: Von Neumann paradox In the Euclidean plane, two figures that are equidecomposable with respect to the group of Euclidean motions are necessarily of the same area, and therefore, a paradoxical decomposition of a square or disk of Banach–Tarski type that uses only Euclidean congruences is impossible. A conceptual explanation of the distinction between the planar and higher-dimensional cases was given by John von Neumann: unlike the group SO(3) of rotations in three dimensions, the group E(2) of Euclidean motions of the plane is solvable, which implies the existence of a finitely-additive measure on E(2) and R2 which is invariant under translations and rotations, and rules out paradoxical decompositions of non-negligible sets. Von Neumann then posed the following question: can such a paradoxical decomposition be constructed if one allows a larger group of equivalences? It is clear that if one permits similarities, any two squares in the plane become equivalent even without further subdivision. This motivates restricting one's attention to the group SA2 of area-preserving affine transformations. Since the area is preserved, any paradoxical decomposition of a square with respect to this group would be counterintuitive for the same reasons as the Banach–Tarski decomposition of a ball. In fact, the group SA2 contains as a subgroup the special linear group SL(2,R), which in its turn contains the free group F2 with two generators as a subgroup. This makes it plausible that the proof of Banach–Tarski paradox can be imitated in the plane. The main difficulty here lies in the fact that the unit square is not invariant under the action of the linear group SL(2, R), hence one cannot simply transfer a paradoxical decomposition from the group to the square, as in the third step of the above proof of the Banach–Tarski paradox. Moreover, the fixed points of the group present difficulties (for example, the origin is fixed under all linear transformations). This is why von Neumann used the larger group SA2 including the translations, and he constructed a paradoxical decomposition of the unit square with respect to the enlarged group (in 1929). Applying the Banach–Tarski method, the paradox for the square can be strengthened as follows: Any two bounded subsets of the Euclidean plane with non-empty interiors are equidecomposable with respect to the area-preserving affine maps. As von Neumann notes:[14] "Infolgedessen gibt es bereits in der Ebene kein nichtnegatives additives Maß (wo das Einheitsquadrat das Maß 1 hat), das gegenüber allen Abbildungen von A2 invariant wäre." "In accordance with this, already in the plane there is no non-negative additive measure (for which the unit square has a measure of 1), which is invariant with respect to all transformations belonging to A2 [the group of area-preserving affine transformations]." To explain further, the question of whether a finitely additive measure (that is preserved under certain transformations) exists or not depends on what transformations are allowed. The Banach measure of sets in the plane, which is preserved by translations and rotations, is not preserved by non-isometric transformations even when they do preserve the area of polygons. The points of the plane (other than the origin) can be divided into two dense sets which may be called A and B. If the A points of a given polygon are transformed by a certain area-preserving transformation and the B points by another, both sets can become subsets of the A points in two new polygons. The new polygons have the same area as the old polygon, but the two transformed sets cannot have the same measure as before (since they contain only part of the A points), and therefore there is no measure that "works". The class of groups isolated by von Neumann in the course of study of Banach–Tarski phenomenon turned out to be very important for many areas of Mathematics: these are amenable groups, or groups with an invariant mean, and include all finite and all solvable groups. Generally speaking, paradoxical decompositions arise when the group used for equivalences in the definition of equidecomposability is not amenable. Recent progress • 2000: Von Neumann's paper left open the possibility of a paradoxical decomposition of the interior of the unit square with respect to the linear group SL(2,R) (Wagon, Question 7.4). In 2000, Miklós Laczkovich proved that such a decomposition exists.[15] More precisely, let A be the family of all bounded subsets of the plane with non-empty interior and at a positive distance from the origin, and B the family of all planar sets with the property that a union of finitely many translates under some elements of SL(2, R) contains a punctured neighborhood of the origin. Then all sets in the family A are SL(2, R)-equidecomposable, and likewise for the sets in B. It follows that both families consist of paradoxical sets. • 2003: It had been known for a long time that the full plane was paradoxical with respect to SA2, and that the minimal number of pieces would equal four provided that there exists a locally commutative free subgroup of SA2. In 2003 Kenzi Satô constructed such a subgroup, confirming that four pieces suffice.[16] • 2011: Laczkovich's paper[17] left open the possibility if there exists a free group F of piecewise linear transformations acting on the punctured disk D \{0,0} without fixed points. Grzegorz Tomkowicz constructed such a group,[18] showing that the system of congruences A ≈ B ≈ C ≈ B U C can be realized by means of F and D \{0,0}. • 2017: It has been known for a long time that there exists in the hyperbolic plane H2 a set E that is a third, a fourth and ... and a $2^{\aleph _{0}}$-th part of H2. The requirement was satisfied by orientation-preserving isometries of H2. Analogous results were obtained by John Frank Adams[19] and Jan Mycielski[20] who showed that the unit sphere S2 contains a set E that is a half, a third, a fourth and ... and a $2^{\aleph _{0}}$-th part of S2. Grzegorz Tomkowicz[21] showed that Adams and Mycielski construction can be generalized to obtain a set E of H2 with the same properties as in S2. • 2017: Von Neumann's paradox concerns the Euclidean plane, but there are also other classical spaces where the paradoxes are possible. For example, one can ask if there is a Banach–Tarski paradox in the hyperbolic plane H2. This was shown by Jan Mycielski and Grzegorz Tomkowicz.[22][23] Tomkowicz[24] proved also that most of the classical paradoxes are an easy consequence of a graph theoretical result and the fact that the groups in question are rich enough. • 2018: In 1984, Jan Mycielski and Stan Wagon [25] constructed a paradoxical decomposition of the hyperbolic plane H2 that uses Borel sets. The paradox depends on the existence of a properly discontinuous subgroup of the group of isometries of H2. Similar paradox is obtained by Grzegorz Tomkowicz [26] who constructed a free properly discontinuous subgroup G of the affine group SA(3,Z). The existence of such a group implies the existence of a subset E of Z3 such that for any finite F of Z3 there exists an element g of G such that $g(E)=E\triangle F$, where $E\,\triangle \,F$ denotes the symmetric difference of E and F. • 2019: Banach–Tarski paradox uses finitely many pieces in the duplication. In the case of countably many pieces, any two sets with non-empty interiors are equidecomposable using translations. But allowing only Lebesgue measurable pieces one obtains: If A and B are subsets of Rn with non-empty interiors, then they have equal Lebesgue measures if and only if they are countably equidecomposable using Lebesgue measurable pieces. Jan Mycielski and Grzegorz Tomkowicz [27] extended this result to finite dimensional Lie groups and second countable locally compact topological groups that are totally disconnected or have countably many connected components. See also • Hausdorff paradox • Nikodym set • Paradoxes of set theory • Tarski's circle-squaring problem – Problem of cutting and reassembling a disk into a square • Von Neumann paradox Notes 1. Tao, Terence (2011). An introduction to measure theory (PDF). p. 3. Archived from the original (PDF) on 6 May 2021. 2. Wagon, Corollary 13.3 3. Wilson, Trevor M. (September 2005). "A continuous movement version of the Banach–Tarski paradox: A solution to De Groot's problem". Journal of Symbolic Logic. 70 (3): 946–952. CiteSeerX 10.1.1.502.6600. doi:10.2178/jsl/1122038921. JSTOR 27588401. S2CID 15825008. 4. Olivier, Leroy (1995). Théorie de la mesure dans les lieux réguliers. ou : Les intersections cachées dans le paradoxe de Banach-Tarski (Report). arXiv:1303.5631. 5. Simpson, Alex (1 November 2012). "Measure, randomness and sublocales". Annals of Pure and Applied Logic. 163 (11): 1642–1659. doi:10.1016/j.apal.2011.12.014. 6. Banach, Stefan; Tarski, Alfred (1924). "Sur la décomposition des ensembles de points en parties respectivement congruentes" (PDF). Fundamenta Mathematicae (in French). 6: 244–277. doi:10.4064/fm-6-1-244-277. 7. Robinson, Raphael M. (1947). "On the Decomposition of Spheres". Fund. Math. 34: 246–260. doi:10.4064/fm-34-1-246-260. This article, based on an analysis of the Hausdorff paradox, settled a question put forth by von Neumann in 1929: 8. Wagon, Corollary 13.3 9. Foreman, M.; Wehrung, F. (1991). "The Hahn–Banach theorem implies the existence of a non-Lebesgue measurable set" (PDF). Fundamenta Mathematicae. 138: 13–19. doi:10.4064/fm-138-1-13-19. 10. Pawlikowski, Janusz (1991). "The Hahn–Banach theorem implies the Banach–Tarski paradox" (PDF). Fundamenta Mathematicae. 138: 21–22. doi:10.4064/fm-138-1-21-22. 11. Wagon, p. 16. 12. INVARIANT MEASURES, EXPANDERS AND PROPERTY T MAXIME BERGERON 13. Churkin, V. A. (2010). "A continuous version of the Hausdorff–Banach–Tarski paradox". Algebra and Logic. 49 (1): 81–89. doi:10.1007/s10469-010-9080-y. S2CID 122711859. Full text in Russian is available from the Mathnet.ru page. 14. On p. 85. Neumann, J. v. (1929). "Zur allgemeinen Theorie des Masses" (PDF). Fundamenta Mathematicae. 13: 73–116. doi:10.4064/fm-13-1-73-116. 15. Laczkovich, Miklós (1999). "Paradoxical sets under SL2(R)". Ann. Univ. Sci. Budapest. Eötvös Sect. Math. 42: 141–145. 16. Satô, Kenzi (2003). "A locally commutative free group acting on the plane". Fundamenta Mathematicae. 180 (1): 25–34. doi:10.4064/fm180-1-3. 17. Laczkovich, Miklós (1999). "Paradoxical sets under SL2(R)". Ann. Univ. Sci. Budapest. Eötvös Sect. Math. 42: 141–145. 18. Tomkowicz, Grzegorz (2011). "A free group of piecewise linear transformations". Colloquium Mathematicum. 125 (2): 141–146. doi:10.4064/cm125-2-1. 19. Adams, John Frank (1954). "On decompositions of the sphere". J. London Math. Soc. 29: 96–99. doi:10.1112/jlms/s1-29.1.96. 20. Mycielski, Jan (1955). "On the paradox of the sphere". Fund. Math. 42 (2): 348–355. doi:10.4064/fm-42-2-348-355. 21. Tomkowicz, Grzegorz (2017). "On decompositions of the hyperbolic plane satisfying many congruences". Bulletin of the London Mathematical Society. 49: 133–140. doi:10.1112/blms.12024. S2CID 125603157. 22. Mycielski, Jan (1989). "The Banach-Tarski paradox for the hyperbolic plane". Fund. Math. 132 (2): 143–149. doi:10.4064/fm-132-2-143-149. 23. Mycielski, Jan; Tomkowicz, Grzegorz (2013). "The Banach-Tarski paradox for the hyperbolic plane (II)". Fund. Math. 222 (3): 289–290. doi:10.4064/fm222-3-5. 24. Tomkowicz, Grzegorz (2017). "Banach-Tarski paradox in some complete manifolds". Proc. Amer. Math. Soc. 145 (12): 5359–5362. doi:10.1090/proc/13657. 25. Mycielski, Jan; Wagon, Stan (1984). "Large free groups of isometries and their geometrical uses". Ens. Math. 30: 247–267. 26. Tomkowicz, Grzegorz (2018). "A properly discontinuous free group of affine transformations". Geom. Dedicata. 197: 91–95. doi:10.1007/s10711-018-0320-y. S2CID 126151042. 27. Mycielski, Jan; Tomkowicz, Grzegorz (2019). "On the equivalence of sets of equal measures by countable decomposition". Bulletin of the London Mathematical Society. 51: 961–966. doi:10.1112/blms.12289. S2CID 209936338. References • Banach, Stefan; Tarski, Alfred (1924). "Sur la décomposition des ensembles de points en parties respectivement congruentes" (PDF). Fundamenta Mathematicae. 6: 244–277. doi:10.4064/fm-6-1-244-277. • Churkin, V. A. (2010). "A continuous version of the Hausdorff–Banach–Tarski paradox". Algebra and Logic. 49 (1): 91–98. doi:10.1007/s10469-010-9080-y. S2CID 122711859. • Edward Kasner & James Newman (1940) Mathematics and the Imagination, pp 205–7, Simon & Schuster. • Stromberg, Karl (March 1979). "The Banach–Tarski paradox". The American Mathematical Monthly. Mathematical Association of America. 86 (3): 151–161. doi:10.2307/2321514. JSTOR 2321514. • Su, Francis E. "The Banach–Tarski Paradox" (PDF). • von Neumann, John (1929). "Zur allgemeinen Theorie des Masses" (PDF). Fundamenta Mathematicae. 13: 73–116. doi:10.4064/fm-13-1-73-116. • Wagon, Stan (1994). The Banach–Tarski Paradox. Cambridge: Cambridge University Press. ISBN 0-521-45704-1. • Wapner, Leonard M. (2005). The Pea and the Sun: A Mathematical Paradox. Wellesley, Massachusetts: A.K. Peters. ISBN 1-56881-213-2. • Tomkowicz, Grzegorz; Wagon, Stan (2016). The Banach–Tarski Paradox 2nd Edition. Cambridge: Cambridge University Press. ISBN 9781107042599. External links Wikimedia Commons has media related to Banach-Tarski paradox. • Banach–Tarski paradox at ProofWiki • The Banach-Tarski Paradox by Stan Wagon (Macalester College), the Wolfram Demonstrations Project. • Irregular Webcomic! #2339 by David Morgan-Mar provides a non-technical explanation of the paradox. It includes a step-by-step demonstration of how to create two spheres from one. • Vsauce. "The Banach–Tarski Paradox" – via YouTube gives an overview on the fundamental basics of the paradox. • Banach-Tarski and the Paradox of Infinite Cloning Authority control International • FAST National • France • BnF data • Germany • Israel • United States Other • IdRef	Wikipedia
The Banach–Tarski Paradox (book) The Banach–Tarski Paradox is a book in mathematics on the Banach–Tarski paradox, the fact that a unit ball can be partitioned into a finite number of subsets and reassembled to form two unit balls. It was written by Stan Wagon and published in 1985 by the Cambridge University Press as volume 24 of their Encyclopedia of Mathematics and its Applications book series.[1][2][3][4][5] A second printing in 1986 added two pages as an addendum, and a 1993 paperback printing added a new preface.[6] In 2016 the Cambridge University Press published a second edition, adding Grzegorz Tomkowicz as a co-author, as volume 163 of the same series.[7][8] The Basic Library List Committee of the Mathematical Association of America has recommended its inclusion in undergraduate mathematics libraries.[8] The Banach–Tarski Paradox First edition AuthorStan Wagon SeriesEncyclopedia of Mathematics and its Applications SubjectThe Banach-Tarski paradox PublisherCambridge University Press Publication date 1985 Topics The Banach–Tarski paradox, proved by Stefan Banach and Alfred Tarski in 1924, states that it is possible to partition a three-dimensional unit ball into finitely many pieces and reassemble them into two unit balls, a single ball of larger or smaller area, or any other bounded set with a non-empty interior. Although it is a mathematical theorem, it is called a paradox because it is so counter-intuitive; in the preface to the book, Jan Mycielski calls it the most surprising result in mathematics. It is closely related to measure theory and the non-existence of a measure on all subsets of three-dimensional space, invariant under all congruences of space, and to the theory of paradoxical sets in free groups and the representation of these groups by three-dimensional rotations, used in the proof of the paradox. The topic of the book is the Banach–Tarski paradox, its proof, and the many related results that have since become known.[3][5] The book is divided into two parts, the first on the existence of paradoxical decompositions and the second on conditions that prevent their existence.[1][7] After two chapters of background material, the first part proves the Banach–Tarski paradox itself, considers higher-dimensional spaces and non-Euclidean geometry, studies the number of pieces necessary for a paradoxical decomposition, and finds analogous results to the Banach–Tarski paradox for one- and two-dimensional sets. The second part includes a related theorem of Tarski that congruence-invariant finitely-additive measures prevent the existence of paradoxical decompositions, a theorem that Lebesgue measure is the only such measure on the Lebesgue measurable sets, material on amenable groups, connections to the axiom of choice and the Hahn–Banach theorem.[3][7] Three appendices describe Euclidean groups, Jordan measure, and a collection of open problems.[1] The second edition adds material on several recent results in this area, in many cases inspired by the first edition of the book. Trevor Wilson proved the existence of a continuous motion from the one-ball assembly to the two-ball assembly, keeping the sets of the partition disjoint at all times; this question had been posed by de Groot in the first edition of the book.[7][9] Miklós Laczkovich solved Tarski's circle-squaring problem, asking for a dissection of a disk to a square of the same area, in 1990.[7][8][10] And Edward Marczewski had asked in 1930 whether the Banach–Tarski paradox could be achieved using only Baire sets; a positive answer was found in 1994 by Randall Dougherty and Matthew Foreman.[8][11] Audience and reception The book is written at a level accessible to mathematics graduate students, but provides a survey of research in this area that should also be useful to more advanced researchers.[3] The beginning parts of the book, including its proof of the Banach–Tarski paradox, should also be readable by undergraduate mathematicians.[4] Reviewer Włodzimierz Bzyl writes that "this beautiful book is written with care and is certainly worth reading".[2] Reviewer John J. Watkins writes that the first edition of the book "became the classic text on paradoxical mathematics" and that the second edition "exceeds any possible expectation I might have had for expanding a book I already deeply treasured".[8] See also • List of paradoxes • History of mathematics References 1. Luxemburg, W. A. J., "Review of The Banach–Tarski Paradox (1st ed.)", zbMATH, Zbl 0569.43001 2. Bzyl, Włodzimierz (1987), "Review of The Banach–Tarski Paradox (1st ed.)", Mathematical Reviews, MR 0803509 3. Gardner, R. J. (March 1986), "Review of The Banach–Tarski Paradox (1st ed.)", Bulletin of the London Mathematical Society, 18 (2): 207–208, doi:10.1112/blms/18.2.207 4. Henson, C. Ward (July–August 1987), American Scientist, 75 (4): 436, JSTOR 27854763{{citation}}: CS1 maint: untitled periodical (link) 5. Mycielski, Jan (August–September 1987), American Mathematical Monthly, 94 (7): 698–700, doi:10.2307/2322243, JSTOR 2322243{{citation}}: CS1 maint: untitled periodical (link) 6. Foreman, Matthew (June 1995), "Review of The Banach–Tarski Paradox (1993 paperback ed.)", Journal of Symbolic Logic, 60 (2): 698, doi:10.2307/2275867, JSTOR 2275867 7. Hart, Klaas Pieter, "Review of The Banach–Tarski Paradox (2nd ed.)", Mathematical Reviews, MR 3616119 8. Watkins, John J. (July 2017), "Review of The Banach–Tarski Paradox (2nd ed.)", MAA Reviews, Mathematical Association of America 9. Wilson, Trevor M. (2005), "A continuous movement version of the Banach–Tarski paradox: a solution to de Groot's problem", Journal of Symbolic Logic, 70 (3): 946–952, doi:10.2178/jsl/1122038921, MR 2155273, S2CID 15825008 10. Laczkovich, M. (1990), "Equidecomposability and discrepancy; a solution of Tarski's circle-squaring problem", Journal für die Reine und Angewandte Mathematik, 1990 (404): 77–117, doi:10.1515/crll.1990.404.77, MR 1037431, S2CID 117762563 11. Dougherty, Randall; Foreman, Matthew (1994), "Banach–Tarski decompositions using sets with the property of Baire", Journal of the American Mathematical Society, 7 (1): 75–124, doi:10.2307/2152721, JSTOR 2152721, MR 1227475	Wikipedia
Birthday problem In probability theory, the birthday problem asks for the probability that, in a set of n randomly chosen people, at least two will share a birthday. The birthday paradox refers to the counterintuitive fact that only 23 people are needed for that probability to exceed 50%. The birthday paradox is a veridical paradox: it seems wrong at first glance but is, in fact, true. While it may seem surprising that only 23 individuals are required to reach a 50% probability of a shared birthday, this result is made more intuitive by considering that the birthday comparisons will be made between every possible pair of individuals. With 23 individuals, there are 23 × 22/2 = 253 pairs to consider, far more than half the number of days in a year. Real-world applications for the birthday problem include a cryptographic attack called the birthday attack, which uses this probabilistic model to reduce the complexity of finding a collision for a hash function, as well as calculating the approximate risk of a hash collision existing within the hashes of a given size of population. The problem is generally attributed to Harold Davenport in about 1927, though he did not publish it at the time. Davenport did not claim to be its discoverer "because he could not believe that it had not been stated earlier".[1][2] The first publication of a version of the birthday problem was by Richard von Mises in 1939.[3] Calculating the probability From a permutations perspective, let the event A be the probability of finding a group of 23 people without any repeated birthdays. Where the event B is the probability of finding a group of 23 people with at least two people sharing same birthday, P(B) = 1 − P(A). P(A) is the ratio of the total number of birthdays, $V_{nr}$, without repetitions and order matters (e.g. for a group of 2 people, mm/dd birthday format, one possible outcome is $\left\{\left\{01/02,05/20\right\},\left\{05/20,01/02\right\},\left\{10/02,08/04\right\},...\right\}$ divided by the total number of birthdays with repetition and order matters, $V_{t}$, as it is the total space of outcomes from the experiment (e.g. 2 people, one possible outcome is $\left\{\left\{01/02,01/02\right\},\left\{10/02,08/04\right\},...\right\}$. Therefore $V_{nr}$ and $V_{t}$ are permutations. ${\begin{aligned}V_{nr}&={\frac {n!}{(n-k)!}}={\frac {365!}{(365-23)!}}\\V_{t}&=n^{k}=365^{23}\\P(A)&={\frac {V_{nr}}{V_{t}}}\approx 0.492703\\P(B)&=1-P(A)\approx 1-0.492703\approx 0.507297(50.7297\%)\end{aligned}}$ Another way the birthday problem can be solved is by asking for an approximate probability that in a group of n people at least two have the same birthday. For simplicity, leap years, twins, selection bias, and seasonal and weekly variations in birth rates[4] are generally disregarded, and instead it is assumed that there are 365 possible birthdays, and that each person's birthday is equally likely to be any of these days, independent of the other people in the group. For independent birthdays, the uniform distribution on birthdays is the distribution that minimizes the probability of two people with the same birthday; any unevenness increases this probability.[5][6] The problem of a non-uniform number of births occurring during each day of the year was first addressed by Murray Klamkin in 1967.[7] As it happens, the real-world distribution yields a critical size of 23 to reach 50%.[8] The goal is to compute P(A), the probability that at least two people in the room have the same birthday. However, it is simpler to calculate P(A′), the probability that no two people in the room have the same birthday. Then, because A and A′ are the only two possibilities and are also mutually exclusive, P(A) = 1 − P(A′). Here is the calculation of P(A) for 23 people. Let the 23 people be numbered 1 to 23. The event that all 23 people have different birthdays is the same as the event that person 2 does not have the same birthday as person 1, and that person 3 does not have the same birthday as either person 1 or person 2, and so on, and finally that person 23 does not have the same birthday as any of persons 1 through 22. Let these events be called Event 2, Event 3, and so on. Event 1 is the event of person 1 having a birthday, which occurs with probability 1. This conjunction of events may be computed using conditional probability: the probability of Event 2 is 364/365, as person 2 may have any birthday other than the birthday of person 1. Similarly, the probability of Event 3 given that Event 2 occurred is 363/365, as person 3 may have any of the birthdays not already taken by persons 1 and 2. This continues until finally the probability of Event 23 given that all preceding events occurred is 343/365. Finally, the principle of conditional probability implies that P(A′) is equal to the product of these individual probabilities: $P(A')={\frac {365}{365}}\times {\frac {364}{365}}\times {\frac {363}{365}}\times {\frac {362}{365}}\times \cdots \times {\frac {343}{365}}$ (1) The terms of equation (1) can be collected to arrive at: $P(A')=\left({\frac {1}{365}}\right)^{23}\times (365\times 364\times 363\times \cdots \times 343)$ (2) Evaluating equation (2) gives P(A′) ≈ 0.492703 Therefore, P(A) ≈ 1 − 0.492703 = 0.507297 (50.7297%). This process can be generalized to a group of n people, where p(n) is the probability of at least two of the n people sharing a birthday. It is easier to first calculate the probability p(n) that all n birthdays are different. According to the pigeonhole principle, p(n) is zero when n > 365. When n ≤ 365: ${\begin{aligned}{\bar {p}}(n)&=1\times \left(1-{\frac {1}{365}}\right)\times \left(1-{\frac {2}{365}}\right)\times \cdots \times \left(1-{\frac {n-1}{365}}\right)\\[6pt]&={\frac {365\times 364\times \cdots \times (365-n+1)}{365^{n}}}\\[6pt]&={\frac {365!}{365^{n}(365-n)!}}={\frac {n!\cdot {\binom {365}{n}}}{365^{n}}}={\frac {_{365}P_{n}}{365^{n}}}\end{aligned}}$ where ! is the factorial operator, (365 n ) is the binomial coefficient and kPr denotes permutation. The equation expresses the fact that the first person has no one to share a birthday, the second person cannot have the same birthday as the first (364/365), the third cannot have the same birthday as either of the first two (363/365), and in general the nth birthday cannot be the same as any of the n − 1 preceding birthdays. The event of at least two of the n persons having the same birthday is complementary to all n birthdays being different. Therefore, its probability p(n) is $p(n)=1-{\bar {p}}(n).$ The following table shows the probability for some other values of n (for this table, the existence of leap years is ignored, and each birthday is assumed to be equally likely): np(n) 100.0% 502.7% 1011.7% 2041.1% 2350.7% 3070.6% 4089.1% 5097.0% 6099.4% 7099.9% 7599.97% 10099.99997% 20099.9999999999999999999999999998% 300(100 − 6×10−80)% 350(100 − 3×10−129)% 365(100 − 1.45×10−155)% ≥ 366100% Approximations The Taylor series expansion of the exponential function (the constant e ≈ 2.718281828) $e^{x}=1+x+{\frac {x^{2}}{2!}}+\cdots $ provides a first-order approximation for ex for $\|x\|\ll 1$: $e^{x}\approx 1+x.$ To apply this approximation to the first expression derived for p(n), set x = −a/365. Thus, $e^{-{\frac {a}{365}}}\approx 1-{\frac {a}{365}}.$ Then, replace a with non-negative integers for each term in the formula of p(n) until a = n − 1, for example, when a = 1, $e^{-{\frac {1}{365}}}\approx 1-{\frac {1}{365}}.$ The first expression derived for p(n) can be approximated as ${\begin{aligned}{\bar {p}}(n)&\approx 1\cdot e^{-{\frac {1}{365}}}\cdot e^{-{\frac {2}{365}}}\cdots e^{-{\frac {n-1}{365}}}\\[6pt]&=e^{-{\frac {1+2+\,\cdots \,+(n-1)}{365}}}\\[6pt]&=e^{-{\frac {n(n-1)/2}{365}}}=e^{-{\frac {n(n-1)}{730}}}.\end{aligned}}$ Therefore, $p(n)=1-{\bar {p}}(n)\approx 1-e^{-{\frac {n(n-1)}{730}}}.$ An even coarser approximation is given by $p(n)\approx 1-e^{-{\frac {n^{2}}{730}}},$ which, as the graph illustrates, is still fairly accurate. According to the approximation, the same approach can be applied to any number of "people" and "days". If rather than 365 days there are d, if there are n persons, and if n ≪ d, then using the same approach as above we achieve the result that if p(n, d) is the probability that at least two out of n people share the same birthday from a set of d available days, then: ${\begin{aligned}p(n,d)&\approx 1-e^{-{\frac {n(n-1)}{2d}}}\\[6pt]&\approx 1-e^{-{\frac {n^{2}}{2d}}}.\end{aligned}}$ Simple exponentiation The probability of any two people not having the same birthday is 364/365. In a room containing n people, there are (n 2 ) = n(n − 1)/2 pairs of people, i.e. (n 2 ) events. The probability of no two people sharing the same birthday can be approximated by assuming that these events are independent and hence by multiplying their probability together. Being independent would be equivalent to picking with replacement, any pair of people in the world, not just in a room. In short 364/365 can be multiplied by itself (n 2 ) times, which gives us ${\bar {p}}(n)\approx \left({\frac {364}{365}}\right)^{\binom {n}{2}}.$ Since this is the probability of no one having the same birthday, then the probability of someone sharing a birthday is $p(n)\approx 1-\left({\frac {364}{365}}\right)^{\binom {n}{2}}.$ And for the group of 23 people, the probability of sharing is $p(23)\approx 1-\left({\frac {364}{365}}\right)^{\binom {23}{2}}=1-\left({\frac {364}{365}}\right)^{253}\approx 0.500477.$ Poisson approximation Applying the Poisson approximation for the binomial on the group of 23 people, $\operatorname {Poi} \left({\frac {\binom {23}{2}}{365}}\right)=\operatorname {Poi} \left({\frac {253}{365}}\right)\approx \operatorname {Poi} (0.6932)$ so $\Pr(X>0)=1-\Pr(X=0)\approx 1-e^{-0.6932}\approx 1-0.499998=0.500002.$ The result is over 50% as previous descriptions. This approximation is the same as the one above based on the Taylor expansion that uses ex ≈ 1 + x. Square approximation A good rule of thumb which can be used for mental calculation is the relation $p(n)\approx {\frac {n^{2}}{2m}}$ which can also be written as $n\approx {\sqrt {2m\times p(n)}}$ which works well for probabilities less than or equal to 1/2. In these equations, m is the number of days in a year. For instance, to estimate the number of people required for a 1/2 chance of a shared birthday, we get $n\approx {\sqrt {2\times 365\times {\tfrac {1}{2}}}}={\sqrt {365}}\approx 19$ Which is not too far from the correct answer of 23. Approximation of number of people This can also be approximated using the following formula for the number of people necessary to have at least a 1/2 chance of matching: $n\geq {\tfrac {1}{2}}+{\sqrt {{\tfrac {1}{4}}+2\times \ln(2)\times 365}}=22.999943.$ This is a result of the good approximation that an event with 1/k probability will have a 1/2 chance of occurring at least once if it is repeated k ln 2 times.[9] Probability table length of hex string no. of bits (b) hash space size (2b) Number of hashed elements such that probability of at least one hash collision ≥ p p = 10−18 p = 10−15 p = 10−12 p = 10−9 p = 10−6 p = 0.001 p = 0.01 p = 0.25 p = 0.50 p = 0.75 8 32 4.3×109 2 2 2 2.9 93 2.9×103 9.3×103 5.0×104 7.7×104 1.1×105 (10) (40) (1.1×1012) 2 2 2 47 1.5×103 4.7×104 1.5×105 8.0×105 1.2×106 1.7×106 (12) (48) (2.8×1014) 2 2 24 7.5×102 2.4×104 7.5×105 2.4×106 1.3×107 2.0×107 2.8×107 16 64 1.8×1019 6.1 1.9×102 6.1×103 1.9×105 6.1×106 1.9×108 6.1×108 3.3×109 5.1×109 7.2×109 (24) (96) (7.9×1028) 4.0×105 1.3×107 4.0×108 1.3×1010 4.0×1011 1.3×1013 4.0×1013 2.1×1014 3.3×1014 4.7×1014 32 128 3.4×1038 2.6×1010 8.2×1011 2.6×1013 8.2×1014 2.6×1016 8.3×1017 2.6×1018 1.4×1019 2.2×1019 3.1×1019 (48) (192) (6.3×1057) 1.1×1020 3.5×1021 1.1×1023 3.5×1024 1.1×1026 3.5×1027 1.1×1028 6.0×1028 9.3×1028 1.3×1029 64 256 1.2×1077 4.8×1029 1.5×1031 4.8×1032 1.5×1034 4.8×1035 1.5×1037 4.8×1037 2.6×1038 4.0×1038 5.7×1038 (96) (384) (3.9×10115) 8.9×1048 2.8×1050 8.9×1051 2.8×1053 8.9×1054 2.8×1056 8.9×1056 4.8×1057 7.4×1057 1.0×1058 128 512 1.3×10154 1.6×1068 5.2×1069 1.6×1071 5.2×1072 1.6×1074 5.2×1075 1.6×1076 8.8×1076 1.4×1077 1.9×1077 The lighter fields in this table show the number of hashes needed to achieve the given probability of collision (column) given a hash space of a certain size in bits (row). Using the birthday analogy: the "hash space size" resembles the "available days", the "probability of collision" resembles the "probability of shared birthday", and the "required number of hashed elements" resembles the "required number of people in a group". One could also use this chart to determine the minimum hash size required (given upper bounds on the hashes and probability of error), or the probability of collision (for fixed number of hashes and probability of error). For comparison, 10−18 to 10−15 is the uncorrectable bit error rate of a typical hard disk.[10] In theory, 128-bit hash functions, such as MD5, should stay within that range until about 8.2×1011 documents, even if its possible outputs are many more. An upper bound on the probability and a lower bound on the number of people The argument below is adapted from an argument of Paul Halmos.[nb 1] As stated above, the probability that no two birthdays coincide is $1-p(n)={\bar {p}}(n)=\prod _{k=1}^{n-1}\left(1-{\frac {k}{365}}\right).$ As in earlier paragraphs, interest lies in the smallest n such that p(n) > 1/2; or equivalently, the smallest n such that p(n) < 1/2. Using the inequality 1 − x < e−x in the above expression we replace 1 − k/365 with e−k⁄365. This yields ${\bar {p}}(n)=\prod _{k=1}^{n-1}\left(1-{\frac {k}{365}}\right)<\prod _{k=1}^{n-1}\left(e^{-{\frac {k}{365}}}\right)=e^{-{\frac {n(n-1)}{730}}}.$ Therefore, the expression above is not only an approximation, but also an upper bound of p(n). The inequality $e^{-{\frac {n(n-1)}{730}}}<{\frac {1}{2}}$ implies p(n) < 1/2. Solving for n gives $n^{2}-n>730\ln 2.$ Now, 730 ln 2 is approximately 505.997, which is barely below 506, the value of n2 − n attained when n = 23. Therefore, 23 people suffice. Incidentally, solving n2 − n = 730 ln 2 for n gives the approximate formula of Frank H. Mathis cited above. This derivation only shows that at most 23 people are needed to ensure a birthday match with even chance; it leaves open the possibility that n is 22 or less could also work. Generalizations Arbitrary number of days Given a year with d days, the generalized birthday problem asks for the minimal number n(d) such that, in a set of n randomly chosen people, the probability of a birthday coincidence is at least 50%. In other words, n(d) is the minimal integer n such that $1-\left(1-{\frac {1}{d}}\right)\left(1-{\frac {2}{d}}\right)\cdots \left(1-{\frac {n-1}{d}}\right)\geq {\frac {1}{2}}.$ The classical birthday problem thus corresponds to determining n(365). The first 99 values of n(d) are given here (sequence A033810 in the OEIS): d 1–23–56–910–1617–2324–3233–4243–5455–6869–8283–99 n(d) 23456789101112 A similar calculation shows that n(d) = 23 when d is in the range 341–372. A number of bounds and formulas for n(d) have been published.[11] For any d ≥ 1, the number n(d) satisfies[12] ${\frac {3-2\ln 2}{6}}<n(d)-{\sqrt {2d\ln 2}}\leq 9-{\sqrt {86\ln 2}}.$ These bounds are optimal in the sense that the sequence n(d) − √2d ln 2 gets arbitrarily close to ${\frac {3-2\ln 2}{6}}\approx 0.27,$ while it has $9-{\sqrt {86\ln 2}}\approx 1.28$ as its maximum, taken for d = 43. The bounds are sufficiently tight to give the exact value of n(d) in most of the cases. For example, for d = 365 these bounds imply that 22.7633 < n(365) < 23.7736 and 23 is the only integer in that range. In general, it follows from these bounds that n(d) always equals either $\left\lceil {\sqrt {2d\ln 2}}\,\right\rceil \quad {\text{or}}\quad \left\lceil {\sqrt {2d\ln 2}}\,\right\rceil +1$ where ⌈ · ⌉ denotes the ceiling function. The formula $n(d)=\left\lceil {\sqrt {2d\ln 2}}\,\right\rceil $ holds for 73% of all integers d.[13] The formula $n(d)=\left\lceil {\sqrt {2d\ln 2}}+{\frac {3-2\ln 2}{6}}\right\rceil $ holds for almost all d, i.e., for a set of integers d with asymptotic density 1.[13] The formula $n(d)=\left\lceil {\sqrt {2d\ln 2}}+{\frac {3-2\ln 2}{6}}+{\frac {9-4(\ln 2)^{2}}{72{\sqrt {2d\ln 2}}}}\right\rceil $ holds for all d ≤ 1018, but it is conjectured that there are infinitely many counterexamples to this formula.[14] The formula $n(d)=\left\lceil {\sqrt {2d\ln 2}}+{\frac {3-2\ln 2}{6}}+{\frac {9-4(\ln 2)^{2}}{72{\sqrt {2d\ln 2}}}}-{\frac {2(\ln 2)^{2}}{135d}}\right\rceil $ holds for all d ≤ 1018, and it is conjectured that this formula holds for all d.[14] More than two people sharing a birthday It is possible to extend the problem to ask how many people in a group are necessary for there to be a greater than 50% probability that at least 3, 4, 5, etc. of the group share the same birthday. The first few values are as follows: >50% probability of 3 people sharing a birthday - 88 people; >50% probability of 4 people sharing a birthday - 187 people (sequence A014088 in the OEIS).[15] Probability of a shared birthday (collision) The birthday problem can be generalized as follows: Given n random integers drawn from a discrete uniform distribution with range [1,d], what is the probability p(n; d) that at least two numbers are the same? (d = 365 gives the usual birthday problem.)[16] The generic results can be derived using the same arguments given above. ${\begin{aligned}p(n;d)&={\begin{cases}1-\displaystyle \prod _{k=1}^{n-1}\left(1-{\frac {k}{d}}\right)&n\leq d\\1&n>d\end{cases}}\\[8px]&\approx 1-e^{-{\frac {n(n-1)}{2d}}}\\&\approx 1-\left({\frac {d-1}{d}}\right)^{\frac {n(n-1)}{2}}\end{aligned}}$ Conversely, if n(p; d) denotes the number of random integers drawn from [1,d] to obtain a probability p that at least two numbers are the same, then $n(p;d)\approx {\sqrt {2d\cdot \ln \left({\frac {1}{1-p}}\right)}}.$ The birthday problem in this more generic sense applies to hash functions: the expected number of N-bit hashes that can be generated before getting a collision is not 2N, but rather only 2N⁄2. This is exploited by birthday attacks on cryptographic hash functions and is the reason why a small number of collisions in a hash table are, for all practical purposes, inevitable. The theory behind the birthday problem was used by Zoe Schnabel[17] under the name of capture-recapture statistics to estimate the size of fish population in lakes. Generalization to multiple types of people The basic problem considers all trials to be of one "type". The birthday problem has been generalized to consider an arbitrary number of types.[18] In the simplest extension there are two types of people, say m men and n women, and the problem becomes characterizing the probability of a shared birthday between at least one man and one woman. (Shared birthdays between two men or two women do not count.) The probability of no shared birthdays here is $p_{0}={\frac {1}{d^{m+n}}}\sum _{i=1}^{m}\sum _{j=1}^{n}S_{2}(m,i)S_{2}(n,j)\prod _{k=0}^{i+j-1}d-k$ where d = 365 and S2 are Stirling numbers of the second kind. Consequently, the desired probability is 1 − p0. This variation of the birthday problem is interesting because there is not a unique solution for the total number of people m + n. For example, the usual 50% probability value is realized for both a 32-member group of 16 men and 16 women and a 49-member group of 43 women and 6 men. Other birthday problems First match A related question is, as people enter a room one at a time, which one is most likely to be the first to have the same birthday as someone already in the room? That is, for what n is p(n) − p(n − 1) maximum? The answer is 20—if there is a prize for first match, the best position in line is 20th. Same birthday as you In the birthday problem, neither of the two people is chosen in advance. By contrast, the probability q(n) that someone in a room of n other people has the same birthday as a particular person (for example, you) is given by $q(n)=1-\left({\frac {365-1}{365}}\right)^{n}$ and for general d by $q(n;d)=1-\left({\frac {d-1}{d}}\right)^{n}.$ In the standard case of d = 365, substituting n = 23 gives about 6.1%, which is less than 1 chance in 16. For a greater than 50% chance that one person in a roomful of n people has the same birthday as you, n would need to be at least 253. This number is significantly higher than 365/2 = 182.5: the reason is that it is likely that there are some birthday matches among the other people in the room. Number of people with a shared birthday For any one person in a group of n people the probability that he or she shares his birthday with someone else is $q(n-1;d)$, as explained above. The expected number of people with a shared (non-unique) birthday can now be calculated easily by multiplying that probability by the number of people (n), so it is: $n\left(1-\left({\frac {d-1}{d}}\right)^{n-1}\right)$ (This multiplication can be done this way because of the linearity of the expected value of indicator variables). This implies that the expected number of people with a non-shared (unique) birthday is: $n\left({\frac {d-1}{d}}\right)^{n-1}$ Similar formulas can be derived for the expected number of people who share with three, four, etc. other people. Number of people until every birthday is achieved The expected number of people needed until every birthday is achieved is called the Coupon collector's problem. It can be calculated by nHn, where Hn is the nth harmonic number. For 365 possible dates (the birthday problem), the answer is 2365. Near matches Another generalization is to ask for the probability of finding at least one pair in a group of n people with birthdays within k calendar days of each other, if there are d equally likely birthdays.[19] ${\begin{aligned}p(n,k,d)&=1-{\frac {(d-nk-1)!}{d^{n-1}{\bigl (}d-n(k+1){\bigr )}!}}\end{aligned}}$ The number of people required so that the probability that some pair will have a birthday separated by k days or fewer will be higher than 50% is given in the following table: kn for d = 365 023 114 211 39 48 58 67 77 Thus in a group of just seven random people, it is more likely than not that two of them will have a birthday within a week of each other.[19] Number of days with at least one birthday The expected number of different birthdays, i.e. the number of days that are at least one person's birthday, is: $d-d\left({\frac {d-1}{d}}\right)^{n}$ This follows from the expected number of days that are no one's birthday: $d\left({\frac {d-1}{d}}\right)^{n}$ which follows from the probability that a particular day is no one's birthday, (d − 1/d)n , easily summed because of the linearity of the expected value. For instance, with d = 365, you should expect about 21 different birthdays when there are 22 people, or 46 different birthdays when there are 50 people. When there are 1000 people, there will be around 341 different birthdays (24 unclaimed birthdays). Number of days with at least two birthdays The above can be generalized from the distribution of the number of people with their birthday on any particular day, which is a Binomial distribution with probability 1/d. Multiplying the relevant probability by d will then give the expected number of days. For example, the expected number of days which are shared; i.e. which are at least two (i.e. not zero and not one) people's birthday is: $d-d\left({\frac {d-1}{d}}\right)^{n}-d\cdot {\binom {n}{1}}\left({\frac {1}{d}}\right)^{1}\left({\frac {d-1}{d}}\right)^{n-1}=d-d\left({\frac {d-1}{d}}\right)^{n}-n\left({\frac {d-1}{d}}\right)^{n-1}$ Number of people who repeat a birthday The probability that the kth integer randomly chosen from [1,d] will repeat at least one previous choice equals q(k − 1; d) above. The expected total number of times a selection will repeat a previous selection as n such integers are chosen equals[20] $\sum _{k=1}^{n}q(k-1;d)=n-d+d\left({\frac {d-1}{d}}\right)^{n}$ This can be seen to equal the number of people minus the expected number of different birthdays. Average number of people to get at least one shared birthday In an alternative formulation of the birthday problem, one asks the average number of people required to find a pair with the same birthday. If we consider the probability function Pr[n people have at least one shared birthday], this average is determining the mean of the distribution, as opposed to the customary formulation, which asks for the median. The problem is relevant to several hashing algorithms analyzed by Donald Knuth in his book The Art of Computer Programming. It may be shown[21][22] that if one samples uniformly, with replacement, from a population of size M, the number of trials required for the first repeated sampling of some individual has expected value n = 1 + Q(M), where $Q(M)=\sum _{k=1}^{M}{\frac {M!}{(M-k)!M^{k}}}.$ The function $Q(M)=1+{\frac {M-1}{M}}+{\frac {(M-1)(M-2)}{M^{2}}}+\cdots +{\frac {(M-1)(M-2)\cdots 1}{M^{M-1}}}$ has been studied by Srinivasa Ramanujan and has asymptotic expansion: $Q(M)\sim {\sqrt {\frac {\pi M}{2}}}-{\frac {1}{3}}+{\frac {1}{12}}{\sqrt {\frac {\pi }{2M}}}-{\frac {4}{135M}}+\cdots .$ With M = 365 days in a year, the average number of people required to find a pair with the same birthday is n = 1 + Q(M) ≈ 24.61659, somewhat more than 23, the number required for a 50% chance. In the best case, two people will suffice; at worst, the maximum possible number of M + 1 = 366 people is needed; but on average, only 25 people are required An analysis using indicator random variables can provide a simpler but approximate analysis of this problem.[23] For each pair (i, j) for k people in a room, we define the indicator random variable Xij, for $1\leq i\leq j\leq k$, by ${\begin{alignedat}{2}X_{ij}&=I\{{\text{person }}i{\text{ and person }}j{\text{ have the same birthday}}\}\\&={\begin{cases}1,&{\text{if person }}i{\text{ and person }}j{\text{ have the same birthday;}}\\0,&{\text{otherwise.}}\end{cases}}\end{alignedat}}$ ${\begin{alignedat}{2}E[X_{ij}]&=\Pr\{{\text{person }}i{\text{ and person }}j{\text{ have the same birthday}}\}\\&={\frac {1}{n}}\end{alignedat}}$ Let X be a random variable counting the pairs of individuals with the same birthday. $X=\sum _{i=1}^{k}\sum _{j=i+1}^{k}X_{ij}$ ${\begin{alignedat}{3}E[X]&=\sum _{i=1}^{k}\sum _{j=i+1}^{k}E[X_{ij}]\\&={\binom {k}{2}}{\frac {1}{n}}\\&={\frac {k(k-1)}{2n}}\\\end{alignedat}}$ For n = 365, if k = 28, the expected number of pairs of individuals with the same birthday is 28 × 27/2 × 365 ≈ 1.0356. Therefore, we can expect at least one matching pair with at least 28 people. An informal demonstration of the problem can be made from the list of prime ministers of Australia, of which there have been 29 as of 2017, in which Paul Keating, the 24th prime minister, and Edmund Barton, the first prime minister, share the same birthday, 18 January. In the 2014 FIFA World Cup, each of the 32 squads had 23 players. An analysis of the official squad lists suggested that 16 squads had pairs of players sharing birthdays, and of these 5 squads had two pairs: Argentina, France, Iran, South Korea and Switzerland each had two pairs, and Australia, Bosnia and Herzegovina, Brazil, Cameroon, Colombia, Honduras, Netherlands, Nigeria, Russia, Spain and USA each with one pair.[24] Voracek, Tran and Formann showed that the majority of people markedly overestimate the number of people that is necessary to achieve a given probability of people having the same birthday, and markedly underestimate the probability of people having the same birthday when a specific sample size is given.[25] Further results showed that psychology students and women did better on the task than casino visitors/personnel or men, but were less confident about their estimates. Reverse problem The reverse problem is to find, for a fixed probability p, the greatest n for which the probability p(n) is smaller than the given p, or the smallest n for which the probability p(n) is greater than the given p. Taking the above formula for d = 365, one has $n(p;365)\approx {\sqrt {730\ln \left({\frac {1}{1-p}}\right)}}.$ The following table gives some sample calculations. pn n↓p(n↓)n↑p(n↑) 0.01 0.14178√365 = 2.70864 20.002743 0.00820 0.050.32029√365 = 6.11916 60.0404670.05624 0.1 0.45904√365 = 8.77002 80.074349 0.09462 0.2 0.66805√365 = 12.76302 120.1670213 0.19441 0.30.84460√365 = 16.13607 160.28360170.31501 0.51.17741√365 = 22.49439 220.47570230.50730 0.71.55176√365 = 29.64625 290.68097300.70632 0.81.79412√365 = 34.27666 340.79532350.81438 0.92.14597√365 = 40.99862 400.89123410.90315 0.952.44775√365 = 46.76414 460.94825470.95477 0.99 3.03485√365 = 57.98081 57 0.99012 580.99166 Some values falling outside the bounds have been colored to show that the approximation is not always exact. Partition problem A related problem is the partition problem, a variant of the knapsack problem from operations research. Some weights are put on a balance scale; each weight is an integer number of grams randomly chosen between one gram and one million grams (one tonne). The question is whether one can usually (that is, with probability close to 1) transfer the weights between the left and right arms to balance the scale. (In case the sum of all the weights is an odd number of grams, a discrepancy of one gram is allowed.) If there are only two or three weights, the answer is very clearly no; although there are some combinations which work, the majority of randomly selected combinations of three weights do not. If there are very many weights, the answer is clearly yes. The question is, how many are just sufficient? That is, what is the number of weights such that it is equally likely for it to be possible to balance them as it is to be impossible? Often, people's intuition is that the answer is above 100000. Most people's intuition is that it is in the thousands or tens of thousands, while others feel it should at least be in the hundreds. The correct answer is 23. The reason is that the correct comparison is to the number of partitions of the weights into left and right. There are 2N − 1 different partitions for N weights, and the left sum minus the right sum can be thought of as a new random quantity for each partition. The distribution of the sum of weights is approximately Gaussian, with a peak at 500000N and width 1000000√N, so that when 2N − 1 is approximately equal to 1000000√N the transition occurs. 223 − 1 is about 4 million, while the width of the distribution is only 5 million.[26] In fiction Arthur C. Clarke's 1961 novel A Fall of Moondust contains a section where the main characters, trapped underground for an indefinite amount of time, are celebrating a birthday and find themselves discussing the validity of the birthday problem. As stated by a physicist passenger: "If you have a group of more than twenty-four people, the odds are better than even that two of them have the same birthday." Eventually, out of 22 present, it is revealed that two characters share the same birthday, May 23. Notes 1. In his autobiography, Halmos criticized the form in which the birthday paradox is often presented, in terms of numerical computation. He believed that it should be used as an example in the use of more abstract mathematical concepts. He wrote: The reasoning is based on important tools that all students of mathematics should have ready access to. The birthday problem used to be a splendid illustration of the advantages of pure thought over mechanical manipulation; the inequalities can be obtained in a minute or two, whereas the multiplications would take much longer, and be much more subject to error, whether the instrument is a pencil or an old-fashioned desk computer. What calculators do not yield is understanding, or mathematical facility, or a solid basis for more advanced, generalized theories. References 1. David Singmaster, Sources in Recreational Mathematics: An Annotated Bibliography, Eighth Preliminary Edition, 2004, section 8.B 2. H.S.M. Coxeter, "Mathematical Recreations and Essays, 11th edition", 1940, p 45, as reported in I. J. Good, Probability and the weighing of evidence, 1950, p. 38 3. Richard Von Mises, "Über Aufteilungs- und Besetzungswahrscheinlichkeiten", Revue de la faculté des sciences de l'Université d'Istanbul 4:145-163, 1939, reprinted in Frank, P.; Goldstein, S.; Kac, M.; Prager, W.; Szegö, G.; Birkhoff, G., eds. (1964). Selected Papers of Richard von Mises. Vol. 2. Providence, Rhode Island: Amer. Math. Soc. pp. 313–334. 4. see Birthday#Distribution through the year 5. (Bloom 1973) 6. Steele, J. Michael (2004). The Cauchy‑Schwarz Master Class. Cambridge: Cambridge University Press. pp. 206, 277. ISBN 9780521546775. 7. Klamkin & Newman 1967. 8. Mario Cortina Borja; John Haigh (September 2007). "The Birthday Problem". Significance. Royal Statistical Society. 4 (3): 124–127. doi:10.1111/j.1740-9713.2007.00246.x. 9. Mathis, Frank H. (June 1991). "A Generalized Birthday Problem". SIAM Review. 33 (2): 265–270. doi:10.1137/1033051. ISSN 0036-1445. JSTOR 2031144. OCLC 37699182. 10. Jim Gray, Catharine van Ingen. Empirical Measurements of Disk Failure Rates and Error Rates 11. D. Brink, A (probably) exact solution to the Birthday Problem, Ramanujan Journal, 2012, . 12. Brink 2012, Theorem 2 13. Brink 2012, Theorem 3 14. Brink 2012, Table 3, Conjecture 1 15. "Minimal number of people to give a 50% probability of having at least n coincident birthdays in one year". The On-line Encyclopedia of Integer Sequences. OEIS. Retrieved 17 February 2020. 16. Suzuki, K.; Tonien, D.; et al. (2006). "Birthday Paradox for Multi-collisions". In Rhee M.S., Lee B. (ed.). Lecture Notes in Computer Science, vol 4296. Berlin: Springer. doi:10.1007/11927587_5. Information Security and Cryptology – ICISC 2006. 17. Z. E. Schnabel (1938) The Estimation of the Total Fish Population of a Lake, American Mathematical Monthly 45, 348–352. 18. M. C. Wendl (2003) Collision Probability Between Sets of Random Variables, Statistics and Probability Letters 64(3), 249–254. 19. M. Abramson and W. O. J. Moser (1970) More Birthday Surprises, American Mathematical Monthly 77, 856–858 20. Might, Matt. "Collision hash collisions with the birthday paradox". Matt Might's blog. Retrieved 17 July 2015. 21. Knuth, D. E. (1973). The Art of Computer Programming. Vol. 3, Sorting and Searching. Reading, Massachusetts: Addison-Wesley. ISBN 978-0-201-03803-3. 22. Flajolet, P.; Grabner, P. J.; Kirschenhofer, P.; Prodinger, H. (1995). "On Ramanujan's Q-Function". Journal of Computational and Applied Mathematics. 58: 103–116. doi:10.1016/0377-0427(93)E0258-N. 23. Cormen; et al. Introduction to Algorithms. 24. Fletcher, James (16 June 2014). "The birthday paradox at the World Cup". bbc.com. BBC. Retrieved 27 August 2015. 25. Voracek, M.; Tran, U. S.; Formann, A. K. (2008). "Birthday and birthmate problems: Misconceptions of probability among psychology undergraduates and casino visitors and personnel". Perceptual and Motor Skills. 106 (1): 91–103. doi:10.2466/pms.106.1.91-103. PMID 18459359. S2CID 22046399. 26. Borgs, C.; Chayes, J.; Pittel, B. (2001). "Phase Transition and Finite Size Scaling in the Integer Partition Problem". Random Structures and Algorithms. 19 (3–4): 247–288. doi:10.1002/rsa.10004. S2CID 6819493. Bibliography • Abramson, M.; Moser, W. O. J. (1970). "More Birthday Surprises". American Mathematical Monthly. 77 (8): 856–858. doi:10.2307/2317022. JSTOR 2317022. • Bloom, D. (1973). "A Birthday Problem". American Mathematical Monthly. 80 (10): 1141–1142. doi:10.2307/2318556. JSTOR 2318556. • Kemeny, John G.; Snell, J. Laurie; Thompson, Gerald (1957). Introduction to Finite Mathematics (First ed.). • Klamkin, M.; Newman, D. (1967). "Extensions of the Birthday Surprise". Journal of Combinatorial Theory. 3 (3): 279–282. doi:10.1016/s0021-9800(67)80075-9. • McKinney, E. H. (1966). "Generalized Birthday Problem". American Mathematical Monthly. 73 (5): 385–387. doi:10.2307/2315408. JSTOR 2315408. • Mosteller, F. (1962). "Understanding the birthday problem". The Mathematics Teacher. Springer Series in Statistics. 55 (5): 322–325. doi:10.1007/978-0-387-44956-2_21. ISBN 978-0-387-20271-6. JSTOR 27956609. • Schneps, Leila; Colmez, Coralie (2013). "Math error number 5. The case of Diana Sylvester: cold hit analysis". Math on Trial. How Numbers Get Used and Abused in the Courtroom. Basic Books. ISBN 978-0-465-03292-1. • Sy M. Blinder (2013). Guide to Essential Math: A Review for Physics, Chemistry and Engineering Students. Elsevier. pp. 5–6. ISBN 978-0-12-407163-6. External links • The Birthday Paradox accounting for leap year birthdays • Weisstein, Eric W. "Birthday Problem". MathWorld. • A humorous article explaining the paradox • SOCR EduMaterials activities birthday experiment • Understanding the Birthday Problem (Better Explained) • Eurobirthdays 2012. A birthday problem. A practical football example of the birthday paradox. • Grime, James. "23: Birthday Probability". Numberphile. Brady Haran. Archived from the original on 2017-02-25. Retrieved 2013-04-02. • Computing the probabilities of the Birthday Problem at WolframAlpha	Wikipedia
Bunyakovsky conjecture The Bunyakovsky conjecture (or Bouniakowsky conjecture) gives a criterion for a polynomial $f(x)$ in one variable with integer coefficients to give infinitely many prime values in the sequence$f(1),f(2),f(3),\ldots .$ It was stated in 1857 by the Russian mathematician Viktor Bunyakovsky. The following three conditions are necessary for $f(x)$ to have the desired prime-producing property: 1. the leading coefficient is positive, 2. the polynomial is irreducible over the rationals (and integers), and 3. the values $f(1),f(2),f(3),\ldots $ have no common factor. (In particular, the coefficients of $f(x)$ should be relatively prime.) Bunyakovsky conjecture FieldAnalytic number theory Conjectured byViktor Bunyakovsky Conjectured in1857 Known casesPolynomials of degree 1 GeneralizationsBateman–Horn conjecture Generalized Dickson conjecture Schinzel's hypothesis H ConsequencesTwin prime conjecture Bunyakovsky's conjecture is that these conditions are sufficient: if $f(x)$ satisfies (1)–(3), then $f(n)$ is prime for infinitely many positive integers $n$. A seemingly weaker yet equivalent statement to Bunyakovsky's conjecture is that for every integer polynomial $f(x)$ that satisfies (1)–(3), $f(n)$ is prime for at least one positive integer $n$: but then, since the translated polynomial $f(x+n)$ still satisfies (1)–(3), in view of the weaker statement $f(m)$ is prime for at least one positive integer $m>n$, so that $f(n)$ is indeed prime for infinitely many positive integers $n$. Bunyakovsky's conjecture is a special case of Schinzel's hypothesis H, one of the most famous open problems in number theory. Discussion of three conditions The first condition is necessary because if the leading coefficient is negative then $f(x)<0$ for all large $x$, and thus $f(n)$ is not a (positive) prime number for large positive integers $n$. (This merely satisfies the sign convention that primes are positive.) The second condition is necessary because if $f(x)=g(x)h(x)$ where the polynomials $g(x)$ and $h(x)$ have integer coefficients, then we have $f(n)=g(n)h(n)$ for all integers $n$; but $g(x)$ and $h(x)$ take the values 0 and $\pm 1$ only finitely many times, so $f(n)$ is composite for all large $n$. The second condition also fails for the polynomials reducible over the rationals. For example, the integer-valued polynomial $P(x)=(1/12)\cdot x^{4}+(11/12)\cdot x^{2}+2$ doesn't satisfy the condition (2) since $P(x)=(1/12)\cdot (x^{4}+11x^{2}+24)=(1/12)\cdot (x^{2}+3)\cdot (x^{2}+8)$, so at least one of the latter two factors must be a divisor of $12$ in order to have $P(x)$ prime, which holds only if $\|x\|\leq 3$. The corresponding values are $2,3,7,17$, so these are the only such primes for integral $x$ since all of these numbers are prime. This isn't a counterexample to Bunyakovsky conjecture since the condition (2) fails. The third condition, that the numbers $f(n)$ have gcd 1, is obviously necessary, but is somewhat subtle, and is best understood by a counterexample. Consider $f(x)=x^{2}+x+2$, which has positive leading coefficient and is irreducible, and the coefficients are relatively prime; however $f(n)$ is even for all integers $n$, and so is prime only finitely many times (namely at $n=0,-1$, when $f(n)=2$). In practice, the easiest way to verify the third condition is to find one pair of positive integers $m$ and $n$ such that $f(m)$ and $f(n)$ are relatively prime. In general, for any integer-valued polynomial $f(x)=c_{0}+c_{1}x+\cdots +c_{d}x^{d}$ we can use $\gcd\{f(n)\}_{n\geq 1}=\gcd(f(m),f(m+1),\dots ,f(m+d))$ for any integer $m$, so the gcd is given by values of $f(x)$ at any consecutive $d+1$ integers.[1] In the example above, we have $f(-1)=2,f(0)=2,f(1)=4$ and so the gcd is $2$, which implies that $x^{2}+x+2$ has even values on the integers. Alternatively, when an integer polynomial $f(x)$ is written in the basis of binomial coefficient polynomials $f(x)=a_{0}+a_{1}{\binom {x}{1}}+\cdots +a_{d}{\binom {x}{d}},$ each coefficient $a_{i}$ is an integer and $\gcd\{f(n)\}_{n\geq 1}=\gcd(a_{0},a_{1},\dots ,a_{d}).$ In the example above, this is $x^{2}+x+2=2{\binom {x}{2}}+2{\binom {x}{1}}+2,$ and the coefficients in the right side of the equation have gcd 2. Using this gcd formula, it can be proved $\gcd\{f(n)\}_{n\geq 1}=1$ if and only if there are positive integers $m$ and $n$ such that $f(m)$ and $f(n)$ are relatively prime. Examples A simple quadratic polynomial Some prime values of the polynomial $f(x)=x^{2}+1$ are listed in the following table. (Values of $x$ form OEIS sequence A005574; those of $x^{2}+1$ form A002496.) $x$ 1246101416202426364054566674849094110116120 $x^{2}+1$ 251737101197257401577677129716012917313743575477705781018837121011345714401 That $n^{2}+1$ should be prime infinitely often is a problem first raised by Euler, and it is also the fifth Hardy–Littlewood conjecture and the fourth of Landau's problems. Despite the extensive numerical evidence [2] it is not known that this sequence extends indefinitely. Cyclotomic polynomials The cyclotomic polynomials $\Phi _{k}(x)$ for $k=1,2,3,\ldots $ satisfy the three conditions of Bunyakovsky's conjecture, so for all k, there should be infinitely many natural numbers n such that $\Phi _{k}(n)$ is prime. It can be shown that if for all k, there exists an integer n > 1 with $\Phi _{k}(n)$ prime, then for all k, there are infinitely many natural numbers n with $\Phi _{k}(n)$ prime. The following sequence gives the smallest natural number n > 1 such that $\Phi _{k}(n)$ is prime, for $k=1,2,3,\ldots $: 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 5, 2, 2, 2, 2, 2, 2, 6, 2, 4, 3, 2, 10, 2, 22, 2, 2, 4, 6, 2, 2, 2, 2, 2, 14, 3, 61, 2, 10, 2, 14, 2, 15, 25, 11, 2, 5, 5, 2, 6, 30, 11, 24, 7, 7, 2, 5, 7, 19, 3, 2, 2, 3, 30, 2, 9, 46, 85, 2, 3, 3, 3, 11, 16, 59, 7, 2, 2, 22, 2, 21, 61, 41, 7, 2, 2, 8, 5, 2, 2, ... (sequence A085398 in the OEIS). This sequence is known to contain some large terms: the 545th term is 2706, the 601st is 2061, and the 943rd is 2042. This case of Bunyakovsky's conjecture is widely believed, but again it is not known that the sequence extends indefinitely. Usually, there is an integer $n$ between 2 and $\phi (k)$ (where $\phi $ is Euler's totient function, so $\phi (k)$ is the degree of $\Phi _{k}(n)$) such that $\Phi _{k}(n)$ is prime, but there are exceptions; the first few are 1, 2, 25, 37, 44, 68, 75, 82, 99, 115, 119, 125, 128, 159, 162, 179, 183, 188, 203, 213, 216, 229, 233, 243, 277, 289, 292, .... Partial results: only Dirichlet's theorem To date, the only case of Bunyakovsky's conjecture that has been proved is that of polynomials of degree 1. This is Dirichlet's theorem, which states that when $a$ and $m$ are relatively prime integers there are infinitely many prime numbers $p\equiv a{\pmod {m}}$. This is Bunyakovsky's conjecture for $f(x)=a+mx$ (or $a-mx$ if $m<0$). The third condition in Bunyakovsky's conjecture for a linear polynomial $mx+a$ is equivalent to $a$ and $m$ being relatively prime. No single case of Bunyakovsky's conjecture for degree greater than 1 is proved, although numerical evidence in higher degree is consistent with the conjecture. Generalized Bunyakovsky conjecture Given $k\geq 1$ polynomials with positive degrees and integer coefficients, each satisfying the three conditions, assume that for any prime $p$ there is an $n$ such that none of the values of the $k$ polynomials at $n$ are divisible by $p$. Given these assumptions, it is conjectured that there are infinitely many positive integers $n$ such that all values of these $k$ polynomials at $x=n$ are prime. Note that the polynomials $\{x,x+2,x+4\}$ do not satisfy the assumption, since one of their values must be divisible by 3 for any integer $x=n$. Neither do $\{x,x^{2}+2\}$, since one of the values must be divisible by 3 for any $x=n$. On the other hand, $\{x^{2}+1,3x-1,x^{2}+x+41\}$ do satisfy the assumption, and the conjecture implies the polynomials have simultaneous prime values for infinitely many positive integers $x=n$. This conjecture includes as special cases the twin prime conjecture (when $k=2$, and the two polynomials are $x$ and $x+2$) as well as the infinitude of prime quadruplets (when $k=4$, and the four polynomials are $x,x+2,x+6$, and $x+8$), sexy primes (when $k=2$, and the two polynomials are $x$ and $x+6$), Sophie Germain primes (when $k=2$, and the two polynomials are $x$ and $2x+1$), and Polignac's conjecture (when $k=2$, and the two polynomials are $x$ and $x+a$, with $a$ any even number). When all the polynomials have degree 1 this is Dickson's conjecture. In fact, this conjecture is equivalent to the Generalized Dickson conjecture. Except for Dirichlet's theorem, no case of the conjecture has been proved, including the above cases. See also • Integer-valued polynomial • Cohn's irreducibility criterion • Schinzel's hypothesis H • Bateman–Horn conjecture • Hardy and Littlewood's conjecture F References 1. Hensel, Kurt (1896). "Ueber den grössten gemeinsamen Theiler aller Zahlen, welche durch eine ganze Function von n Veränderlichen darstellbar sind". Journal für die reine und angewandte Mathematik. 1896 (116): 350–356. doi:10.1515/crll.1896.116.350. S2CID 118266353. 2. Wolf, Marek (2013), "Some Conjectures On Primes Of The Form m2 + 1" (PDF), Journal of Combinatorics and Number Theory, 5: 103–132 Bibliography • Ed Pegg, Jr. "Bouniakowsky conjecture". MathWorld. • Rupert, Wolfgang M. (1998-08-05). "Reducibility of polynomials f(x, y) modulo p". arXiv:math/9808021. • Bouniakowsky, V. (1857). "Sur les diviseurs numériques invariables des fonctions rationnelles entières". Mém. Acad. Sc. St. Pétersbourg. 6: 305–329. Prime number conjectures • Hardy–Littlewood • 1st • 2nd • Agoh–Giuga • Andrica's • Artin's • Bateman–Horn • Brocard's • Bunyakovsky • Chinese hypothesis • Cramér's • Dickson's • Elliott–Halberstam • Firoozbakht's • Gilbreath's • Grimm's • Landau's problems • Goldbach's • weak • Legendre's • Twin prime • Legendre's constant • Lemoine's • Mersenne • Oppermann's • Polignac's • Pólya • Schinzel's hypothesis H • Waring's prime number	Wikipedia
Bowyer–Watson algorithm In computational geometry, the Bowyer–Watson algorithm is a method for computing the Delaunay triangulation of a finite set of points in any number of dimensions. The algorithm can be also used to obtain a Voronoi diagram of the points, which is the dual graph of the Delaunay triangulation. Description The Bowyer–Watson algorithm is an incremental algorithm. It works by adding points, one at a time, to a valid Delaunay triangulation of a subset of the desired points. After every insertion, any triangles whose circumcircles contain the new point are deleted, leaving a star-shaped polygonal hole which is then re-triangulated using the new point. By using the connectivity of the triangulation to efficiently locate triangles to remove, the algorithm can take O(N log N) operations to triangulate N points, although special degenerate cases exist where this goes up to O(N2).[1] • First step: insert a node in an enclosing "super"-triangle • Insert second node • Insert third node • Insert fourth node • Insert fifth (and last) node • Remove edges with extremes in the super-triangle History The algorithm is sometimes known just as the Bowyer Algorithm or the Watson Algorithm. Adrian Bowyer and David Watson devised it independently of each other at the same time, and each published a paper on it in the same issue of The Computer Journal (see below). Pseudocode The following pseudocode describes a basic implementation of the Bowyer-Watson algorithm. Its time complexity is $O(n^{2})$. Efficiency can be improved in a number of ways. For example, the triangle connectivity can be used to locate the triangles which contain the new point in their circumcircle, without having to check all of the triangles - by doing so we can decrease time complexity to $O(n\log n)$. Pre-computing the circumcircles can save time at the expense of additional memory usage. And if the points are uniformly distributed, sorting them along a space filling Hilbert curve prior to insertion can also speed point location.[2] function BowyerWatson (pointList) // pointList is a set of coordinates defining the points to be triangulated triangulation := empty triangle mesh data structure add super-triangle to triangulation // must be large enough to completely contain all the points in pointList for each point in pointList do // add all the points one at a time to the triangulation badTriangles := empty set for each triangle in triangulation do // first find all the triangles that are no longer valid due to the insertion if point is inside circumcircle of triangle add triangle to badTriangles polygon := empty set for each triangle in badTriangles do // find the boundary of the polygonal hole for each edge in triangle do if edge is not shared by any other triangles in badTriangles add edge to polygon for each triangle in badTriangles do // remove them from the data structure remove triangle from triangulation for each edge in polygon do // re-triangulate the polygonal hole newTri := form a triangle from edge to point add newTri to triangulation for each triangle in triangulation // done inserting points, now clean up if triangle contains a vertex from original super-triangle remove triangle from triangulation return triangulation References 1. Rebay, S. Efficient Unstructured Mesh Generation by Means of Delaunay Triangulation and Bowyer-Watson Algorithm. Journal of Computational Physics Volume 106 Issue 1, May 1993, p. 127. 2. Liu, Yuanxin, and Jack Snoeyink. "A comparison of five implementations of 3D Delaunay tessellation." Combinatorial and Computational Geometry 52 (2005): 439-458. Further reading • Bowyer, Adrian (1981). "Computing Dirichlet tessellations". Comput. J. 24 (2): 162–166. doi:10.1093/comjnl/24.2.162. • Watson, David F. (1981). "Computing the n-dimensional Delaunay tessellation with application to Voronoi polytopes". Comput. J. 24 (2): 167–172. doi:10.1093/comjnl/24.2.167. • Efficient Triangulation Algorithm Suitable for Terrain Modelling generic explanations with source code examples in several languages. External links • pyDelaunay2D : A didactic Python implementation of Bowyer–Watson algorithm. • Bl4ckb0ne/delaunay-triangulation : C++ implementation of Bowyer–Watson algorithm.	Wikipedia
Calculus of moving surfaces The calculus of moving surfaces (CMS) [1] is an extension of the classical tensor calculus to deforming manifolds. Central to the CMS is the Tensorial Time Derivative ${\dot {\nabla }}$ whose original definition [2] was put forth by Jacques Hadamard. It plays the role analogous to that of the covariant derivative $\nabla _{\alpha }$ on differential manifolds in that it produces a tensor when applied to a tensor. Suppose that $\Sigma _{t}$ is the evolution of the surface $\Sigma $ indexed by a time-like parameter $t$. The definitions of the surface velocity $C$ and the operator ${\dot {\nabla }}$ are the geometric foundations of the CMS. The velocity C is the rate of deformation of the surface $\Sigma $ in the instantaneous normal direction. The value of $C$ at a point $P$ is defined as the limit $C=\lim _{h\to 0}{\frac {{\text{Distance}}(P,P^{})}{h}}$ where $P^{}$ is the point on $\Sigma _{t+h}$ that lies on the straight line perpendicular to $\Sigma _{t}$ at point P. This definition is illustrated in the first geometric figure below. The velocity $C$ is a signed quantity: it is positive when ${\overline {PP^{}}}$ points in the direction of the chosen normal, and negative otherwise. The relationship between $\Sigma _{t}$ and $C$ is analogous to the relationship between location and velocity in elementary calculus: knowing either quantity allows one to construct the other by differentiation or integration. The Tensorial Time Derivative ${\dot {\nabla }}$ for a scalar field F defined on $\Sigma _{t}$ is the rate of change in $F$ in the instantaneously normal direction: ${\frac {\delta F}{\delta t}}=\lim _{h\to 0}{\frac {F(P^{})-F(P)}{h}}$ This definition is also illustrated in second geometric figure. The above definitions are geometric. In analytical settings, direct application of these definitions may not be possible. The CMS gives analytical definitions of C and ${\dot {\nabla }}$ in terms of elementary operations from calculus and differential geometry. Analytical definitions For analytical definitions of $C$ and ${\dot {\nabla }}$, consider the evolution of $S$ given by $Z^{i}=Z^{i}\left(t,S\right)$ where $Z^{i}$ are general curvilinear space coordinates and $S^{\alpha }$ are the surface coordinates. By convention, tensor indices of function arguments are dropped. Thus the above equations contains $S$ rather than $S^{\alpha }$. The velocity object ${\textbf {V}}=V^{i}{\textbf {Z}}_{i}$ is defined as the partial derivative $V^{i}={\frac {\partial Z^{i}\left(t,S\right)}{\partial t}}$ The velocity $C$ can be computed most directly by the formula $C=V^{i}N_{i}$ where $N_{i}$ are the covariant components of the normal vector ${\vec {N}}$. Also, defining the shift tensor representation of the Surface's Tangent Space $Z_{i}^{\alpha }={\textbf {S}}^{\alpha }\cdot {\textbf {Z}}_{i}$ and the Tangent Velocity as $V^{\alpha }=Z_{i}^{\alpha }V^{i}$ , then the definition of the ${\dot {\nabla }}$ derivative for an invariant F reads ${\dot {\nabla }}F={\frac {\partial F\left(t,S\right)}{\partial t}}-V^{\alpha }\nabla _{\alpha }F$ where $\nabla _{\alpha }$ is the covariant derivative on S. For tensors, an appropriate generalization is needed. The proper definition for a representative tensor $T_{j\beta }^{i\alpha }$ reads ${\dot {\nabla }}T_{j\beta }^{i\alpha }={\frac {\partial T_{j\beta }^{i\alpha }}{\partial t}}-V^{\eta }\nabla _{\eta }T_{j\beta }^{i\alpha }+V^{m}\Gamma _{mk}^{i}T_{j\beta }^{k\alpha }-V^{m}\Gamma _{mj}^{k}T_{k\beta }^{i\alpha }+{\dot {\Gamma }}_{\eta }^{\alpha }T_{j\beta }^{i\eta }-{\dot {\Gamma }}_{\beta }^{\eta }T_{j\eta }^{i\alpha }$ where $\Gamma _{mj}^{k}$ are Christoffel symbols and ${\dot {\Gamma }}_{\beta }^{\alpha }=\nabla _{\beta }V^{\alpha }-CB_{\beta }^{\alpha }$ is the surface's appropriate temporal symbols ($B_{\beta }^{\alpha }$ is a matrix representation of the surface's curvature shape operator) Properties of the ${\dot {\nabla }}$-derivative The ${\dot {\nabla }}$-derivative commutes with contraction, satisfies the product rule for any collection of indices ${\dot {\nabla }}(S_{\alpha }^{i}T_{j}^{\beta })=T_{j}^{\beta }{\dot {\nabla }}S_{\alpha }^{i}+S_{\alpha }^{i}{\dot {\nabla }}T_{j}^{\beta }$ and obeys a chain rule for surface restrictions of spatial tensors: ${\dot {\nabla }}F_{k}^{j}(Z,t)={\frac {\partial F_{k}^{j}}{\partial t}}+CN^{i}\nabla _{i}F_{k}^{j}$ Chain rule shows that the ${\dot {\nabla }}$-derivatives of spatial "metrics" vanishes ${\dot {\nabla }}\delta _{j}^{i}=0,{\dot {\nabla }}Z_{ij}=0,{\dot {\nabla }}Z^{ij}=0,{\dot {\nabla }}\varepsilon _{ijk}=0,{\dot {\nabla }}\varepsilon ^{ijk}=0$ where $Z_{ij}$ and $Z^{ij}$ are covariant and contravariant metric tensors, $\delta _{j}^{i}$ is the Kronecker delta symbol, and $\varepsilon _{ijk}$ and $\varepsilon ^{ijk}$ are the Levi-Civita symbols. The main article on Levi-Civita symbols describes them for Cartesian coordinate systems. The preceding rule is valid in general coordinates, where the definition of the Levi-Civita symbols must include the square root of the determinant of the covariant metric tensor $Z_{ij}$. Differentiation table for the ${\dot {\nabla }}$-derivative The ${\dot {\nabla }}$ derivative of the key surface objects leads to highly concise and attractive formulas. When applied to the covariant surface metric tensor $S_{\alpha \beta }$ and the contravariant metric tensor $S^{\alpha \beta }$, the following identities result ${\begin{aligned}{\dot {\nabla }}S_{\alpha \beta }&=0\\[8pt]{\dot {\nabla }}S^{\alpha \beta }&=0\end{aligned}}$ where $B_{\alpha \beta }$ and $B^{\alpha \beta }$ are the doubly covariant and doubly contravariant curvature tensors. These curvature tensors, as well as for the mixed curvature tensor $B_{\beta }^{\alpha }$, satisfy ${\begin{aligned}{\dot {\nabla }}B_{\alpha \beta }&=\nabla _{\alpha }\nabla _{\beta }C+CB_{\alpha \gamma }B_{\beta }^{\gamma }\\[8pt]{\dot {\nabla }}B_{\beta }^{\alpha }&=\nabla _{\beta }\nabla ^{\alpha }C+CB_{\gamma }^{\alpha }B_{\beta }^{\gamma }\\[8pt]{\dot {\nabla }}B^{\alpha \beta }&=\nabla ^{\alpha }\nabla ^{\beta }C+CB^{\gamma \alpha }B_{\gamma }^{\beta }\end{aligned}}$ The shift tensor $Z_{\alpha }^{i}$ and the normal$N^{i}$ satisfy ${\begin{aligned}{\dot {\nabla }}Z_{\alpha }^{i}&=N^{i}\nabla _{\alpha }C\\[8pt]{\dot {\nabla }}N^{i}&=-Z_{\alpha }^{i}\nabla ^{\alpha }C\end{aligned}}$ Finally, the surface Levi-Civita symbols $\varepsilon _{\alpha \beta }$ and $\varepsilon ^{\alpha \beta }$ satisfy ${\begin{aligned}{\dot {\nabla }}\varepsilon _{\alpha \beta }&=0\\[8pt]{\dot {\nabla }}\varepsilon ^{\alpha \beta }&=0\end{aligned}}$ Time differentiation of integrals The CMS provides rules for time differentiation of volume and surface integrals. References 1. Grinfeld, P. (2010). "Hamiltonian Dynamic Equations for Fluid Films". Studies in Applied Mathematics. doi:10.1111/j.1467-9590.2010.00485.x. ISSN 0022-2526. 2. J. Hadamard, Leçons Sur La Propagation Des Ondes Et Les Équations de l'Hydrodynamique. Paris: Hermann, 1903.	Wikipedia
Chudnovsky algorithm The Chudnovsky algorithm is a fast method for calculating the digits of π, based on Ramanujan’s π formulae. It was published by the Chudnovsky brothers in 1988.[1] It was used in the world record calculations of 2.7 trillion digits of π in December 2009,[2] 10 trillion digits in October 2011,[3][4] 22.4 trillion digits in November 2016,[5] 31.4 trillion digits in September 2018–January 2019,[6] 50 trillion digits on January 29, 2020,[7] 62.8 trillion digits on August 14, 2021,[8] and 100 trillion digits on March 21, 2022.[9] Algorithm The algorithm is based on the negated Heegner number $d=-163$, the j-function $j\left({\tfrac {1+i{\sqrt {163}}}{2}}\right)=-640320^{3}$, and on the following rapidly convergent generalized hypergeometric series:[2] ${\frac {1}{\pi }}=12\sum _{q=0}^{\infty }{\frac {(-1)^{q}(6q)!(545140134q+13591409)}{(3q)!(q!)^{3}\left(640320\right)^{3q+{\frac {3}{2}}}}}$ A detailed proof of this formula can be found here:[10] For a high performance iterative implementation, this can be simplified to ${\frac {(640320)^{\frac {3}{2}}}{12\pi }}={\frac {426880{\sqrt {10005}}}{\pi }}=\sum _{q=0}^{\infty }{\frac {(6q)!(545140134q+13591409)}{(3q)!(q!)^{3}\left(-262537412640768000\right)^{q}}}$ There are 3 big integer terms (the multinomial term Mq, the linear term Lq, and the exponential term Xq) that make up the series and π equals the constant C divided by the sum of the series, as below: $\pi =C\left(\sum _{q=0}^{\infty }{\frac {M_{q}\cdot L_{q}}{X_{q}}}\right)^{-1}$, where: $C=426880{\sqrt {10005}}$, $M_{q}={\frac {(6q)!}{(3q)!(q!)^{3}}}$, $L_{q}=545140134q+13591409$, $X_{q}=(-262537412640768000)^{q}$. The terms Mq, Lq, and Xq satisfy the following recurrences and can be computed as such: ${\begin{alignedat}{4}L_{q+1}&=L_{q}+545140134\,\,&&{\textrm {where}}\,\,L_{0}&&=13591409\\[4pt]X_{q+1}&=X_{q}\cdot (-262537412640768000)&&{\textrm {where}}\,\,X_{0}&&=1\\[4pt]M_{q+1}&=M_{q}\cdot \left({\frac {(12q+2)(12q+6)(12q+10)}{(q+1)^{3}}}\right)\,\,&&{\textrm {where}}\,\,M_{0}&&=1\\[4pt]\end{alignedat}}$ The computation of Mq can be further optimized by introducing an additional term Kq as follows: ${\begin{alignedat}{4}K_{q+1}&=K_{q}+12\,\,&&{\textrm {where}}\,\,K_{0}&&=-6\\[4pt]M_{q+1}&=M_{q}\cdot \left({\frac {K_{q+1}^{3}-16K_{q+1}}{\left(q+1\right)^{3}}}\right)\,\,&&{\textrm {where}}\,\,M_{0}&&=1\\[12pt]\end{alignedat}}$ Note that $e^{\pi {\sqrt {163}}}\approx 640320^{3}+743.99999999999925\dots $ and $640320^{3}=262537412640768000$ $545140134=163\cdot 127\cdot 19\cdot 11\cdot 7\cdot 3^{2}\cdot 2$ $13591409=13\cdot 1045493$ This identity is similar to some of Ramanujan's formulas involving π,[2] and is an example of a Ramanujan–Sato series. The time complexity of the algorithm is $O\left(n(\log n)^{3}\right)$.[11] See also • Borwein's algorithm • Numerical approximations of π • Ramanujan–Sato series References 1. Chudnovsky, David; Chudnovsky, Gregory (1988), Approximation and complex multiplication according to ramanujan, Ramanujan revisited: proceedings of the centenary conference 2. Baruah, Nayandeep Deka; Berndt, Bruce C.; Chan, Heng Huat (2009), "Ramanujan's series for 1/π: a survey", American Mathematical Monthly, 116 (7): 567–587, doi:10.4169/193009709X458555, JSTOR 40391165, MR 2549375 3. Yee, Alexander; Kondo, Shigeru (2011), 10 Trillion Digits of Pi: A Case Study of summing Hypergeometric Series to high precision on Multicore Systems, Technical Report, Computer Science Department, University of Illinois, hdl:2142/28348 4. Aron, Jacob (March 14, 2012), "Constants clash on pi day", New Scientist 5. "22.4 Trillion Digits of Pi". www.numberworld.org. 6. "Google Cloud Topples the Pi Record". www.numberworld.org/. 7. "The Pi Record Returns to the Personal Computer". www.numberworld.org/. 8. "Pi-Challenge - Weltrekordversuch der FH Graubünden - FH Graubünden". www.fhgr.ch. Retrieved 2021-08-17. 9. "Calculating 100 trillion digits of pi on Google Cloud". cloud.google.com. Retrieved 2022-06-10. 10. Milla, Lorenz (2018), A detailed proof of the Chudnovsky formula with means of basic complex analysis, arXiv:1809.00533 11. "y-cruncher - Formulas". www.numberworld.org. Retrieved 2018-02-25.	Wikipedia
Cistercian numerals The medieval Cistercian numerals, or "ciphers" in nineteenth-century parlance, were developed by the Cistercian monastic order in the early thirteenth century at about the time that Arabic numerals were introduced to northwestern Europe. They are more compact than Arabic or Roman numerals, with a single glyph able to indicate any integer from 1 to 9,999. Part of a series on Numeral systems Place-value notation Hindu-Arabic numerals • Western Arabic • Eastern Arabic • Bengali • Devanagari • Gujarati • Gurmukhi • Odia • Sinhala • Tamil • Malayalam • Telugu • Kannada • Dzongkha • Tibetan • Balinese • Burmese • Javanese • Khmer • Lao • Mongolian • Sundanese • Thai East Asian systems Contemporary • Chinese • Suzhou • Hokkien • Japanese • Korean • Vietnamese Historic • Counting rods • Tangut Other systems • History Ancient • Babylonian Post-classical • Cistercian • Mayan • Muisca • Pentadic • Quipu • Rumi Contemporary • Cherokee • Kaktovik (Iñupiaq) By radix/base Common radices/bases • 2 • 3 • 4 • 5 • 6 • 8 • 10 • 12 • 16 • 20 • 60 • (table) Non-standard radices/bases • Bijective (1) • Signed-digit (balanced ternary) • Mixed (factorial) • Negative • Complex (2i) • Non-integer (φ) • Asymmetric Sign-value notation Non-alphabetic • Aegean • Attic • Aztec • Brahmi • Chuvash • Egyptian • Etruscan • Kharosthi • Prehistoric counting • Proto-cuneiform • Roman • Tally marks Alphabetic • Abjad • Armenian • Alphasyllabic • Akṣarapallī • Āryabhaṭa • Kaṭapayādi • Coptic • Cyrillic • Geʽez • Georgian • Glagolitic • Greek • Hebrew List of numeral systems Digits are based on a horizontal or vertical stave, with the position of the digit on the stave indicating its place value (units, tens, hundreds or thousands). These digits are compounded on a single stave to indicate more complex numbers. The Cistercians eventually abandoned the system in favor of the Arabic numerals, but marginal use outside the order continued until the early twentieth century. History The digits and idea of forming them into ligatures were apparently based on a two-place (1–99) numeral system introduced into the Cistercian Order by John of Basingstoke, archdeacon of Leicester, who it seems based them on a twelfth-century English shorthand (ars notaria).[nb 1] In its earliest attestations, in the monasteries of the County of Hainaut, the Cistercian system was not used for numbers greater than 99, but it was soon expanded to four places, enabling numbers up to 9,999.[2] The two dozen or so surviving Cistercian manuscripts that use the system date from the thirteenth to the fifteenth century, and cover an area from England to Italy, Normandy to Sweden. The numbers were not used for arithmetic, fractions or accounting, but indicated years, foliation (numbering pages), divisions of texts, the numbering of notes and other lists, indexes and concordances, arguments in Easter tables, and the lines of a staff in musical notation.[3] Although mostly confined to the Cistercian order, there was some usage outside it. A late-fifteenth-century Norman treatise on arithmetic used both Cistercian and Indo-Arabic numerals. In one known case, Cistercian numerals were inscribed on a physical object, indicating the calendrical, angular and other numbers on the fourteenth-century astrolabe of Berselius, which was made in French Picardy.[4] After the Cistercians had abandoned the system, marginal use continued outside the order. In 1533, Heinrich Cornelius Agrippa von Nettesheim included a description of these ciphers in his Three Books of Occult Philosophy.[5] The numerals were used by wine-gaugers in the Bruges area at least until the early eighteenth century.[6][7][8] In the late eighteenth century, Chevaliers de la Rose-Croix of Paris briefly adopted the numerals for mystical use, and in the early twentieth century Nazis considered using the numerals as Aryan symbolism.[3][9][10][11] The modern definitive expert on Cistercian numerals is the mathematician and historian of astronomy, David A. King.[12][1] Form A horizontal stave was most common while the numerals were in use among the Cistercians. A vertical stave was attested only in Northern France in the fourteenth and fifteenth centuries. However, eighteenth- and twentieth-century revivals of the system in France and Germany used a vertical stave. There is also some historical variation as to which corner of the number represented which place value. The place-values shown here were the most common among the Cistercians and the only ones used later.[3][13] Using graphic substitutes with a vertical stave,[nb 2] the first five digits are ꜒ 1, ꜓ 2, ꜒꜓ 3, ꜓꜒ 4, ꜍ 5. Reversing them forms the tens, ˥ 10, ˦ 20, ˦˥ 30, ˥˦ 40, ꜈ 50. Inverting them forms the hundreds, ꜖ 100, ꜕ 200, ꜖꜕ 300, ꜕꜖ 400, ꜑ 500, and doing both forms the thousands, ˩ 1,000, ˨ 2,000, ˨˩ 3,000, ˩˨ 4,000, ꜌ 5,000. Thus ⌶ (a digit 1 at each corner) is the number 1,111. (The exact forms varied by date and by monastery. For example, the digits shown here for 3 and 4 were in some manuscripts swapped with those for 7 and 8, and the 5's may be written with a lower dot (꜎ etc.), with a short vertical stroke in place of the dot, or even with a triangle joining to the stave, which in other manuscripts indicated a 9.)[13][1] • The vertical forms of the digits (1–9, 10–90, 100–900 and 1,000–9,000), with an innovative form of 5 as engraved on an early-sixteenth-century Norman astrolabe. • All Cistercian numerals from 1 to 9999[15] (open to enlarge). • A fourteenth-century Norman manuscript that used only Cistercian numerals. These were horizontal to fit the flow of the text. Note the round form of the digit 9. Numbers were later retranscribed with Hindu-Arabic digits in the margin notes: here we see 4,484, 715 and 5,199. Horizontal numbers were the same, but rotated 90 degrees counter-clockwise. (That is, ⌙ for 1, ⌐ for 10, ⏗ for 100—thus ⏘ for 101—and ¬ for 1,000, as seen above.)[2][1] Omitting a digit from a corner meant a value of zero for that power of ten, but there was no digit zero. (That is, an empty stave was not defined.)[16] Higher numbers When the system spread outside the order in the fifteenth and sixteenth centuries, numbers into the millions were enabled by compounding with the digit for "thousand". For example, a late-fifteenth century Norman treatise on arithmetic indicated 10,000 as a ligature of ⌋ "1,000" wrapped under and around ⌉ "10" (and similarly for higher numbers), and Noviomagus in 1539 wrote "million" by subscripting ¬ "1,000" under another ¬ "1,000".[17] A late-thirteenth-century Cistercian doodle had differentiated horizontal digits for lower powers of ten from vertical digits for higher powers of ten, but that potentially productive convention is not known to have been exploited at the time; it could have covered numbers into the tens of millions (horizontal 100 to 103, vertical 104 to 107).[18] A sixteenth-century mathematician used vertical digits for the traditional values, horizontal digits for millions, and rotated them a further 45° counter-clockwise for billions and another 90° for trillions, but it is not clear how the intermediate powers of ten were to be indicated and this convention was not adopted by others.[19] The Ciphers of the Monks The Ciphers of the Monks: A Forgotten Number-notation of the Middle Ages, by David A. King and published in 2001, describes the Cistercian numeral system.[20] The book[21] received mixed reviews. Historian Ann Moyer lauded King for re-introducing the numerical system to a larger audience, since many had forgotten about it.[22] Mathematician Detlef Spalt claimed that King exaggerated the system's importance and made mistakes in applying the system in the book devoted to it.[23] Moritz Wedell, however, called the book a "lucid description" and a "comprehensive review of the history of research" concerning the monks' ciphers.[24] Notes 1. Basingstoke's biographer claimed that he learned his system from his teacher in Athens. However, there is no known parallel among Greek numbering systems. It seems more likely that Basingstoke picked up the idea of alphabetic numerical notation in Greece and applied it to an English ars notaria, such as the one at right, commonly attributed to John of Tilbury.[1] 2. Cistercian numerals are not supported by Unicode, and are here substituted with Chao tone letters. Depending on the fonts you have installed, it may be that only the ones and twos will display properly. (The Under-ConScript Unicode Registry has tentatively assigned the units to PUA values U+EBA1 to U+EBAF.)[14] References 1. Chrisomalis, Stephen (2010). Numerical notation : a comparative history. Cambridge: Cambridge University Press. p. 350. doi:10.1017/CBO9780511676062. ISBN 978-0-511-67683-3. OCLC 630115876. 2. King, David A. (2001). The Ciphers of the Monks : a forgotten number-notation of the Middle Ages. Stuttgart: F. Steiner. pp. 16, 29, 34, 41. ISBN 3-515-07640-9. OCLC 48254993. 3. King, David (1993). "Rewriting history through instruments: The secrets of a medieval astrolabe from Picardy". In Anderson, R. G. W.; Bennett, J. A. & Ryan, W. F. (eds.). Making Instruments Count: Essays on Historical Scientific Instruments Presented to Gerard L'Estrange Turner. University of Michigan. ISBN 978-0860783947. 4. King, David A. (1992). "The Ciphers of the Monks and the Astrolabe of Berselius Reconsidered". In Demidov, Sergei S.; Rowe, David; Folkerts, Menso & Scriba, Christoph J. (eds.). Amphora. Basel: Birkhäuser. pp. 375–388. doi:10.1007/978-3-0348-8599-7_18. ISBN 978-3-0348-8599-7. 5. Agrippa von Nettesheim, Heinrich Cornelius (1533). "De notis Hebraeorum et Chaldaeorum". De Occulta Philosophia (in Latin). p. 141. 6. Meskens, Ad; Bonte, Germain; De Groot, Jacques; De Jonghe, Mieke & King, David A. (1999). "Wine-Gauging at Damme [The evidence of a late medieval manuscript]". Histoire & Mesure. 14 (1): 51–77. doi:10.3406/hism.1999.1501. 7. Beaujouan, Guy (1950). "Les soi-disant chiffres grecs ou chaldéens (XIIe – XVIe siècle)". Revue d'histoire des sciences (in French). 3 (2): 170–174. doi:10.3406/rhs.1950.2795. 8. Sesiano, Jacques (1985). "Un système artificiel de numérotation au Moyen Age". In Folkerts, Menso & Lindgren, Uta (eds.). Mathemata : Festschrift für Helmuth Gericke (in French). Stuttgart: F. Steiner Verlag. ISBN 3-515-04324-1. OCLC 12644728. 9. King (2001:243, 251) 10. De Laurence, Lauron William (1915). The Great Book of Magical Art, Hindu Magic and East Indian Occultism. Chicago: De Laurence Co. p. 174. 11. Beard, Daniel Carter (1918). The American boys' book of signs, signals and symbols. New York Public Library. Philadelphia : Lippincott. p. 92. 12. King, David (1995). "A forgotten Cistercian system of numerical notation". Citeaux Commentarii Cistercienses. 46 (3–4): 183–217. 13. King (2001:39) 14. "Character Encodings - Private Use Agreements - Under-ConScript Unicode Registry - Cistercian Numerals". www.kreativekorp.com. Retrieved 6 April 2021. 15. R.Ugalde, Laurence. "Cistercian numerals in Fōrmulæ programming language". Fōrmulæ. Retrieved July 29, 2021. 16. King (2001:427) 17. King (2001:156, 214) 18. King (2001:182–185) 19. King (2001:210) 20. Høyrup, Jens (2008). "Book review". Annals of Science. 65 (2): 306–308. 21. King, D.A. (2001). The Ciphers of the Monks: A Forgotten Number-notation of the Middle Ages. F. Steiner. ISBN 9783515076401. Retrieved 2015-08-13. 22. Moyer, Ann (2003). "Book review". Speculum. 78 (3): 919–921. doi:10.1017/S0038713400132002. JSTOR 20060835. Retrieved 2021-01-08. 23. Spalt, Detlef (2004). "Book review". Sudhoffs Archiv (in German). 88 (1): 108–109. JSTOR 20777934. Retrieved 2021-01-08. 24. Wedell, Moritz (2003). "Buchbesprechung". Zeitschrift für Germanistik (in German). 13 (3): 671–673. External links • Media related to Cistercian numerals at Wikimedia Commons • FRB Cistercian font (OTF) at GitHub. Uses the Private Use Area, since Unicode has declined to assign character codes. Font characters are segments, to be combined into the complete numerals. • Cistercian number generator at dCode. Uses digit shapes similar to the astrolabe (vertical stave, triangular 5). • L2/20-290 Background for Unicode consideration of Cistercian numerals • Cistercian Web Component for use on web pages. Includes a live updating Cistercian numeral clock.	Wikipedia
The Classical Groups The Classical Groups: Their Invariants and Representations is a mathematics book by Hermann Weyl (1939), which describes classical invariant theory in terms of representation theory. It is largely responsible for the revival of interest in invariant theory, which had been almost killed off by David Hilbert's solution of its main problems in the 1890s. In Weyl's wonderful and terrible1 book The Classical Groups [W] one may discern two main themes: first, the study of the polynomial invariants for an arbitrary number of (contravariant or covariant) variables for a standard classical group action; second, the isotypic decomposition of the full tensor algebra for such an action. 1Most people who know the book feel the material in it is wonderful. Many also feel the presentation is terrible. (The author is not among these latter.) Howe (1989, p.539) Weyl (1939a) gave an informal talk about the topic of his book. There was a second edition in 1946. Contents Chapter I defines invariants and other basic ideas and describes the relation to Felix Klein's Erlangen program in geometry. Chapter II describes the invariants of the special and general linear group of a vector space V on the polynomials over a sum of copies of V and its dual. It uses the Capelli identity to find an explicit set of generators for the invariants. Chapter III studies the group ring of a finite group and its decomposition into a sum of matrix algebras. Chapter IV discusses Schur–Weyl duality between representations of the symmetric and general linear groups. Chapters V and VI extend the discussion of invariants of the general linear group in chapter II to the orthogonal and symplectic groups, showing that the ring of invariants is generated by the obvious ones. Chapter VII describes the Weyl character formula for the characters of representations of the classical groups. Chapter VIII on invariant theory proves Hilbert's theorem that invariants of the special linear group are finitely generated. Chapter IX and X give some supplements to the previous chapters. References • Howe, Roger (1988), "The classical groups and invariants of binary forms", in Wells, R. O. Jr. (ed.), The mathematical heritage of Hermann Weyl (Durham, NC, 1987), Proc. Sympos. Pure Math., vol. 48, Providence, R.I.: American Mathematical Society, pp. 133–166, ISBN 978-0-8218-1482-6, MR 0974333 • Howe, Roger (1989), "Remarks on classical invariant theory.", Transactions of the American Mathematical Society, American Mathematical Society, 313 (2): 539–570, doi:10.2307/2001418, ISSN 0002-9947, JSTOR 2001418, MR 0986027 • Jacobson, Nathan (1940), "Book Review: The Classical Groups", Bulletin of the American Mathematical Society, 46 (7): 592–595, doi:10.1090/S0002-9904-1940-07236-2, ISSN 0002-9904, MR 1564136 • Weyl, Hermann (1939), The Classical Groups. Their Invariants and Representations, Princeton University Press, ISBN 978-0-691-05756-9, MR 0000255 • Weyl, Hermann (1939a), "Invariants", Duke Mathematical Journal, 5 (3): 489–502, doi:10.1215/S0012-7094-39-00540-5, ISSN 0012-7094, MR 0000030	Wikipedia
The Crest of the Peacock The Crest of the Peacock: Non-European Roots of Mathematics is a book authored by George Gheverghese Joseph, and was first published by Princeton University Press in 1991. The book was brought out as a response to view of the history of mathematics epitomized by Morris Kline's statement that, comparing to what the Greeks achieved, "the mathematics of Egyptians and Babylonians is the scrawling of children just learning to write, as opposed to great literature",[1] criticised by Joseph as "Eurocentric".[2] The third edition of the book was released in 2011.[3] The Crest of the Peacock: Non-European Roots of Mathematics AuthorGeorge Gheverghese Joseph LanguageEnglish SubjectHistory of mathematics PublisherPrinceton University Press Publication date 1991 Pages592 ISBN9780691135267 The book is divided into 11 chapters. Chapter 1 provides a lengthy justification for the book. Chapter 2 is devoted to a discussion of the mathematics of Native Americans and Chapter 3 to the mathematics of ancient Egyptians. The next two chapters consider the mathematics of Mesopotamia, then there are two chapters on Chinese mathematics, three chapters on Indian mathematics, and the final chapter discusses Islamic mathematics. Plagiarism C. K. Raju accused Joseph and Dennis Almerida of plagiarism [4] of his decade long scholastic work [5] that began in 1998 for the Project of History of Indian Science, Philosophy and Culture funded by the Indian Academy of Sciences concerning Indian mathematics and its possible knowledge transfer. An ethics investigation of the research team of George Gheverghese Joseph and Dennis Almeida led to the dismissal of Dennis Almeida by University of Exeter [6] and the University of Manchester posting an erratum and acknowledgement of C.K. Raju's work.[7] G. G. Joseph denies the charges.[8][9] Reviews • A review of the first edition of the book: Victor J Katz (August 1992). "Book Review: The crest of the peacock: Non-European roots of mathematics: By George Gheverghese Joseph" (PDF). Historia Mathematica. 19 (3): 310–315. doi:10.1016/0315-0860(92)90042-a. Retrieved 5 March 2016. • A review of the book by European Mathematical Information Service: Abdul Karim Bangura (Summer 2001). "Review of The Crest of the Peacock: Non-European Roots of Mathematics. 2nd. ed". Nexus Mathematical Journal. 3 (3). Retrieved 5 March 2016. • A review of the book by David Pingree: David Pingree (September 1993). "Reviewed Work: The Crest of the Peacock: Non-European Roots of Mathematics by George Gheverghese Joseph". Isis. 84 (3): 548–549. doi:10.1086/356553. JSTOR 235648. • For a critical assessment of some of the claims and arguments of the author: Clemency Montelle (December 2013). "Review of The crest of the peacock: Non-European roots of mathematics" (PDF). Notices of the AMS: 1459–1463. Retrieved 5 March 2016. References 1. Joseph (1991), p. 177 misquotes Kline by replacing "Babylonians" by "Mesopotamians". The actual quote from Kline is "Compared with the accomplishments of their immediate successors, the Greeks, the mathematics of the Egyptians and Babylonians is the scrawling of children just learning to write as opposed to great literature." Morris Kline (1962). Mathematics: A Cultural Approach. Addison-Wesley Pub. Co. p. 14., Mathematics for Liberal Arts (1967), re-published as Mathematics for the Nonmathematician (1985), p. 14. 2. Clemency Montelle (December 2013). "Book Review" (PDF). Notices of the AMS: 1459–1463. Retrieved 5 March 2016. 3. The Crest of the Peacock: Non-European Roots of Mathematics Third Edition. 24 October 2010. ISBN 9780691135267. Retrieved 5 March 2016. {{cite book}}: \|website= ignored (help) 4. http://ckraju.net/Joseph/Summary-table-of-plagiarism-by-Joseph-and-Almeida.pdf 5. C.K. Raju. (2007). Cultural Foundations of Mathematics. Pearson Longman. ISBN 978-81-317-0871-2 6. "Prof Raju's Charge of Plagiarism Found Correct" Times of India/ Hindustan Times (Nov 7, 2004) http://ckraju.net/press/HT8Nov2004p1.pdf 7. "Indians predated Newton 'discovery' by 250 years". 8. "Indian claims Brit study on calculus was plagiarised" Times of India/ Hindustan Times (Aug 20, 2007) 9. Hindustan Times Correction: http://ckraju.net/press/HT25Aug07.jpg (Aug 25, 2007)	Wikipedia
The Cube Made Interesting The Cube Made Interesting is a geometry book aimed at high school mathematics students, on the geometry of the cube. It was originally written in Polish by Aniela Ehrenfeucht (née Miklaszewska, 1905–2000), titled Ciekawy Sześcian [the interesting cube], and published by Polish Scientific Publishers PWN in 1960. Wacław Zawadowski translated it into English, and the translation was published in 1964 by the Pergamon Press and Macmillan Inc., in their "Popular Lectures in Mathematics" series.[1][2] Topics The book begins with Euler's polyhedral formula,[3] and includes material on the symmetries of their cube and their realization as geometric rotations,[4] and on the shape of plane sections through cubes.[4] It describes the 30 combinatorially distinct colorings of the six faces of the cube by six different colors, and discusses arrangements of colored cubes that match pairs of equal-colored faces.[1] The final chapter of the book concerns Prince Rupert's cube, the problem of fitting a cube through a hole drilled through a smaller cube without breaking the smaller cube into pieces.[5] An unusual feature of the book is its heavy illustration with red-blue anaglyphs;[1][2][3][4] provided with the book are red-blue glasses allowing readers to see these images as three-dimensional shapes.[5] Audience and reception This book is based on talks given by Ehrenfeucht to students and teachers,[2] and is aimed at a secondary-school audience.[3][4] Reviewer A. A. Kosinski writes that it "would contribute profitably to the development of the geometric imagination of a student", and Martyn Cundy writes that "the claim of the title is certainly justified". However, H. S. M. Coxeter notes that some of the terminology has become incorrect or nonstandard in the translation, suggesting that copyediting by someone more familiar with English mathematical terminology would have helped avoid these problems.[1] Cundy complains that the material on Prince Rupert's cube does not provide its optimal solution, and suggests that the color printing and inclusion of 3D viewing glasses made it unnecessarily expensive.[5] References 1. Coxeter, H. S. M., "Review of The Cube Made Interesting", Mathematical Reviews, MR 0170242 2. Gillespie, R. P. (December 1966), "Review of The Cube Made Interesting", Proceedings of the Edinburgh Mathematical Society, 15 (2): 164–164, doi:10.1017/s0013091500011652 3. Sekanina, M., "Review of Ciekawy Sześcian", Zentralblatt für Mathematik, Zbl 0093.33505 4. Kosinski, A. A., "Review of Ciekawy Sześcian", Mathematical Reviews, MR 0145385 5. Cundy, H. M. (December 1965), "Review of The Cube Made Interesting", The Mathematical Gazette, 49 (370): 452, doi:10.2307/3612197, JSTOR 3612197	Wikipedia
Cunningham Project The Cunningham Project is a collaborative effort started in 1925 to factor numbers of the form bn ± 1 for b = 2, 3, 5, 6, 7, 10, 11, 12 and large n. The project is named after Allan Joseph Champneys Cunningham, who published the first version of the table together with Herbert J. Woodall.[1] There are three printed versions of the table, the most recent published in 2002,[2] as well as an online version by Samuel Wagstaff.[3] The current limits of the exponents are: Base 2 3 5 6 7 10 11 12 Limit 1500 900 600 550 500 450 400 400 Aurifeuillean (LM) limit 3000 1800 1200 1100 1000 900 800 800 Factors of Cunningham number Two types of factors can be derived from a Cunningham number without having to use a factorization algorithm: algebraic factors of binomial numbers (e.g. difference of two squares and sum of two cubes), which depend on the exponent, and aurifeuillean factors, which depend on both the base and the exponent. Algebraic factors Main article: Binomial number § Factorization From elementary algebra, $(b^{kn}-1)=(b^{n}-1)\sum _{r=0}^{k-1}b^{rn}$ for all k, and $(b^{kn}+1)=(b^{n}+1)\sum _{r=0}^{k-1}(-1)^{r}\cdot b^{rn}$ for odd k. In addition, b2n − 1 = (bn − 1)(bn + 1). Thus, when m divides n, bm − 1 and bm + 1 are factors of bn − 1 if the quotient of n over m is even; only the first number is a factor if the quotient is odd. bm + 1 is a factor of bn − 1, if m divides n and the quotient is odd. In fact, $b^{n}-1=\prod _{d\mid n}\Phi _{d}(b)$ and $b^{n}+1=\prod _{d\mid 2n,\,d\nmid n}\Phi _{d}(b)$ See this page for more information. Aurifeuillean factors Main article: Aurifeuillean factorization When the number is of a particular form (the exact expression varies with the base), aurifeuillean factorization may be used, which gives a product of two or three numbers. The following equations give aurifeuillean factors for the Cunningham project bases as a product of F, L and M:[4] Let b = s2 × k with squarefree k, if one of the conditions holds, then $\Phi _{n}(b)$ have aurifeuillean factorization. (i) $k\equiv 1\mod 4$ and $n\equiv k{\pmod {2k}};$ (ii) $k\equiv 2,3{\pmod {4}}$ and $n\equiv 2k{\pmod {4k}}.$ b Number F L M Other definitions 2 24k+2 + 1 1 22k +1 − 2k +1 + 1 22k +1 + 2k +1 + 1 3 36k+3 + 1 32k +1 + 1 32k +1 − 3k +1 + 1 32k +1 + 3k +1 + 1 5 510k+5 − 1 52k +1 − 1 T 2 − 5k +1T + 52k +1 T 2 + 5k +1T + 52k +1 T = 52k +1 + 1 6 612k+6 + 1 64k+2 + 1 T 2 − 6k +1T + 62k +1 T 2 + 6k +1T + 62k +1 T = 62k +1 + 1 7 714k+7 + 1 72k +1 + 1 A − B A + B A = 76k+3 + 3(74k+2) + 3(72k +1) + 1 B = 75k+3 + 73k+2 + 7k +1 10 1020k +10 + 1 104k+2 + 1 A − B A + B A = 108k+4 + 5(106k+3) + 7(104k+2) + 5(102k +1) + 1 B = 107k+4 + 2(105k+3) + 2(103k+2) + 10k +1 11 1122k +11 + 1 112k +1 + 1 A − B A + B A = 1110k+5 + 5(118k+4) − 116k+3 − 114k+2 + 5(112k +1) + 1 B = 119k+5 + 117k+4 − 115k+3 + 113k+2 + 11k +1 12 126k+3 + 1 122k +1 + 1 122k +1 − 6(12k) + 1 122k +1 + 6(12k) + 1 Other factors Once the algebraic and aurifeuillean factors are removed, the other factors of bn ± 1 are always of the form 2kn + 1, since they are all factors of $\Phi _{n}(b)$. When n is prime, both algebraic and aurifeuillean factors are not possible, except the trivial factors (b − 1 for bn − 1 and b + 1 for bn + 1). For Mersenne numbers, the trivial factors are not possible for prime n, so all factors are of the form 2kn + 1. In general, all factors of (bn − 1) /(b − 1) are of the form 2kn + 1, where b ≥ 2 and n is prime, except when n divides b − 1, in which case (bn − 1)/(b − 1) is divisible by n itself. Cunningham numbers of the form bn − 1 can only be prime if b = 2 and n is prime, assuming that n ≥ 2; these are the Mersenne numbers. Numbers of the form bn + 1 can only be prime if b is even and n is a power of 2, again assuming n ≥ 2; these are the generalized Fermat numbers, which are Fermat numbers when b = 2. Any factor of a Fermat number 22n + 1 is of the form k2n+2 + 1. Notation bn − 1 is denoted as b,n−. Similarly, bn + 1 is denoted as b,n+. When dealing with numbers of the form required for aurifeuillean factorization, b,nL and b,nM are used to denote L and M in the products above.[5] References to b,n− and b,n+ are to the number with all algebraic and aurifeuillean factors removed. For example, Mersenne numbers are of the form 2,n− and Fermat numbers are of the form 2,2n+; the number Aurifeuille factored in 1871 was the product of 2,58L and 2,58M. See also • Cunningham number • ECMNET and NFS@Home, two collaborations working for the Cunningham project References 1. Cunningham, Allan J. C.; Woodall, H. J. (1925). Factorization of yn ± 1, y = 2, 3, 5, 6, 7, 10, 11, 12, up to high powers n. Hodgson. 2. Brillhart, John; Lehmer, Derrick H.; Selfridge, John L.; Tuckerman, Bryant; Wagstaff, Samuel S. (2002). Factorizations of bn ± 1, b = 2, 3, 5, 6, 7, 10, 11, 12 up to high powers. Contemporary Mathematics. Vol. 22. AMS. doi:10.1090/conm/022. ISBN 9780821850787. 3. "The Cunningham Project". Retrieved 21 July 2023. 4. "Main Cunningham Tables". Retrieved 21 July 2023. At the end of tables 2LM, 3+, 5-, 6+, 7+, 10+, 11+ and 12+ there are formulae detailing the aurifeuillean factorizations. 5. "Explanation of the notation on the Pages". Retrieved 21 July 2023. External links • Cunningham project homepage • Factorizations of bn±1, b = 2, 3, 5, 6, 7, 10, 11, 12 Up to High Powers, second edition • Factorizations of bn±1, b = 2, 3, 5, 6, 7, 10, 11, 12 Up to High Powers, third edition • Machine-readable Cunningham tables • The Cunningham Project • Brent-Montgomery-te Riele table (Cunningham tables for higher bases (bases 13 ≤ b ≤ 99, perfect powers excluded, since a power of bn is also a power of b)) • Online factor collection • Cunningham project on Prime Wiki • Cunningham project on PrimePages	Wikipedia
Decline Decline may refer to: • Decadence, involves a perceived decay in standards, morals, dignity, religious faith, or skill over time • "Decline" (song), 2017 song by Raye and Mr Eazi • The Decline (EP), an EP by NOFX • The Decline (band), Australian skate punk band from Perth • The Decline (film), a 2020 Canadian thriller drama film Look up decline in Wiktionary, the free dictionary. See also • Declination (disambiguation) • Declinism • Decline and Fall (disambiguation) • Decline of the Roman Empire • Decline of Detroit • Ottoman decline thesis • The Decline of the West by Oswald Spengler • Social disintegration, • Societal collapse • Withering away of the state	Wikipedia
Euclid's Elements The Elements (Ancient Greek: Στοιχεῖα Stoikheîa) is a mathematical treatise consisting of 13 books attributed to the ancient Greek mathematician Euclid in Alexandria, Ptolemaic Egypt c. 300 BC. It is a collection of definitions, postulates, propositions (theorems and constructions), and mathematical proofs of the propositions. The books cover plane and solid Euclidean geometry, elementary number theory, and incommensurable lines. Elements is the oldest extant large-scale deductive treatment of mathematics. It has proven instrumental in the development of logic and modern science, and its logical rigor was not surpassed until the 19th century. Elements Title page of Sir Henry Billingsley's first English version of Euclid's Elements, 1570. Billingsley erroneously attributed the original work to Euclid of Megara. AuthorEuclid LanguageAncient Greek SubjectEuclidean geometry, elementary number theory, incommensurable lines GenreMathematics Publication date c. 300 BC Pages13 books Euclid's Elements has been referred to as the most successful[lower-alpha 1][lower-alpha 2] and influential[lower-alpha 3] textbook ever written. It was one of the very earliest mathematical works to be printed after the invention of the printing press and has been estimated to be second only to the Bible in the number of editions published since the first printing in 1482,[1] the number reaching well over one thousand.[lower-alpha 4] For centuries, when the quadrivium was included in the curriculum of all university students, knowledge of at least part of Euclid's Elements was required of all students. Not until the 20th century, by which time its content was universally taught through other school textbooks, did it cease to be considered something all educated people had read. History Basis in earlier work Scholars believe that the Elements is largely a compilation of propositions based on books by earlier Greek mathematicians.[3] Proclus (412–485 AD), a Greek mathematician who lived around seven centuries after Euclid, wrote in his commentary on the Elements: "Euclid, who put together the Elements, collecting many of Eudoxus' theorems, perfecting many of Theaetetus', and also bringing to irrefragable demonstration the things which were only somewhat loosely proved by his predecessors". Pythagoras (c. 570–495 BC) was probably the source for most of books I and II, Hippocrates of Chios (c. 470–410 BC, not the better known Hippocrates of Kos) for book III, and Eudoxus of Cnidus (c. 408–355 BC) for book V, while books IV, VI, XI, and XII probably came from other Pythagorean or Athenian mathematicians.[4] The Elements may have been based on an earlier textbook by Hippocrates of Chios, who also may have originated the use of letters to refer to figures.[5] Other similar works are also reported to have been written by Theudius of Magnesia, Leon, and Hermotimus of Colophon.[6][7] Transmission of the text In the 4th century AD, Theon of Alexandria produced an edition of Euclid which was so widely used that it became the only surviving source until François Peyrard's 1808 discovery at the Vatican of a manuscript not derived from Theon's. This manuscript, the Heiberg manuscript, is from a Byzantine workshop around 900 and is the basis of modern editions.[8] Papyrus Oxyrhynchus 29 is a tiny fragment of an even older manuscript, but only contains the statement of one proposition. Although Euclid was known to Cicero, for instance, no record exists of the text having been translated into Latin prior to Boethius in the fifth or sixth century.[2] The Arabs received the Elements from the Byzantines around 760; this version was translated into Arabic under Harun al Rashid (c. 800).[2] The Byzantine scholar Arethas commissioned the copying of one of the extant Greek manuscripts of Euclid in the late ninth century.[9] Although known in Byzantium, the Elements was lost to Western Europe until about 1120, when the English monk Adelard of Bath translated it into Latin from an Arabic translation.[lower-alpha 5] A relatively recent discovery was made of a Greek-to-Latin translation from the 12th century at Palermo, Sicily. The name of the translator is not known other than he was an anonymous medical student from Salerno who was visiting Palermo in order to translate the Almagest to Latin. The Euclid manuscript is extant and quite complete.[11] The first printed edition appeared in 1482 (based on Campanus of Novara's 1260 edition),[12] and since then it has been translated into many languages and published in about a thousand different editions. Theon's Greek edition was recovered in 1533. In 1570, John Dee provided a widely respected "Mathematical Preface", along with copious notes and supplementary material, to the first English edition by Henry Billingsley. Copies of the Greek text still exist, some of which can be found in the Vatican Library and the Bodleian Library in Oxford. The manuscripts available are of variable quality, and invariably incomplete. By careful analysis of the translations and originals, hypotheses have been made about the contents of the original text (copies of which are no longer available). Ancient texts which refer to the Elements itself, and to other mathematical theories that were current at the time it was written, are also important in this process. Such analyses are conducted by J. L. Heiberg and Sir Thomas Little Heath in their editions of the text. Also of importance are the scholia, or annotations to the text. These additions, which often distinguished themselves from the main text (depending on the manuscript), gradually accumulated over time as opinions varied upon what was worthy of explanation or further study. Influence The Elements is still considered a masterpiece in the application of logic to mathematics. In historical context, it has proven enormously influential in many areas of science. Scientists Nicolaus Copernicus, Johannes Kepler, Galileo Galilei, Albert Einstein and Sir Isaac Newton were all influenced by the Elements, and applied their knowledge of it to their work.[13][14] Mathematicians and philosophers, such as Thomas Hobbes, Baruch Spinoza, Alfred North Whitehead, and Bertrand Russell, have attempted to create their own foundational "Elements" for their respective disciplines, by adopting the axiomatized deductive structures that Euclid's work introduced. The austere beauty of Euclidean geometry has been seen by many in western culture as a glimpse of an otherworldly system of perfection and certainty. Abraham Lincoln kept a copy of Euclid in his saddlebag, and studied it late at night by lamplight; he related that he said to himself, "You never can make a lawyer if you do not understand what demonstrate means; and I left my situation in Springfield, went home to my father's house, and stayed there till I could give any proposition in the six books of Euclid at sight".[15][16] Edna St. Vincent Millay wrote in her sonnet "Euclid alone has looked on Beauty bare", "O blinding hour, O holy, terrible day, When first the shaft into his vision shone Of light anatomized!". Albert Einstein recalled a copy of the Elements and a magnetic compass as two gifts that had a great influence on him as a boy, referring to the Euclid as the "holy little geometry book".[17][18] The success of the Elements is due primarily to its logical presentation of most of the mathematical knowledge available to Euclid. Much of the material is not original to him, although many of the proofs are his. However, Euclid's systematic development of his subject, from a small set of axioms to deep results, and the consistency of his approach throughout the Elements, encouraged its use as a textbook for about 2,000 years. The Elements still influences modern geometry books. Furthermore, its logical, axiomatic approach and rigorous proofs remain the cornerstone of mathematics. In modern mathematics One of the most notable influences of Euclid on modern mathematics is the discussion of the parallel postulate. In Book I, Euclid lists five postulates, the fifth of which stipulates If a line segment intersects two straight lines forming two interior angles on the same side that sum to less than two right angles, then the two lines, if extended indefinitely, meet on that side on which the angles sum to less than two right angles. This postulate plagued mathematicians for centuries due to its apparent complexity compared with the other four postulates. Many attempts were made to prove the fifth postulate based on the other four, but they never succeeded. Eventually in 1829, mathematician Nikolai Lobachevsky published a description of acute geometry (or hyperbolic geometry), a geometry which assumed a different form of the parallel postulate. It is in fact possible to create a valid geometry without the fifth postulate entirely, or with different versions of the fifth postulate (elliptic geometry). If one takes the fifth postulate as a given, the result is Euclidean geometry. Contents • Book 1 contains 5 postulates (including the infamous parallel postulate) and 5 common notions, and covers important topics of plane geometry such as the Pythagorean theorem, equality of angles and areas, parallelism, the sum of the angles in a triangle, and the construction of various geometric figures. • Book 2 contains a number of lemmas concerning the equality of rectangles and squares, sometimes referred to as "geometric algebra", and concludes with a construction of the golden ratio and a way of constructing a square equal in area to any rectilineal plane figure. • Book 3 deals with circles and their properties: finding the center, inscribed angles, tangents, the power of a point, Thales' theorem. • Book 4 constructs the incircle and circumcircle of a triangle, as well as regular polygons with 4, 5, 6, and 15 sides. • Book 5, on proportions of magnitudes, gives the highly sophisticated theory of proportion probably developed by Eudoxus, and proves properties such as "alternation" (if a : b :: c : d, then a : c :: b : d). • Book 6 applies proportions to plane geometry, especially the construction and recognition of similar figures. • Book 7 deals with elementary number theory: divisibility, prime numbers and their relation to composite numbers, Euclid's algorithm for finding the greatest common divisor, finding the least common multiple. • Book 8 deals with the construction and existence of geometric sequences of integers. • Book 9 applies the results of the preceding two books and gives the infinitude of prime numbers and the construction of all even perfect numbers. • Book 10 proves the irrationality of the square roots of non-square integers (e.g. ${\sqrt {2}}$) and classifies the square roots of incommensurable lines into thirteen disjoint categories. Euclid here introduces the term "irrational", which has a different meaning than the modern concept of irrational numbers. He also gives a formula to produce Pythagorean triples.[19] • Book 11 generalizes the results of book 6 to solid figures: perpendicularity, parallelism, volumes and similarity of parallelepipeds. • Book 12 studies the volumes of cones, pyramids, and cylinders in detail by using the method of exhaustion, a precursor to integration, and shows, for example, that the volume of a cone is a third of the volume of the corresponding cylinder. It concludes by showing that the volume of a sphere is proportional to the cube of its radius (in modern language) by approximating its volume by a union of many pyramids. • Book 13 constructs the five regular Platonic solids inscribed in a sphere and compares the ratios of their edges to the radius of the sphere. Summary Contents of Euclid's Elements Book I II III IV V VI VII VIII IX X XI XII XIII Totals Definitions 23211718422--1628--131 Postulates 5------------5 Common Notions 5------------5 Propositions 481437162533392736115391818465 Euclid's method and style of presentation • "To draw a straight line from any point to any point." • "To describe a circle with any center and distance." Euclid, Elements, Book I, Postulates 1 & 3.[20] Euclid's axiomatic approach and constructive methods were widely influential. Many of Euclid's propositions were constructive, demonstrating the existence of some figure by detailing the steps he used to construct the object using a compass and straightedge. His constructive approach appears even in his geometry's postulates, as the first and third postulates stating the existence of a line and circle are constructive. Instead of stating that lines and circles exist per his prior definitions, he states that it is possible to 'construct' a line and circle. It also appears that, for him to use a figure in one of his proofs, he needs to construct it in an earlier proposition. For example, he proves the Pythagorean theorem by first inscribing a square on the sides of a right triangle, but only after constructing a square on a given line one proposition earlier.[21] As was common in ancient mathematical texts, when a proposition needed proof in several different cases, Euclid often proved only one of them (often the most difficult), leaving the others to the reader. Later editors such as Theon often interpolated their own proofs of these cases. Euclid's presentation was limited by the mathematical ideas and notations in common currency in his era, and this causes the treatment to seem awkward to the modern reader in some places. For example, there was no notion of an angle greater than two right angles,[22] the number 1 was sometimes treated separately from other positive integers, and as multiplication was treated geometrically he did not use the product of more than 3 different numbers. The geometrical treatment of number theory may have been because the alternative would have been the extremely awkward Alexandrian system of numerals.[23] The presentation of each result is given in a stylized form, which, although not invented by Euclid, is recognized as typically classical. It has six different parts: First is the 'enunciation', which states the result in general terms (i.e., the statement of the proposition). Then comes the 'setting-out', which gives the figure and denotes particular geometrical objects by letters. Next comes the 'definition' or 'specification', which restates the enunciation in terms of the particular figure. Then the 'construction' or 'machinery' follows. Here, the original figure is extended to forward the proof. Then, the 'proof' itself follows. Finally, the 'conclusion' connects the proof to the enunciation by stating the specific conclusions drawn in the proof, in the general terms of the enunciation.[24] No indication is given of the method of reasoning that led to the result, although the Data does provide instruction about how to approach the types of problems encountered in the first four books of the Elements.[4] Some scholars have tried to find fault in Euclid's use of figures in his proofs, accusing him of writing proofs that depended on the specific figures drawn rather than the general underlying logic, especially concerning Proposition II of Book I. However, Euclid's original proof of this proposition, is general, valid, and does not depend on the figure used as an example to illustrate one given configuration.[25] Criticism Euclid's list of axioms in the Elements was not exhaustive, but represented the principles that were the most important. His proofs often invoke axiomatic notions which were not originally presented in his list of axioms. Later editors have interpolated Euclid's implicit axiomatic assumptions in the list of formal axioms.[26] For example, in the first construction of Book 1, Euclid used a premise that was neither postulated nor proved: that two circles with centers at the distance of their radius will intersect in two points.[27] Later, in the fourth construction, he used superposition (moving the triangles on top of each other) to prove that if two sides and their angles are equal, then they are congruent; during these considerations he uses some properties of superposition, but these properties are not described explicitly in the treatise. If superposition is to be considered a valid method of geometric proof, all of geometry would be full of such proofs. For example, propositions I.2 and I.3 can be proved trivially by using superposition.[28] Mathematician and historian W. W. Rouse Ball put the criticisms in perspective, remarking that "the fact that for two thousand years [the Elements] was the usual text-book on the subject raises a strong presumption that it is not unsuitable for that purpose."[22] Apocrypha It was not uncommon in ancient times to attribute to celebrated authors works that were not written by them. It is by these means that the apocryphal books XIV and XV of the Elements were sometimes included in the collection.[29] The spurious Book XIV was probably written by Hypsicles on the basis of a treatise by Apollonius. The book continues Euclid's comparison of regular solids inscribed in spheres, with the chief result being that the ratio of the surfaces of the dodecahedron and icosahedron inscribed in the same sphere is the same as the ratio of their volumes, the ratio being ${\sqrt {\frac {10}{3(5-{\sqrt {5}})}}}={\sqrt {\frac {5+{\sqrt {5}}}{6}}}.$ The spurious Book XV was probably written, at least in part, by Isidore of Miletus. This book covers topics such as counting the number of edges and solid angles in the regular solids, and finding the measure of dihedral angles of faces that meet at an edge.[lower-alpha 6] Editions • 1460s, Regiomontanus (incomplete) • 1482, Erhard Ratdolt (Venice), editio princeps (in Latin)[30][31] • 1533, editio princeps of the Greek text by Simon Grynäus[32] • 1557, by Jean Magnien and Pierre de Montdoré, reviewed by Stephanus Gracilis (only propositions, no full proofs, includes original Greek and the Latin translation) • 1572, Commandinus Latin edition • 1574, Christoph Clavius Translations • 1505, Bartolomeo Zamberti (Latin) • 1543, Nicolo Tartaglia (Italian) • 1557, Jean Magnien and Pierre de Montdoré, reviewed by Stephanus Gracilis (Greek to Latin) • 1558, Johann Scheubel (German) • 1562, Jacob Kündig (German) • 1562, Wilhelm Holtzmann (German) • 1564–1566, Pierre Forcadel de Béziers (French) • 1570, Henry Billingsley (English) • 1572, Commandinus (Latin) • 1575, Commandinus (Italian) • 1576, Rodrigo de Zamorano (Spanish) • 1594, Typographia Medicea (edition of the Arabic translation of The Recension of Euclid's "Elements"[33] • 1604, Jean Errard de Bar-le-Duc (French) • 1606, Jan Pieterszoon Dou (Dutch) • 1607, Matteo Ricci, Xu Guangqi (Chinese) • 1613, Pietro Cataldi (Italian) • 1615, Denis Henrion (French) • 1617, Frans van Schooten (Dutch) • 1637, L. Carduchi (Spanish) • 1639, Pierre Hérigone (French) • 1651, Heinrich Hoffmann (German) • 1651, Thomas Rudd (English) • 1660, Isaac Barrow (English) • 1661, John Leeke and Geo. Serle (English) • 1663, Domenico Magni (Italian from Latin) • 1672, Claude François Milliet Dechales (French) • 1680, Vitale Giordano (Italian) • 1685, William Halifax (English) • 1689, Jacob Knesa (Spanish) • 1690, Vincenzo Viviani (Italian) • 1694, Ant. Ernst Burkh v. Pirckenstein (German) • 1695, Claes Jansz Vooght (Dutch) • 1697, Samuel Reyher (German) • 1702, Hendrik Coets (Dutch) • 1705, Charles Scarborough (English) • 1708, John Keill (English) • 1714, Chr. Schessler (German) • 1714, W. Whiston (English) • 1720s, Jagannatha Samrat (Sanskrit, based on the Arabic translation of Nasir al-Din al-Tusi)[34] • 1731, Guido Grandi (abbreviation to Italian) • 1738, Ivan Satarov (Russian from French) • 1744, Mårten Strömer (Swedish) • 1749, Dechales (Italian) • 1749, Methodios Anthrakitis (Μεθόδιος Ανθρακίτης) (Greek) • 1745, Ernest Gottlieb Ziegenbalg (Danish) • 1752, Leonardo Ximenes (Italian) • 1756, Robert Simson (English) • 1763, Pibo Steenstra (Dutch) • 1768, Angelo Brunelli (Portuguese) • 1773, 1781, J. F. Lorenz (German) • 1780, Baruch Schick of Shklov (Hebrew)[35] • 1781, 1788 James Williamson (English) • 1781, William Austin (English) • 1789, Pr. Suvoroff nad Yos. Nikitin (Russian from Greek) • 1795, John Playfair (English) • 1803, H.C. Linderup (Danish) • 1804, François Peyrard (French). Peyrard discovered in 1808 the Vaticanus Graecus 190, which enables him to provide a first definitive version in 1814–1818 • 1807, Józef Czech (Polish based on Greek, Latin and English editions) • 1807, J. K. F. Hauff (German) • 1818, Vincenzo Flauti (Italian) • 1820, Benjamin of Lesbos (Modern Greek) • 1826, George Phillips (English) • 1828, Joh. Josh and Ign. Hoffmann (German) • 1828, Dionysius Lardner (English) • 1833, E. S. Unger (German) • 1833, Thomas Perronet Thompson (English) • 1836, H. Falk (Swedish) • 1844, 1845, 1859, P. R. Bråkenhjelm (Swedish) • 1850, F. A. A. Lundgren (Swedish) • 1850, H. A. Witt and M. E. Areskong (Swedish) • 1862, Isaac Todhunter (English) • 1865, Sámuel Brassai (Hungarian) • 1873, Masakuni Yamada (Japanese) • 1880, Vachtchenko-Zakhartchenko (Russian) • 1897, Thyra Eibe (Danish) • 1901, Max Simon (German) • 1907, František Servít (Czech)[36] • 1908, Thomas Little Heath (English) • 1939, R. Catesby Taliaferro (English) • 1953, 1958, 1975, Evangelos Stamatis (Ευάγγελος Σταµάτης) (Modern Greek) • 1999, Maja Hudoletnjak Grgić (Book I-VI) (Croatian)[37] • 2009, Irineu Bicudo (Portuguese) • 2019, Ali Sinan Sertöz (Turkish)[38] • 2022, Ján Čižmár (Slovak) Currently in print • Euclid's Elements – All thirteen books complete in one volume, Based on Heath's translation, Green Lion Press ISBN 1-888009-18-7. • The Elements: Books I–XIII – Complete and Unabridged, (2006) Translated by Sir Thomas Heath, Barnes & Noble ISBN 0-7607-6312-7. • The Thirteen Books of Euclid's Elements, translation and commentaries by Heath, Thomas L. (1956) in three volumes. Dover Publications. ISBN 0-486-60088-2 (vol. 1), ISBN 0-486-60089-0 (vol. 2), ISBN 0-486-60090-4 (vol. 3) Free versions • Euclid's Elements Redux, Volume 1, contains books I–III, based on John Casey's translation.[39] • Euclid's Elements Redux, Volume 2, contains books IV–VIII, based on John Casey's translation.[39] References Notes 1. Wilson 2006, p. 278 states, "Euclid's Elements subsequently became the basis of all mathematical education, not only in the Roman and Byzantine periods, but right down to the mid-20th century, and it could be argued that it is the most successful textbook ever written." 2. Boyer 1991, p. 100 notes, "As teachers at the school he called a band of leading scholars, among whom was the author of the most fabulously successful mathematics textbook ever written – the Elements (Stoichia) of Euclid". 3. Boyer 1991, p. 119 notes, "The Elements of Euclid not only was the earliest major Greek mathematical work to come down to us, but also the most influential textbook of all times. [...]The first printed versions of the Elements appeared at Venice in 1482, one of the very earliest of mathematical books to be set in type; it has been estimated that since then at least a thousand editions have been published. Perhaps no book other than the Bible can boast so many editions, and certainly no mathematical work has had an influence comparable with that of Euclid's Elements". 4. Bunt, Jones & Bedient 1988, p. 142 state, "the Elements became known to Western Europe via the Arabs and the Moors. There, the Elements became the foundation of mathematical education. More than 1000 editions of the Elements are known. In all probability, it is, next to the Bible, the most widely spread book in the civilization of the Western world." 5. One older work claims Adelard disguised himself as a Muslim student to obtain a copy in Muslim Córdoba.[10] However, more recent biographical work has turned up no clear documentation that Adelard ever went to Muslim-ruled Spain, although he spent time in Norman-ruled Sicily and Crusader-ruled Antioch, both of which had Arabic-speaking populations. Charles Burnett, Adelard of Bath: Conversations with his Nephew (Cambridge, 1999); Charles Burnett, Adelard of Bath (University of London, 1987). 6. Boyer 1991, pp. 118–119 writes, "In ancient times it was not uncommon to attribute to a celebrated author works that were not by him; thus, some versions of Euclid's Elements include a fourteenth and even a fifteenth book, both shown by later scholars to be apocryphal. The so-called Book XIV continues Euclid's comparison of the regular solids inscribed in a sphere, the chief results being that the ratio of the surfaces of the dodecahedron and icosahedron inscribed in the same sphere is the same as the ratio of their volumes, the ratio being that of the edge of the cube to the edge of the icosahedron, that is, ${\sqrt {10/[3(5-{\sqrt {5}})]}}$. It is thought that this book may have been composed by Hypsicles on the basis of a treatise (now lost) by Apollonius comparing the dodecahedron and icosahedron. [...] The spurious Book XV, which is inferior, is thought to have been (at least in part) the work of Isidore of Miletus (fl. ca. A.D. 532), architect of the cathedral of Holy Wisdom (Hagia Sophia) at Constantinople. This book also deals with the regular solids, counting the number of edges and solid angles in the solids, and finding the measures of the dihedral angles of faces meeting at an edge. Citations 1. Boyer 1991, p. 100. 2. Russell 2013, p. 177. 3. Waerden 1975, p. 197. 4. Ball 1915, p. 54. 5. Ball 1915, p. 38. 6. Unguru, S. (1985). Digging for Structure into the Elements: Euclid, Hilbert, and Mueller. Historia Mathematica 12, 176 7. Zhmud, L. (1998). Plato as "Architect of Science". Phonesis 43, 211 8. The Earliest Surviving Manuscript Closest to Euclid's Original Text (Circa 850); an image Archived 2009-12-20 at the Wayback Machine of one page 9. Reynolds & Wilson 1991, p. 57. 10. Ball 1915, p. 165. 11. Murdoch, John E. (1967). "Euclides Graeco-Latinus: A Hitherto Unknown Medieval Latin Translation of the Elements Made Directly from the Greek". Harvard Studies in Classical Philology. 71: 249–302. doi:10.2307/310767. JSTOR 310767. 12. Busard 2005, p. 1. 13. Andrew., Liptak (2 September 2017). "One of the world's most influential math texts is getting a beautiful, minimalist edition". The Verge. 14. Grabiner., Judith. "How Euclid once ruled the world". Plus Magazine. 15. Ketcham 1901. 16. Euclid as Founding Father 17. Herschbach, Dudley. "Einstein as a Student" (PDF). Department of Chemistry and Chemical Biology, Harvard University, Cambridge, MA. p. 3. Archived from the original (PDF) on 2009-02-26.: about Max Talmud visited on Thursdays for six years. 18. Prindle, Joseph. "Albert Einstein - Young Einstein". www.alberteinsteinsite.com. Archived from the original on 10 June 2017. Retrieved 29 April 2018. 19. Joyce, D. E. (June 1997), "Book X, Proposition XXIX", Euclid's Elements, Clark University 20. Hartshorne 2000, p. 18. 21. Hartshorne 2000, pp. 18–20. 22. Ball 1915, p. 55. 23. Ball 1915, pp. 54, 58, 127. 24. Heath 1963, p. 216. 25. Toussaint 1993, pp. 12–23. 26. Heath 1956a, p. 62. 27. Heath 1956a, p. 242. 28. Heath 1956a, p. 249. 29. Boyer 1991, pp. 118–119. 30. Alexanderson & Greenwalt 2012, p. 163 31. "Editio Princeps of Euclid's Elements, the Most Famous Textbook Ever Published : History of Information". www.historyofinformation.com. Retrieved 2023-07-28. 32. "The First Printed Edition of the Greek Text of Euclid is also the First Edition to Include the Diagrams within the Text : History of Information". historyofinformation.com. Retrieved 2023-07-28. 33. Nasir al-Din al-Tusi 1594. 34. Sarma 1997, pp. 460–461. 35. "JNUL Digitized Book Repository". huji.ac.il. 22 June 2009. Archived from the original on 22 June 2009. Retrieved 29 April 2018. 36. Servít 1907. 37. Euklid 1999. 38. Sertöz 2019. 39. Callahan & Casey 2015. Sources • Alexanderson, Gerald L.; Greenwalt, William S. (2012), "About the cover: Billingsley's Euclid in English", Bulletin of the American Mathematical Society, New Series, 49 (1): 163–167, doi:10.1090/S0273-0979-2011-01365-9 • Artmann, Benno: Euclid – The Creation of Mathematics. New York, Berlin, Heidelberg: Springer 1999, ISBN 0-387-98423-2 • Ball, Walter William Rouse (1915) [1st ed. 1888]. A Short Account of the History of Mathematics (6th ed.). MacMillan. • Boyer, Carl B. (1991). "Euclid of Alexandria". A History of Mathematics (Second ed.). John Wiley & Sons. ISBN 0-471-54397-7. • Bunt, Lucas Nicolaas Hendrik; Jones, Phillip S.; Bedient, Jack D. (1988). The Historical Roots of Elementary Mathematics. Dover. • Busard, H.L.L. (2005). "Introduction to the Text". Campanus of Novara and Euclid's Elements. Stuttgart: Franz Steiner Verlag. ISBN 978-3-515-08645-5. • Callahan, Daniel; Casey, John (2015). Euclid's "Elements" Redux. • Dodgson, Charles L.; Hagar, Amit (2009). "Introduction". Euclid and His Modern Rivals. Cambridge University Press. ISBN 978-1-108-00100-7. • Hartshorne, Robin (2000). Geometry: Euclid and Beyond (2nd ed.). New York, NY: Springer. ISBN 9780387986500. • Heath, Thomas L. (1956a). The Thirteen Books of Euclid's Elements. Vol. 1. Books I and II (2nd ed.). New York: Dover Publications. OL 22193354M. • Heath, Thomas L. (1956b). The Thirteen Books of Euclid's Elements. Vol. 2. Books III to IX (2nd ed.). New York: Dover Publications. OL 7650092M. • Heath, Thomas L. (1956c). The Thirteen Books of Euclid's Elements. Vol. 3. Books X to XIII and Appendix (2nd ed.). New York: Dover Publications. OCLC 929205858. Heath's authoritative translation plus extensive historical research and detailed commentary throughout the text. • Heath, Thomas L. (1963). A Manual of Greek Mathematics. Dover Publications. ISBN 978-0-486-43231-1. • Ketcham, Henry (1901). The Life of Abraham Lincoln. New York: Perkins Book Company. • Nasir al-Din al-Tusi (1594). Kitāb taḥrīr uṣūl li-Uqlīdus [The Recension of Euclid's "Elements"] (in Arabic). • Reynolds, Leighton Durham; Wilson, Nigel Guy (9 May 1991). Scribes and scholars: a guide to the transmission of Greek and Latin literature (2nd ed.). Oxford: Clarendon Press. ISBN 978-0-19-872145-1. • Russell, Bertrand (2013). History of Western Philosophy: Collectors Edition. Routledge. ISBN 978-1-135-69284-1. • Sarma, K.V. (1997). Selin, Helaine (ed.). Encyclopaedia of the history of science, technology, and medicine in non-western cultures. Springer. ISBN 978-0-7923-4066-9. • Servít, František (1907). Eukleidovy Zaklady (Elementa) [Euclid's Elements] (PDF) (in Czech). • Sertöz, Ali Sinan (2019). Öklidin Elemanlari: Ciltli [Euclid's Elements] (in Turkish). Tübitak. ISBN 978-605-312-329-3. • Toussaint, Godfried (1993). "A new look at euclid's second proposition". The Mathematical Intelligencer. 15 (3): 12–24. doi:10.1007/BF03024252. ISSN 0343-6993. S2CID 26811463. • Waerden, Bartel Leendert (1975). Science awakening. Noordhoff International. ISBN 978-90-01-93102-5. • Wilson, Nigel Guy (2006). Encyclopedia of Ancient Greece. Routledge. • Euklid (1999). Elementi I-VI. Translated by Hudoletnjak Grgić, Maja. KruZak. ISBN 953-96477-6-2. External links Wikiquote has quotations related to Euclid's Elements. Wikisource has original text related to this article: The Elements of Euclid Wikimedia Commons has media related to Elements of Euclid. • Clark University Euclid's elements • Multilingual edition of Elementa in the Bibliotheca Polyglotta • Euclid (1997) [c. 300 BC]. David E. Joyce (ed.). "Elements". Retrieved 2006-08-30. In HTML with Java-based interactive figures. • Richard Fitzpatrick's bilingual edition (freely downloadable PDF, typeset in a two-column format with the original Greek beside a modern English translation; also available in print as ISBN 979-8589564587) • Heath's English translation (HTML, without the figures, public domain) (accessed February 4, 2010) • Heath's English translation and commentary, with the figures (Google Books): vol. 1, vol. 2, vol. 3, vol. 3 c. 2 • Oliver Byrne's 1847 edition (also hosted at archive.org)– an unusual version by Oliver Byrne who used color rather than labels such as ABC (scanned page images, public domain) • Web adapted version of Byrne’s Euclid designed by Nicholas Rougeux • Video adaptation, animated and explained by Sandy Bultena, contains books I-VII. • The First Six Books of the Elements by John Casey and Euclid scanned by Project Gutenberg. • Reading Euclid – a course in how to read Euclid in the original Greek, with English translations and commentaries (HTML with figures) • Sir Thomas More's manuscript • Latin translation by Aethelhard of Bath • Euclid Elements – The original Greek text Greek HTML • Clay Mathematics Institute Historical Archive – The thirteen books of Euclid's Elements copied by Stephen the Clerk for Arethas of Patras, in Constantinople in 888 AD • Kitāb Taḥrīr uṣūl li-Ūqlīdis Arabic translation of the thirteen books of Euclid's Elements by Nasīr al-Dīn al-Ṭūsī. Published by Medici Oriental Press(also, Typographia Medicea). Facsimile hosted by Islamic Heritage Project. • Euclid's Elements Redux, an open textbook based on the Elements • 1607 Chinese translations reprinted as part of Siku Quanshu, or "Complete Library of the Four Treasuries." Euclid Works • Data • Elements • Optics Topics • Namesakes • Euclidean geometry • Euclidean algorithm • Euclid's theorem • Euclidean relation Scholars • Thomas Heath • Robert Simson • Isaac Todhunter Related • Papyrus Oxyrhynchus 29 • Category Ancient Greek mathematics Mathematicians (timeline) • Anaxagoras • Anthemius • Archytas • Aristaeus the Elder • Aristarchus • Aristotle • Apollonius • Archimedes • Autolycus • Bion • Bryson • Callippus • Carpus • Chrysippus • Cleomedes • Conon • Ctesibius • Democritus • Dicaearchus • Diocles • Diophantus • Dinostratus • Dionysodorus • Domninus • Eratosthenes • Eudemus • Euclid • Eudoxus • Eutocius • Geminus • Heliodorus • Heron • Hipparchus • Hippasus • Hippias • Hippocrates • Hypatia • Hypsicles • Isidore of Miletus • Leon • Marinus • Menaechmus • Menelaus • Metrodorus • Nicomachus • Nicomedes • Nicoteles • Oenopides • Pappus • Perseus • Philolaus • Philon • Philonides • Plato • Porphyry • Posidonius • Proclus • Ptolemy • Pythagoras • Serenus • Simplicius • Sosigenes • Sporus • Thales • Theaetetus • Theano • Theodorus • Theodosius • Theon of Alexandria • Theon of Smyrna • Thymaridas • Xenocrates • Zeno of Elea • Zeno of Sidon • Zenodorus Treatises • Almagest • Archimedes Palimpsest • Arithmetica • Conics (Apollonius) • Catoptrics • Data (Euclid) • Elements (Euclid) • Measurement of a Circle • On Conoids and Spheroids • On the Sizes and Distances (Aristarchus) • On Sizes and Distances (Hipparchus) • On the Moving Sphere (Autolycus) • Optics (Euclid) • On Spirals • On the Sphere and Cylinder • Ostomachion • Planisphaerium • Sphaerics • The Quadrature of the Parabola • The Sand Reckoner Problems • Constructible numbers • Angle trisection • Doubling the cube • Squaring the circle • Problem of Apollonius Concepts and definitions • Angle • Central • Inscribed • Axiomatic system • Axiom • Chord • Circles of Apollonius • Apollonian circles • Apollonian gasket • Circumscribed circle • Commensurability • Diophantine equation • Doctrine of proportionality • Euclidean geometry • Golden ratio • Greek numerals • Incircle and excircles of a triangle • Method of exhaustion • Parallel postulate • Platonic solid • Lune of Hippocrates • Quadratrix of Hippias • Regular polygon • Straightedge and compass construction • Triangle center Results In Elements • Angle bisector theorem • Exterior angle theorem • Euclidean algorithm • Euclid's theorem • Geometric mean theorem • Greek geometric algebra • Hinge theorem • Inscribed angle theorem • Intercept theorem • Intersecting chords theorem • Intersecting secants theorem • Law of cosines • Pons asinorum • Pythagorean theorem • Tangent-secant theorem • Thales's theorem • Theorem of the gnomon Apollonius • Apollonius's theorem Other • Aristarchus's inequality • Crossbar theorem • Heron's formula • Irrational numbers • Law of sines • Menelaus's theorem • Pappus's area theorem • Problem II.8 of Arithmetica • Ptolemy's inequality • Ptolemy's table of chords • Ptolemy's theorem • Spiral of Theodorus Centers • Cyrene • Mouseion of Alexandria • Platonic Academy Related • Ancient Greek astronomy • Attic numerals • Greek numerals • Latin translations of the 12th century • Non-Euclidean geometry • Philosophy of mathematics • Neusis construction History of • A History of Greek Mathematics • by Thomas Heath • algebra • timeline • arithmetic • timeline • calculus • timeline • geometry • timeline • logic • timeline • mathematics • timeline • numbers • prehistoric counting • numeral systems • list Other cultures • Arabian/Islamic • Babylonian • Chinese • Egyptian • Incan • Indian • Japanese Ancient Greece portal • Mathematics portal Authority control International • VIAF • 2 • 3 National • Spain • 2 • France • BnF data • Catalonia • Germany • Israel • United States • Australia • Poland • 2 • Vatican Other • IdRef • 2	Wikipedia
Entropy influence conjecture In mathematics, the entropy influence conjecture is a statement about Boolean functions originally conjectured by Ehud Friedgut and Gil Kalai in 1996.[1] Statement For a function $f:\{-1,1\}^{n}\to \{-1,1\},$ note its Fourier expansion $f(x)=\sum _{S\subset [n]}{\widehat {f}}(S)x_{S},{\text{ where }}x_{S}=\prod _{i\in S}x_{i}.$ The entropy–influence conjecture states that there exists an absolute constant C such that $H(f)\leq CI(f),$ where the total influence $I$ is defined by $I(f)=\sum _{S}\|S\|{\widehat {f}}(S)^{2},$ and the entropy $H$ (of the spectrum) is defined by $H(f)=-\sum _{S}{\widehat {f}}(S)^{2}\log {\widehat {f}}(S)^{2},$ (where x log x is taken to be 0 when x = 0). See also • Analysis of Boolean functions References 1. Friedgut, Ehud; Kalai, Gil (1996). "Every monotone graph property has a sharp threshold". Proceedings of the American Mathematical Society. 124 (10): 2993–3002. doi:10.1090/s0002-9939-96-03732-x. • Unsolved Problems in Number Theory, Logic and Cryptography • The Open Problems Project, discrete and computational geometry problems	Wikipedia
The Equidistribution of Lattice Shapes of Rings of Integers of Cubic, Quartic, and Quintic Number Fields The Equidistribution of Lattice Shapes of Rings of Integers of Cubic, Quartic, and Quintic Number Fields: An Artist's Rendering is a mathematics book by Piper Harron (also known as Piper H), based on her Princeton University doctoral thesis of the same title. It has been described as "feminist",[1] "unique",[2] "honest",[2] "generous",[3] and "refreshing".[4] The Equidistribution of Lattice Shapes of Rings of Integers of Cubic, Quartic, and Quintic Number Fields: An Artist's Rendering AuthorPiper H PublisherBirkhäuser Basel Publication date 2021 ISBN978-3-319-76531-0 Thesis and reception Harron was advised by Fields Medalist Manjul Bhargava, and her thesis deals with the properties of number fields, specifically the shape of their rings of integers.[2][5] Harron and Bhargava showed that, viewed as a lattice in real vector space, the ring of integers of a random number field does not have any special symmetries.[5][6] Rather than simply presenting the proof, Harron intended for the thesis and book to explain both the mathematics and the process (and struggle) that was required to reach this result.[5] The writing is accessible and informal, and the book features sections targeting three different audiences: laypeople, people with general mathematical knowledge, and experts in number theory.[1] Harron intentionally departs from the typical academic format as she is writing for a community of mathematicians who "do not feel that they are encouraged to be themselves".[1] Unusually for a mathematics thesis, Harron intersperses her rigorous analysis and proofs with cartoons, poetry, pop-culture references, and humorous diagrams.[2] Science writer Evelyn Lamb, in Scientific American, expresses admiration for Harron for explaining the process behind the mathematics in a way that is accessible to non-mathematicians, especially "because as a woman of color, she could pay a higher price for doing it."[4] Mathematician Philp Ording calls her approach to communicating mathematical abstractions "generous".[3] Her thesis went viral in late 2015, especially within the mathematical community, in part because of the prologue which begins by stating that "respected research math is dominated by men of a certain attitude".[2][4] Harron had left academia for several years, later saying that she found the atmosphere oppressive and herself miserable and verging on failure.[7] She returned determined that, even if she did not do math the "right way", she "could still contribute to the community".[7] Her prologue states that the community lacks diversity and discourages diversity of thought.[4] "It is not my place to make the system comfortable with itself", she concludes.[4] A concise proof was published in Compositio Mathematica in 2016.[8] Author Harron earned her doctorate from Princeton in 2016. As of 2021, Harron, who also goes by Piper H., is a postdoctoral researcher at the University of Toronto.[9][10] References 1. Molinari, Julia (April 2021). "Re-imagining Doctoral Writings as Emergent Open Systems". Re-imagining doctoral writing (preprint). Colorado Press. 2. Salerno, Adriana (February–March 2019). "Book review: Mathematics for the People" (PDF). MAA Focus. 39 (1): 50–51. 3. Ording, Philip (2016). "Creative Writing in Mathematics and Science" (PDF). Banff International Research Station Proceedings 2016. p. 7. Retrieved June 18, 2021. 4. Lamb, Evelyn (December 28, 2015). "Contrasts in Number Theory". Scientific American. Retrieved June 18, 2021. 5. "The Equidistribution of Lattice Shapes of Rings of Integers of Cubic, Quartic, and Quintic Number Fields". Springer Nature Switzerland AG. Retrieved June 18, 2021. 6. Harron, Piper (June 20–24, 2016). "Contributed Talks" (PDF). 14th Meeting of the Canadian Number Theory Association. University of Calgary. p. 26. 7. Kamanos, Anastasia (2019). The Female Artist in Academia: Home and Away. Rowman & Littlefield. p. 21. ISBN 9781793604118. 8. Bhargava, Manjul; Harron, Piper (June 2016). "The equidistribution of lattice shapes of rings of integers in cubic, quartic, and quintic number fields". Compositio Mathematica. 152 (6): 1111–1120. arXiv:1309.2025. doi:10.1112/S0010437X16007260. MR 3518306. S2CID 118043017. Zbl 1347.11074. 9. "Piper H". University of Toronto. Retrieved June 18, 2021. 10. Dance, Amber (February 9, 2017). "Relationships: Sweethearts in science". Nature. 542 (7640): 261–263. doi:10.1038/nj7640-261a. External links • The Equidistribution of Lattice Shapes of Rings of Integers of Cubic, Quartic, and Quintic Number Fields (Harron's PhD thesis) • The Liberated Mathematician	Wikipedia
Euclidean algorithm In mathematics, the Euclidean algorithm,[note 1] or Euclid's algorithm, is an efficient method for computing the greatest common divisor (GCD) of two integers (numbers), the largest number that divides them both without a remainder. It is named after the ancient Greek mathematician Euclid, who first described it in his Elements (c. 300 BC). It is an example of an algorithm, a step-by-step procedure for performing a calculation according to well-defined rules, and is one of the oldest algorithms in common use. It can be used to reduce fractions to their simplest form, and is a part of many other number-theoretic and cryptographic calculations. This article is about an algorithm for the greatest common divisor. For the mathematics of space, see Euclidean geometry. For other uses of "Euclidean", see Euclidean (disambiguation). The Euclidean algorithm is based on the principle that the greatest common divisor of two numbers does not change if the larger number is replaced by its difference with the smaller number. For example, 21 is the GCD of 252 and 105 (as 252 = 21 × 12 and 105 = 21 × 5), and the same number 21 is also the GCD of 105 and 252 − 105 = 147. Since this replacement reduces the larger of the two numbers, repeating this process gives successively smaller pairs of numbers until the two numbers become equal. When that occurs, they are the GCD of the original two numbers. By reversing the steps or using the extended Euclidean algorithm, the GCD can be expressed as a linear combination of the two original numbers, that is the sum of the two numbers, each multiplied by an integer (for example, 21 = 5 × 105 + (−2) × 252). The fact that the GCD can always be expressed in this way is known as Bézout's identity. The version of the Euclidean algorithm described above (and by Euclid) can take many subtraction steps to find the GCD when one of the given numbers is much bigger than the other. A more efficient version of the algorithm shortcuts these steps, instead replacing the larger of the two numbers by its remainder when divided by the smaller of the two (with this version, the algorithm stops when reaching a zero remainder). With this improvement, the algorithm never requires more steps than five times the number of digits (base 10) of the smaller integer. This was proven by Gabriel Lamé in 1844 (Lamé's Theorem),[1][2] and marks the beginning of computational complexity theory. Additional methods for improving the algorithm's efficiency were developed in the 20th century. The Euclidean algorithm has many theoretical and practical applications. It is used for reducing fractions to their simplest form and for performing division in modular arithmetic. Computations using this algorithm form part of the cryptographic protocols that are used to secure internet communications, and in methods for breaking these cryptosystems by factoring large composite numbers. The Euclidean algorithm may be used to solve Diophantine equations, such as finding numbers that satisfy multiple congruences according to the Chinese remainder theorem, to construct continued fractions, and to find accurate rational approximations to real numbers. Finally, it can be used as a basic tool for proving theorems in number theory such as Lagrange's four-square theorem and the uniqueness of prime factorizations. The original algorithm was described only for natural numbers and geometric lengths (real numbers), but the algorithm was generalized in the 19th century to other types of numbers, such as Gaussian integers and polynomials of one variable. This led to modern abstract algebraic notions such as Euclidean domains. Background: greatest common divisor Main article: Greatest common divisor The Euclidean algorithm calculates the greatest common divisor (GCD) of two natural numbers a and b. The greatest common divisor g is the largest natural number that divides both a and b without leaving a remainder. Synonyms for GCD include greatest common factor (GCF), highest common factor (HCF), highest common divisor (HCD), and greatest common measure (GCM). The greatest common divisor is often written as gcd(a, b) or, more simply, as (a, b),[3] although the latter notation is ambiguous, also used for concepts such as an ideal in the ring of integers, which is closely related to GCD. If gcd(a, b) = 1, then a and b are said to be coprime (or relatively prime).[4] This property does not imply that a or b are themselves prime numbers.[5] For example, 6 and 35 factor as 6 = 2 × 3 and 35 = 5 × 7, so they are not prime, but their prime factors are different, so 6 and 35 are coprime, with no common factors other than 1. Let g = gcd(a, b). Since a and b are both multiples of g, they can be written a = mg and b = ng, and there is no larger number G > g for which this is true. The natural numbers m and n must be coprime, since any common factor could be factored out of m and n to make g greater. Thus, any other number c that divides both a and b must also divide g. The greatest common divisor g of a and b is the unique (positive) common divisor of a and b that is divisible by any other common divisor c.[6] The greatest common divisor can be visualized as follows.[7] Consider a rectangular area a by b, and any common divisor c that divides both a and b exactly. The sides of the rectangle can be divided into segments of length c, which divides the rectangle into a grid of squares of side length c. The GCD g is the largest value of c for which this is possible. For illustration, a 24×60 rectangular area can be divided into a grid of: 1×1 squares, 2×2 squares, 3×3 squares, 4×4 squares, 6×6 squares or 12×12 squares. Therefore, 12 is the GCD of 24 and 60. A 24×60 rectangular area can be divided into a grid of 12×12 squares, with two squares along one edge (24/12 = 2) and five squares along the other (60/12 = 5). The greatest common divisor of two numbers a and b is the product of the prime factors shared by the two numbers, where each prime factor can be repeated as many times as divides both a and b.[8] For example, since 1386 can be factored into 2 × 3 × 3 × 7 × 11, and 3213 can be factored into 3 × 3 × 3 × 7 × 17, the GCD of 1386 and 3213 equals 63 = 3 × 3 × 7, the product of their shared prime factors (with 3 repeated since 3 × 3 divides both). If two numbers have no common prime factors, their GCD is 1 (obtained here as an instance of the empty product), in other words they are coprime. A key advantage of the Euclidean algorithm is that it can find the GCD efficiently without having to compute the prime factors.[9][10] Factorization of large integers is believed to be a computationally very difficult problem, and the security of many widely used cryptographic protocols is based upon its infeasibility.[11] Another definition of the GCD is helpful in advanced mathematics, particularly ring theory.[12] The greatest common divisor g of two nonzero numbers a and b is also their smallest positive integral linear combination, that is, the smallest positive number of the form ua + vb where u and v are integers. The set of all integral linear combinations of a and b is actually the same as the set of all multiples of g (mg, where m is an integer). In modern mathematical language, the ideal generated by a and b is the ideal generated by g alone (an ideal generated by a single element is called a principal ideal, and all ideals of the integers are principal ideals). Some properties of the GCD are in fact easier to see with this description, for instance the fact that any common divisor of a and b also divides the GCD (it divides both terms of ua + vb). The equivalence of this GCD definition with the other definitions is described below. The GCD of three or more numbers equals the product of the prime factors common to all the numbers,[13] but it can also be calculated by repeatedly taking the GCDs of pairs of numbers.[14] For example, gcd(a, b, c) = gcd(a, gcd(b, c)) = gcd(gcd(a, b), c) = gcd(gcd(a, c), b). Thus, Euclid's algorithm, which computes the GCD of two integers, suffices to calculate the GCD of arbitrarily many integers. Description Procedure The Euclidean algorithm proceeds in a series of steps, with the output of each step used as the input for the next. Track the steps using an integer counter k, so the initial step corresponds to k = 0, the next step to k = 1, and so on. Each step begins with two nonnegative remainders rk−2 and rk−1, with rk−2 > rk−1. The kth step performs division-with-remainder to find the quotient qk and remainder rk so that: $r_{k-2}=q_{k}r_{k-1}+r_{k}\ {\text{ with }}\ r_{k-1}>r_{k}\geq 0.$ That is, multiples of the smaller number rk−1 are subtracted from the larger number rk−2 until the remainder rk is smaller than rk−1. Then the algorithm proceeds to the (k+1)th step starting with rk−1 and rk. In the initial step k = 0, the remainders are set to r−2 = a and r−1 = b, the numbers for which the GCD is sought. In the next step k = 1, the remainders are r−1 = b and the remainder r0 of the initial step, and so on. The algorithm proceeds in a sequence of equations ${\begin{aligned}a&=q_{0}b+r_{0}\\b&=q_{1}r_{0}+r_{1}\\r_{0}&=q_{2}r_{1}+r_{2}\\r_{1}&=q_{3}r_{2}+r_{3}\\&\,\,\,\vdots \end{aligned}}$ The algorithm need not be modified if a < b: in that case, the initial quotient is q0 = 0, the first remainder is r0 = a, and henceforth rk−2 > rk−1 for all k ≥ 1. Since the remainders are non-negative integers that decrease with every step, the sequence $r_{-1}>r_{0}>r_{1}>r_{2}>\cdots \geq 0$ cannot be infinite, so the algorithm must eventually fail to produce the next step; but the division algorithm can always proceed to the (N+1)th step provided rN > 0. Thus the algorithm must eventually produce a zero remainder rN = 0.[15] The final nonzero remainder is the greatest common divisor of a and b: $r_{N-1}=\gcd(a,b).$ Proof of validity The validity of the Euclidean algorithm can be proven by a two-step argument.[16] In the first step, the final nonzero remainder rN−1 is shown to divide both a and b. Since it is a common divisor, it must be less than or equal to the greatest common divisor g. In the second step, it is shown that any common divisor of a and b, including g, must divide rN−1; therefore, g must be less than or equal to rN−1. These two opposite inequalities imply rN−1 = g. To demonstrate that rN−1 divides both a and b (the first step), rN−1 divides its predecessor rN−2 rN−2 = qN rN−1 since the final remainder rN is zero. rN−1 also divides its next predecessor rN−3 rN−3 = qN−1 rN−2 + rN−1 because it divides both terms on the right-hand side of the equation. Iterating the same argument, rN−1 divides all the preceding remainders, including a and b. None of the preceding remainders rN−2, rN−3, etc. divide a and b, since they leave a remainder. Since rN−1 is a common divisor of a and b, rN−1 ≤ g. In the second step, any natural number c that divides both a and b (in other words, any common divisor of a and b) divides the remainders rk. By definition, a and b can be written as multiples of c : a = mc and b = nc, where m and n are natural numbers. Therefore, c divides the initial remainder r0, since r0 = a − q0b = mc − q0nc = (m − q0n)c. An analogous argument shows that c also divides the subsequent remainders r1, r2, etc. Therefore, the greatest common divisor g must divide rN−1, which implies that g ≤ rN−1. Since the first part of the argument showed the reverse (rN−1 ≤ g), it follows that g = rN−1. Thus, g is the greatest common divisor of all the succeeding pairs:[17][18] g = gcd(a, b) = gcd(b, r0) = gcd(r0, r1) = … = gcd(rN−2, rN−1) = rN−1. Worked example For illustration, the Euclidean algorithm can be used to find the greatest common divisor of a = 1071 and b = 462. To begin, multiples of 462 are subtracted from 1071 until the remainder is less than 462. Two such multiples can be subtracted (q0 = 2), leaving a remainder of 147: 1071 = 2 × 462 + 147. Then multiples of 147 are subtracted from 462 until the remainder is less than 147. Three multiples can be subtracted (q1 = 3), leaving a remainder of 21: 462 = 3 × 147 + 21. Then multiples of 21 are subtracted from 147 until the remainder is less than 21. Seven multiples can be subtracted (q2 = 7), leaving no remainder: 147 = 7 × 21 + 0. Since the last remainder is zero, the algorithm ends with 21 as the greatest common divisor of 1071 and 462. This agrees with the gcd(1071, 462) found by prime factorization above. In tabular form, the steps are: Step kEquationQuotient and remainder 01071 = q0 462 + r0q0 = 2 and r0 = 147 1462 = q1 147 + r1q1 = 3 and r1 = 21 2147 = q2 21 + r2q2 = 7 and r2 = 0; algorithm ends Visualization The Euclidean algorithm can be visualized in terms of the tiling analogy given above for the greatest common divisor.[19] Assume that we wish to cover an a×b rectangle with square tiles exactly, where a is the larger of the two numbers. We first attempt to tile the rectangle using b×b square tiles; however, this leaves an r0×b residual rectangle untiled, where r0 < b. We then attempt to tile the residual rectangle with r0×r0 square tiles. This leaves a second residual rectangle r1×r0, which we attempt to tile using r1×r1 square tiles, and so on. The sequence ends when there is no residual rectangle, i.e., when the square tiles cover the previous residual rectangle exactly. The length of the sides of the smallest square tile is the GCD of the dimensions of the original rectangle. For example, the smallest square tile in the adjacent figure is 21×21 (shown in red), and 21 is the GCD of 1071 and 462, the dimensions of the original rectangle (shown in green). Euclidean division Main article: Euclidean division At every step k, the Euclidean algorithm computes a quotient qk and remainder rk from two numbers rk−1 and rk−2 rk−2 = qk rk−1 + rk where the rk is non-negative and is strictly less than the absolute value of rk−1. The theorem which underlies the definition of the Euclidean division ensures that such a quotient and remainder always exist and are unique.[20] In Euclid's original version of the algorithm, the quotient and remainder are found by repeated subtraction; that is, rk−1 is subtracted from rk−2 repeatedly until the remainder rk is smaller than rk−1. After that rk and rk−1 are exchanged and the process is iterated. Euclidean division reduces all the steps between two exchanges into a single step, which is thus more efficient. Moreover, the quotients are not needed, thus one may replace Euclidean division by the modulo operation, which gives only the remainder. Thus the iteration of the Euclidean algorithm becomes simply rk = rk−2 mod rk−1. Implementations Implementations of the algorithm may be expressed in pseudocode. For example, the division-based version may be programmed as[21] function gcd(a, b) while b ≠ 0 t := b b := a mod b a := t return a At the beginning of the kth iteration, the variable b holds the latest remainder rk−1, whereas the variable a holds its predecessor, rk−2. The step b := a mod b is equivalent to the above recursion formula rk ≡ rk−2 mod rk−1. The temporary variable t holds the value of rk−1 while the next remainder rk is being calculated. At the end of the loop iteration, the variable b holds the remainder rk, whereas the variable a holds its predecessor, rk−1. (If negative inputs are allowed, or if the mod function may return negative values, the last line must be changed into return abs(a).) In the subtraction-based version, which was Euclid's original version, the remainder calculation (b := a mod b) is replaced by repeated subtraction.[22] Contrary to the division-based version, which works with arbitrary integers as input, the subtraction-based version supposes that the input consists of positive integers and stops when a = b: function gcd(a, b) while a ≠ b if a > b a := a − b else b := b − a return a The variables a and b alternate holding the previous remainders rk−1 and rk−2. Assume that a is larger than b at the beginning of an iteration; then a equals rk−2, since rk−2 > rk−1. During the loop iteration, a is reduced by multiples of the previous remainder b until a is smaller than b. Then a is the next remainder rk. Then b is reduced by multiples of a until it is again smaller than a, giving the next remainder rk+1, and so on. The recursive version[23] is based on the equality of the GCDs of successive remainders and the stopping condition gcd(rN−1, 0) = rN−1. function gcd(a, b) if b = 0 return a else return gcd(b, a mod b) (As above, if negative inputs are allowed, or if the mod function may return negative values, the instruction "return a" must be changed into "return max(a, −a)".) For illustration, the gcd(1071, 462) is calculated from the equivalent gcd(462, 1071 mod 462) = gcd(462, 147). The latter GCD is calculated from the gcd(147, 462 mod 147) = gcd(147, 21), which in turn is calculated from the gcd(21, 147 mod 21) = gcd(21, 0) = 21. Method of least absolute remainders In another version of Euclid's algorithm, the quotient at each step is increased by one if the resulting negative remainder is smaller in magnitude than the typical positive remainder.[24][25] Previously, the equation rk−2 = qk rk−1 + rk assumed that \|rk−1\| > rk > 0. However, an alternative negative remainder ek can be computed: rk−2 = (qk + 1) rk−1 + ek if rk−1 > 0 or rk−2 = (qk − 1) rk−1 + ek if rk−1 < 0. If rk is replaced by ek. when \|ek\| < \|rk\|, then one gets a variant of Euclidean algorithm such that \|rk\| ≤ \|rk−1\| / 2 at each step. Leopold Kronecker has shown that this version requires the fewest steps of any version of Euclid's algorithm.[24][25] More generally, it has been proven that, for every input numbers a and b, the number of steps is minimal if and only if qk is chosen in order that $\left\|{\frac {r_{k+1}}{r_{k}}}\right\|<{\frac {1}{\varphi }}\sim 0.618,$ where $\varphi $ is the golden ratio.[26] Historical development The Euclidean algorithm is one of the oldest algorithms in common use.[27] It appears in Euclid's Elements (c. 300 BC), specifically in Book 7 (Propositions 1–2) and Book 10 (Propositions 2–3). In Book 7, the algorithm is formulated for integers, whereas in Book 10, it is formulated for lengths of line segments. (In modern usage, one would say it was formulated there for real numbers. But lengths, areas, and volumes, represented as real numbers in modern usage, are not measured in the same units and there is no natural unit of length, area, or volume; the concept of real numbers was unknown at that time.) The latter algorithm is geometrical. The GCD of two lengths a and b corresponds to the greatest length g that measures a and b evenly; in other words, the lengths a and b are both integer multiples of the length g. The algorithm was probably not discovered by Euclid, who compiled results from earlier mathematicians in his Elements.[28][29] The mathematician and historian B. L. van der Waerden suggests that Book VII derives from a textbook on number theory written by mathematicians in the school of Pythagoras.[30] The algorithm was probably known by Eudoxus of Cnidus (about 375 BC).[27][31] The algorithm may even pre-date Eudoxus,[32][33] judging from the use of the technical term ἀνθυφαίρεσις (anthyphairesis, reciprocal subtraction) in works by Euclid and Aristotle.[34] Centuries later, Euclid's algorithm was discovered independently both in India and in China,[35] primarily to solve Diophantine equations that arose in astronomy and making accurate calendars. In the late 5th century, the Indian mathematician and astronomer Aryabhata described the algorithm as the "pulverizer",[36] perhaps because of its effectiveness in solving Diophantine equations.[37] Although a special case of the Chinese remainder theorem had already been described in the Chinese book Sunzi Suanjing,[38] the general solution was published by Qin Jiushao in his 1247 book Shushu Jiuzhang (數書九章 Mathematical Treatise in Nine Sections).[39] The Euclidean algorithm was first described numerically and popularized in Europe in the second edition of Bachet's Problèmes plaisants et délectables (Pleasant and enjoyable problems, 1624).[36] In Europe, it was likewise used to solve Diophantine equations and in developing continued fractions. The extended Euclidean algorithm was published by the English mathematician Nicholas Saunderson,[40] who attributed it to Roger Cotes as a method for computing continued fractions efficiently.[41] In the 19th century, the Euclidean algorithm led to the development of new number systems, such as Gaussian integers and Eisenstein integers. In 1815, Carl Gauss used the Euclidean algorithm to demonstrate unique factorization of Gaussian integers, although his work was first published in 1832.[42] Gauss mentioned the algorithm in his Disquisitiones Arithmeticae (published 1801), but only as a method for continued fractions.[35] Peter Gustav Lejeune Dirichlet seems to have been the first to describe the Euclidean algorithm as the basis for much of number theory.[43] Lejeune Dirichlet noted that many results of number theory, such as unique factorization, would hold true for any other system of numbers to which the Euclidean algorithm could be applied.[44] Lejeune Dirichlet's lectures on number theory were edited and extended by Richard Dedekind, who used Euclid's algorithm to study algebraic integers, a new general type of number. For example, Dedekind was the first to prove Fermat's two-square theorem using the unique factorization of Gaussian integers.[45] Dedekind also defined the concept of a Euclidean domain, a number system in which a generalized version of the Euclidean algorithm can be defined (as described below). In the closing decades of the 19th century, the Euclidean algorithm gradually became eclipsed by Dedekind's more general theory of ideals.[46] "[The Euclidean algorithm] is the granddaddy of all algorithms, because it is the oldest nontrivial algorithm that has survived to the present day." Donald Knuth, The Art of Computer Programming, Vol. 2: Seminumerical Algorithms, 2nd edition (1981), p. 318. Other applications of Euclid's algorithm were developed in the 19th century. In 1829, Charles Sturm showed that the algorithm was useful in the Sturm chain method for counting the real roots of polynomials in any given interval.[47] The Euclidean algorithm was the first integer relation algorithm, which is a method for finding integer relations between commensurate real numbers. Several novel integer relation algorithms have been developed, such as the algorithm of Helaman Ferguson and R.W. Forcade (1979)[48] and the LLL algorithm.[49][50] In 1969, Cole and Davie developed a two-player game based on the Euclidean algorithm, called The Game of Euclid,[51] which has an optimal strategy.[52] The players begin with two piles of a and b stones. The players take turns removing m multiples of the smaller pile from the larger. Thus, if the two piles consist of x and y stones, where x is larger than y, the next player can reduce the larger pile from x stones to x − my stones, as long as the latter is a nonnegative integer. The winner is the first player to reduce one pile to zero stones.[53][54] Mathematical applications Bézout's identity Bézout's identity states that the greatest common divisor g of two integers a and b can be represented as a linear sum of the original two numbers a and b.[55] In other words, it is always possible to find integers s and t such that g = sa + tb.[56][57] The integers s and t can be calculated from the quotients q0, q1, etc. by reversing the order of equations in Euclid's algorithm.[58] Beginning with the next-to-last equation, g can be expressed in terms of the quotient qN−1 and the two preceding remainders, rN−2 and rN−3: g = rN−1 = rN−3 − qN−1 rN−2 . Those two remainders can be likewise expressed in terms of their quotients and preceding remainders, rN−2 = rN−4 − qN−2 rN−3 and rN−3 = rN−5 − qN−3 rN−4 . Substituting these formulae for rN−2 and rN−3 into the first equation yields g as a linear sum of the remainders rN−4 and rN−5. The process of substituting remainders by formulae involving their predecessors can be continued until the original numbers a and b are reached: r2 = r0 − q2 r1 r1 = b − q1 r0 r0 = a − q0 b. After all the remainders r0, r1, etc. have been substituted, the final equation expresses g as a linear sum of a and b, so that g = sa + tb. The Euclidean algorithm, and thus Bezout's identity, can be generalized to the context of Euclidean domains. Principal ideals and related problems Bézout's identity provides yet another definition of the greatest common divisor g of two numbers a and b.[12] Consider the set of all numbers ua + vb, where u and v are any two integers. Since a and b are both divisible by g, every number in the set is divisible by g. In other words, every number of the set is an integer multiple of g. This is true for every common divisor of a and b. However, unlike other common divisors, the greatest common divisor is a member of the set; by Bézout's identity, choosing u = s and v = t gives g. A smaller common divisor cannot be a member of the set, since every member of the set must be divisible by g. Conversely, any multiple m of g can be obtained by choosing u = ms and v = mt, where s and t are the integers of Bézout's identity. This may be seen by multiplying Bézout's identity by m, mg = msa + mtb. Therefore, the set of all numbers ua + vb is equivalent to the set of multiples m of g. In other words, the set of all possible sums of integer multiples of two numbers (a and b) is equivalent to the set of multiples of gcd(a, b). The GCD is said to be the generator of the ideal of a and b. This GCD definition led to the modern abstract algebraic concepts of a principal ideal (an ideal generated by a single element) and a principal ideal domain (a domain in which every ideal is a principal ideal). Certain problems can be solved using this result.[59] For example, consider two measuring cups of volume a and b. By adding/subtracting u multiples of the first cup and v multiples of the second cup, any volume ua + vb can be measured out. These volumes are all multiples of g = gcd(a, b). Extended Euclidean algorithm Main article: Extended Euclidean algorithm The integers s and t of Bézout's identity can be computed efficiently using the extended Euclidean algorithm. This extension adds two recursive equations to Euclid's algorithm[60] sk = sk−2 − qksk−1 tk = tk−2 − qktk−1 with the starting values s−2 = 1, t−2 = 0 s−1 = 0, t−1 = 1. Using this recursion, Bézout's integers s and t are given by s = sN and t = tN, where N+1 is the step on which the algorithm terminates with rN+1 = 0. The validity of this approach can be shown by induction. Assume that the recursion formula is correct up to step k − 1 of the algorithm; in other words, assume that rj = sj a + tj b for all j less than k. The kth step of the algorithm gives the equation rk = rk−2 − qkrk−1. Since the recursion formula has been assumed to be correct for rk−2 and rk−1, they may be expressed in terms of the corresponding s and t variables rk = (sk−2 a + tk−2 b) − qk(sk−1 a + tk−1 b). Rearranging this equation yields the recursion formula for step k, as required rk = sk a + tk b = (sk−2 − qksk−1) a + (tk−2 − qktk−1) b. Matrix method The integers s and t can also be found using an equivalent matrix method.[61] The sequence of equations of Euclid's algorithm ${\begin{aligned}a&=q_{0}b+r_{0}\\b&=q_{1}r_{0}+r_{1}\\&\,\,\,\vdots \\r_{N-2}&=q_{N}r_{N-1}+0\end{aligned}}$ can be written as a product of 2×2 quotient matrices multiplying a two-dimensional remainder vector ${\begin{pmatrix}a\\b\end{pmatrix}}={\begin{pmatrix}q_{0}&1\\1&0\end{pmatrix}}{\begin{pmatrix}b\\r_{0}\end{pmatrix}}={\begin{pmatrix}q_{0}&1\\1&0\end{pmatrix}}{\begin{pmatrix}q_{1}&1\\1&0\end{pmatrix}}{\begin{pmatrix}r_{0}\\r_{1}\end{pmatrix}}=\cdots =\prod _{i=0}^{N}{\begin{pmatrix}q_{i}&1\\1&0\end{pmatrix}}{\begin{pmatrix}r_{N-1}\\0\end{pmatrix}}\,.$ Let M represent the product of all the quotient matrices $\mathbf {M} ={\begin{pmatrix}m_{11}&m_{12}\\m_{21}&m_{22}\end{pmatrix}}=\prod _{i=0}^{N}{\begin{pmatrix}q_{i}&1\\1&0\end{pmatrix}}={\begin{pmatrix}q_{0}&1\\1&0\end{pmatrix}}{\begin{pmatrix}q_{1}&1\\1&0\end{pmatrix}}\cdots {\begin{pmatrix}q_{N}&1\\1&0\end{pmatrix}}\,.$ This simplifies the Euclidean algorithm to the form ${\begin{pmatrix}a\\b\end{pmatrix}}=\mathbf {M} {\begin{pmatrix}r_{N-1}\\0\end{pmatrix}}=\mathbf {M} {\begin{pmatrix}g\\0\end{pmatrix}}\,.$ To express g as a linear sum of a and b, both sides of this equation can be multiplied by the inverse of the matrix M.[61][62] The determinant of M equals (−1)N+1, since it equals the product of the determinants of the quotient matrices, each of which is negative one. Since the determinant of M is never zero, the vector of the final remainders can be solved using the inverse of M ${\begin{pmatrix}g\\0\end{pmatrix}}=\mathbf {M} ^{-1}{\begin{pmatrix}a\\b\end{pmatrix}}=(-1)^{N+1}{\begin{pmatrix}m_{22}&-m_{12}\\-m_{21}&m_{11}\end{pmatrix}}{\begin{pmatrix}a\\b\end{pmatrix}}\,.$ Since the top equation gives g = (−1)N+1 ( m22 a − m12 b), the two integers of Bézout's identity are s = (−1)N+1m22 and t = (−1)Nm12. The matrix method is as efficient as the equivalent recursion, with two multiplications and two additions per step of the Euclidean algorithm. Euclid's lemma and unique factorization Bézout's identity is essential to many applications of Euclid's algorithm, such as demonstrating the unique factorization of numbers into prime factors.[63] To illustrate this, suppose that a number L can be written as a product of two factors u and v, that is, L = uv. If another number w also divides L but is coprime with u, then w must divide v, by the following argument: If the greatest common divisor of u and w is 1, then integers s and t can be found such that 1 = su + tw . by Bézout's identity. Multiplying both sides by v gives the relation v = suv + twv = sL + twv . Since w divides both terms on the right-hand side, it must also divide the left-hand side, v. This result is known as Euclid's lemma.[64] Specifically, if a prime number divides L, then it must divide at least one factor of L. Conversely, if a number w is coprime to each of a series of numbers a1, a2, ..., an, then w is also coprime to their product, a1 × a2 × ... × an.[64] Euclid's lemma suffices to prove that every number has a unique factorization into prime numbers.[65] To see this, assume the contrary, that there are two independent factorizations of L into m and n prime factors, respectively L = p1p2…pm = q1q2…qn . Since each prime p divides L by assumption, it must also divide one of the q factors; since each q is prime as well, it must be that p = q. Iteratively dividing by the p factors shows that each p has an equal counterpart q; the two prime factorizations are identical except for their order. The unique factorization of numbers into primes has many applications in mathematical proofs, as shown below. Linear Diophantine equations Diophantine equations are equations in which the solutions are restricted to integers; they are named after the 3rd-century Alexandrian mathematician Diophantus.[66] A typical linear Diophantine equation seeks integers x and y such that[67] ax + by = c where a, b and c are given integers. This can be written as an equation for x in modular arithmetic: ax ≡ c mod b. Let g be the greatest common divisor of a and b. Both terms in ax + by are divisible by g; therefore, c must also be divisible by g, or the equation has no solutions. By dividing both sides by c/g, the equation can be reduced to Bezout's identity sa + tb = g where s and t can be found by the extended Euclidean algorithm.[68] This provides one solution to the Diophantine equation, x1 = s (c/g) and y1 = t (c/g). In general, a linear Diophantine equation has no solutions, or an infinite number of solutions.[69] To find the latter, consider two solutions, (x1, y1) and (x2, y2), where ax1 + by1 = c = ax2 + by2 or equivalently a(x1 − x2) = b(y2 − y1). Therefore, the smallest difference between two x solutions is b/g, whereas the smallest difference between two y solutions is a/g. Thus, the solutions may be expressed as x = x1 − bu/g y = y1 + au/g. By allowing u to vary over all possible integers, an infinite family of solutions can be generated from a single solution (x1, y1). If the solutions are required to be positive integers (x > 0, y > 0), only a finite number of solutions may be possible. This restriction on the acceptable solutions allows some systems of Diophantine equations with more unknowns than equations to have a finite number of solutions;[70] this is impossible for a system of linear equations when the solutions can be any real number (see Underdetermined system). Multiplicative inverses and the RSA algorithm A finite field is a set of numbers with four generalized operations. The operations are called addition, subtraction, multiplication and division and have their usual properties, such as commutativity, associativity and distributivity. An example of a finite field is the set of 13 numbers {0, 1, 2, ..., 12} using modular arithmetic. In this field, the results of any mathematical operation (addition, subtraction, multiplication, or division) is reduced modulo 13; that is, multiples of 13 are added or subtracted until the result is brought within the range 0–12. For example, the result of 5 × 7 = 35 mod 13 = 9. Such finite fields can be defined for any prime p; using more sophisticated definitions, they can also be defined for any power m of a prime p m. Finite fields are often called Galois fields, and are abbreviated as GF(p) or GF(p m). In such a field with m numbers, every nonzero element a has a unique modular multiplicative inverse, a−1 such that aa−1 = a−1a ≡ 1 mod m. This inverse can be found by solving the congruence equation ax ≡ 1 mod m,[71] or the equivalent linear Diophantine equation[72] ax + my = 1. This equation can be solved by the Euclidean algorithm, as described above. Finding multiplicative inverses is an essential step in the RSA algorithm, which is widely used in electronic commerce; specifically, the equation determines the integer used to decrypt the message.[73] Although the RSA algorithm uses rings rather than fields, the Euclidean algorithm can still be used to find a multiplicative inverse where one exists. The Euclidean algorithm also has other applications in error-correcting codes; for example, it can be used as an alternative to the Berlekamp–Massey algorithm for decoding BCH and Reed–Solomon codes, which are based on Galois fields.[74] Chinese remainder theorem Euclid's algorithm can also be used to solve multiple linear Diophantine equations.[75] Such equations arise in the Chinese remainder theorem, which describes a novel method to represent an integer x. Instead of representing an integer by its digits, it may be represented by its remainders xi modulo a set of N coprime numbers mi:[76] ${\begin{aligned}x_{1}&\equiv x{\pmod {m_{1}}}\\x_{2}&\equiv x{\pmod {m_{2}}}\\&\,\,\,\vdots \\x_{N}&\equiv x{\pmod {m_{N}}}\,.\end{aligned}}$ The goal is to determine x from its N remainders xi. The solution is to combine the multiple equations into a single linear Diophantine equation with a much larger modulus M that is the product of all the individual moduli mi, and define Mi as $M_{i}={\frac {M}{m_{i}}}.$ Thus, each Mi is the product of all the moduli except mi. The solution depends on finding N new numbers hi such that $M_{i}h_{i}\equiv 1{\pmod {m_{i}}}\,.$ With these numbers hi, any integer x can be reconstructed from its remainders xi by the equation $x\equiv (x_{1}M_{1}h_{1}+x_{2}M_{2}h_{2}+\cdots +x_{N}M_{N}h_{N}){\pmod {M}}\,.$ Since these numbers hi are the multiplicative inverses of the Mi, they may be found using Euclid's algorithm as described in the previous subsection. Stern–Brocot tree Main article: Stern–Brocot tree The Euclidean algorithm can be used to arrange the set of all positive rational numbers into an infinite binary search tree, called the Stern–Brocot tree. The number 1 (expressed as a fraction 1/1) is placed at the root of the tree, and the location of any other number a/b can be found by computing gcd(a,b) using the original form of the Euclidean algorithm, in which each step replaces the larger of the two given numbers by its difference with the smaller number (not its remainder), stopping when two equal numbers are reached. A step of the Euclidean algorithm that replaces the first of the two numbers corresponds to a step in the tree from a node to its right child, and a step that replaces the second of the two numbers corresponds to a step in the tree from a node to its left child. The sequence of steps constructed in this way does not depend on whether a/b is given in lowest terms, and forms a path from the root to a node containing the number a/b.[77] This fact can be used to prove that each positive rational number appears exactly once in this tree. For example, 3/4 can be found by starting at the root, going to the left once, then to the right twice: ${\begin{aligned}&\gcd(3,4)&\leftarrow \\={}&\gcd(3,1)&\rightarrow \\={}&\gcd(2,1)&\rightarrow \\={}&\gcd(1,1).\end{aligned}}$ The Euclidean algorithm has almost the same relationship to another binary tree on the rational numbers called the Calkin–Wilf tree. The difference is that the path is reversed: instead of producing a path from the root of the tree to a target, it produces a path from the target to the root. Continued fractions The Euclidean algorithm has a close relationship with continued fractions.[78] The sequence of equations can be written in the form ${\begin{aligned}{\frac {a}{b}}&=q_{0}+{\frac {r_{0}}{b}}\\{\frac {b}{r_{0}}}&=q_{1}+{\frac {r_{1}}{r_{0}}}\\{\frac {r_{0}}{r_{1}}}&=q_{2}+{\frac {r_{2}}{r_{1}}}\\&\,\,\,\vdots \\{\frac {r_{k-2}}{r_{k-1}}}&=q_{k}+{\frac {r_{k}}{r_{k-1}}}\\&\,\,\,\vdots \\{\frac {r_{N-2}}{r_{N-1}}}&=q_{N}\,.\end{aligned}}$ The last term on the right-hand side always equals the inverse of the left-hand side of the next equation. Thus, the first two equations may be combined to form ${\frac {a}{b}}=q_{0}+{\cfrac {1}{q_{1}+{\cfrac {r_{1}}{r_{0}}}}}\,.$ The third equation may be used to substitute the denominator term r1/r0, yielding ${\frac {a}{b}}=q_{0}+{\cfrac {1}{q_{1}+{\cfrac {1}{q_{2}+{\cfrac {r_{2}}{r_{1}}}}}}}\,.$ The final ratio of remainders rk/rk−1 can always be replaced using the next equation in the series, up to the final equation. The result is a continued fraction ${\frac {a}{b}}=q_{0}+{\cfrac {1}{q_{1}+{\cfrac {1}{q_{2}+{\cfrac {1}{\ddots +{\cfrac {1}{q_{N}}}}}}}}}=[q_{0};q_{1},q_{2},\ldots ,q_{N}]\,.$ In the worked example above, the gcd(1071, 462) was calculated, and the quotients qk were 2, 3 and 7, respectively. Therefore, the fraction 1071/462 may be written ${\frac {1071}{462}}=2+{\cfrac {1}{3+{\cfrac {1}{7}}}}=[2;3,7]$ as can be confirmed by calculation. Factorization algorithms Calculating a greatest common divisor is an essential step in several integer factorization algorithms,[79] such as Pollard's rho algorithm,[80] Shor's algorithm,[81] Dixon's factorization method[82] and the Lenstra elliptic curve factorization.[83] The Euclidean algorithm may be used to find this GCD efficiently. Continued fraction factorization uses continued fractions, which are determined using Euclid's algorithm.[84] Algorithmic efficiency The computational efficiency of Euclid's algorithm has been studied thoroughly.[85] This efficiency can be described by the number of division steps the algorithm requires, multiplied by the computational expense of each step. The first known analysis of Euclid's algorithm is due to A. A. L. Reynaud in 1811,[86] who showed that the number of division steps on input (u, v) is bounded by v; later he improved this to v/2 + 2. Later, in 1841, P. J. E. Finck showed[87] that the number of division steps is at most 2 log2 v + 1, and hence Euclid's algorithm runs in time polynomial in the size of the input.[88] Émile Léger, in 1837, studied the worst case, which is when the inputs are consecutive Fibonacci numbers.[88] Finck's analysis was refined by Gabriel Lamé in 1844,[89] who showed that the number of steps required for completion is never more than five times the number h of base-10 digits of the smaller number b.[90][91] In the uniform cost model (suitable for analyzing the complexity of gcd calculation on numbers that fit into a single machine word), each step of the algorithm takes constant time, and Lamé's analysis implies that the total running time is also O(h). However, in a model of computation suitable for computation with larger numbers, the computational expense of a single remainder computation in the algorithm can be as large as O(h2).[92] In this case the total time for all of the steps of the algorithm can be analyzed using a telescoping series, showing that it is also O(h2). Modern algorithmic techniques based on the Schönhage–Strassen algorithm for fast integer multiplication can be used to speed this up, leading to quasilinear algorithms for the GCD.[93][94] Number of steps The number of steps to calculate the GCD of two natural numbers, a and b, may be denoted by T(a, b).[95] If g is the GCD of a and b, then a = mg and b = ng for two coprime numbers m and n. Then T(a, b) = T(m, n) as may be seen by dividing all the steps in the Euclidean algorithm by g.[96] By the same argument, the number of steps remains the same if a and b are multiplied by a common factor w: T(a, b) = T(wa, wb). Therefore, the number of steps T may vary dramatically between neighboring pairs of numbers, such as T(a, b) and T(a, b + 1), depending on the size of the two GCDs. The recursive nature of the Euclidean algorithm gives another equation T(a, b) = 1 + T(b, r0) = 2 + T(r0, r1) = … = N + T(rN−2, rN−1) = N + 1 where T(x, 0) = 0 by assumption.[95] Worst-case If the Euclidean algorithm requires N steps for a pair of natural numbers a > b > 0, the smallest values of a and b for which this is true are the Fibonacci numbers FN+2 and FN+1, respectively.[97] More precisely, if the Euclidean algorithm requires N steps for the pair a > b, then one has a ≥ FN+2 and b ≥ FN+1. This can be shown by induction.[98] If N = 1, b divides a with no remainder; the smallest natural numbers for which this is true is b = 1 and a = 2, which are F2 and F3, respectively. Now assume that the result holds for all values of N up to M − 1. The first step of the M-step algorithm is a = q0b + r0, and the Euclidean algorithm requires M − 1 steps for the pair b > r0. By induction hypothesis, one has b ≥ FM+1 and r0 ≥ FM. Therefore, a = q0b + r0 ≥ b + r0 ≥ FM+1 + FM = FM+2, which is the desired inequality. This proof, published by Gabriel Lamé in 1844, represents the beginning of computational complexity theory,[99] and also the first practical application of the Fibonacci numbers.[97] This result suffices to show that the number of steps in Euclid's algorithm can never be more than five times the number of its digits (base 10).[100] For if the algorithm requires N steps, then b is greater than or equal to FN+1 which in turn is greater than or equal to φN−1, where φ is the golden ratio. Since b ≥ φN−1, then N − 1 ≤ logφb. Since log10φ > 1/5, (N − 1)/5 < log10φ logφb = log10b. Thus, N ≤ 5 log10b. Thus, the Euclidean algorithm always needs less than O(h) divisions, where h is the number of digits in the smaller number b. Average The average number of steps taken by the Euclidean algorithm has been defined in three different ways. The first definition is the average time T(a) required to calculate the GCD of a given number a and a smaller natural number b chosen with equal probability from the integers 0 to a − 1[95] $T(a)={\frac {1}{a}}\sum _{0\leq b<a}T(a,b).$ However, since T(a, b) fluctuates dramatically with the GCD of the two numbers, the averaged function T(a) is likewise "noisy".[101] To reduce this noise, a second average τ(a) is taken over all numbers coprime with a $\tau (a)={\frac {1}{\varphi (a)}}\sum _{\begin{smallmatrix}0\leq b<a\\\gcd(a,b)=1\end{smallmatrix}}T(a,b).$ There are φ(a) coprime integers less than a, where φ is Euler's totient function. This tau average grows smoothly with a[102][103] $\tau (a)={\frac {12}{\pi ^{2}}}\ln 2\ln a+C+O(a^{-1/6-\varepsilon })$ with the residual error being of order a−(1/6) + ε, where ε is infinitesimal. The constant C in this formula is called Porter's constant[104] and equals $C=-{\frac {1}{2}}+{\frac {6\ln 2}{\pi ^{2}}}\left(4\gamma -{\frac {24}{\pi ^{2}}}\zeta '(2)+3\ln 2-2\right)\approx 1.467$ where γ is the Euler–Mascheroni constant and ζ' is the derivative of the Riemann zeta function.[105][106] The leading coefficient (12/π2) ln 2 was determined by two independent methods.[107][108] Since the first average can be calculated from the tau average by summing over the divisors d of a[109] $T(a)={\frac {1}{a}}\sum _{d\mid a}\varphi (d)\tau (d)$ it can be approximated by the formula[110] $T(a)\approx C+{\frac {12}{\pi ^{2}}}\ln 2\left(\ln a-\sum _{d\mid a}{\frac {\Lambda (d)}{d}}\right)$ where Λ(d) is the Mangoldt function.[111] A third average Y(n) is defined as the mean number of steps required when both a and b are chosen randomly (with uniform distribution) from 1 to n[110] $Y(n)={\frac {1}{n^{2}}}\sum _{a=1}^{n}\sum _{b=1}^{n}T(a,b)={\frac {1}{n}}\sum _{a=1}^{n}T(a).$ Substituting the approximate formula for T(a) into this equation yields an estimate for Y(n)[112] $Y(n)\approx {\frac {12}{\pi ^{2}}}\ln 2\ln n+0.06.$ Computational expense per step In each step k of the Euclidean algorithm, the quotient qk and remainder rk are computed for a given pair of integers rk−2 and rk−1 rk−2 = qk rk−1 + rk. The computational expense per step is associated chiefly with finding qk, since the remainder rk can be calculated quickly from rk−2, rk−1, and qk rk = rk−2 − qk rk−1. The computational expense of dividing h-bit numbers scales as O(h(ℓ+1)), where ℓ is the length of the quotient.[92] For comparison, Euclid's original subtraction-based algorithm can be much slower. A single integer division is equivalent to the quotient q number of subtractions. If the ratio of a and b is very large, the quotient is large and many subtractions will be required. On the other hand, it has been shown that the quotients are very likely to be small integers. The probability of a given quotient q is approximately ln \|u/(u − 1)\| where u = (q + 1)2.[113] For illustration, the probability of a quotient of 1, 2, 3, or 4 is roughly 41.5%, 17.0%, 9.3%, and 5.9%, respectively. Since the operation of subtraction is faster than division, particularly for large numbers,[114] the subtraction-based Euclid's algorithm is competitive with the division-based version.[115] This is exploited in the binary version of Euclid's algorithm.[116] Combining the estimated number of steps with the estimated computational expense per step shows that the Euclid's algorithm grows quadratically (h2) with the average number of digits h in the initial two numbers a and b. Let h0, h1, ..., hN−1 represent the number of digits in the successive remainders r0, r1, ..., rN−1. Since the number of steps N grows linearly with h, the running time is bounded by $O{\Big (}\sum _{i<N}h_{i}(h_{i}-h_{i+1}+2){\Big )}\subseteq O{\Big (}h\sum _{i<N}(h_{i}-h_{i+1}+2){\Big )}\subseteq O(h(h_{0}+2N))\subseteq O(h^{2}).$ Alternative methods Euclid's algorithm is widely used in practice, especially for small numbers, due to its simplicity.[117] For comparison, the efficiency of alternatives to Euclid's algorithm may be determined. One inefficient approach to finding the GCD of two natural numbers a and b is to calculate all their common divisors; the GCD is then the largest common divisor. The common divisors can be found by dividing both numbers by successive integers from 2 to the smaller number b. The number of steps of this approach grows linearly with b, or exponentially in the number of digits. Another inefficient approach is to find the prime factors of one or both numbers. As noted above, the GCD equals the product of the prime factors shared by the two numbers a and b.[8] Present methods for prime factorization are also inefficient; many modern cryptography systems even rely on that inefficiency.[11] The binary GCD algorithm is an efficient alternative that substitutes division with faster operations by exploiting the binary representation used by computers.[118][119] However, this alternative also scales like O(h²). It is generally faster than the Euclidean algorithm on real computers, even though it scales in the same way.[93] Additional efficiency can be gleaned by examining only the leading digits of the two numbers a and b.[120][121] The binary algorithm can be extended to other bases (k-ary algorithms),[122] with up to fivefold increases in speed.[123] Lehmer's GCD algorithm uses the same general principle as the binary algorithm to speed up GCD computations in arbitrary bases. A recursive approach for very large integers (with more than 25,000 digits) leads to quasilinear integer GCD algorithms,[124] such as those of Schönhage,[125][126] and Stehlé and Zimmermann.[127] These algorithms exploit the 2×2 matrix form of the Euclidean algorithm given above. These quasilinear methods generally scale as O(h (log h)2 (log log h)).[93][94] Generalizations Although the Euclidean algorithm is used to find the greatest common divisor of two natural numbers (positive integers), it may be generalized to the real numbers, and to other mathematical objects, such as polynomials,[128] quadratic integers[129] and Hurwitz quaternions.[130] In the latter cases, the Euclidean algorithm is used to demonstrate the crucial property of unique factorization, i.e., that such numbers can be factored uniquely into irreducible elements, the counterparts of prime numbers. Unique factorization is essential to many proofs of number theory. Rational and real numbers Euclid's algorithm can be applied to real numbers, as described by Euclid in Book 10 of his Elements. The goal of the algorithm is to identify a real number g such that two given real numbers, a and b, are integer multiples of it: a = mg and b = ng, where m and n are integers.[28] This identification is equivalent to finding an integer relation among the real numbers a and b; that is, it determines integers s and t such that sa + tb = 0. If such an equation is possible, a and b are called commensurable lengths, otherwise they are incommensurable lengths.[131][132] The real-number Euclidean algorithm differs from its integer counterpart in two respects. First, the remainders rk are real numbers, although the quotients qk are integers as before. Second, the algorithm is not guaranteed to end in a finite number N of steps. If it does, the fraction a/b is a rational number, i.e., the ratio of two integers ${\frac {a}{b}}={\frac {mg}{ng}}={\frac {m}{n}},$ and can be written as a finite continued fraction [q0; q1, q2, ..., qN]. If the algorithm does not stop, the fraction a/b is an irrational number and can be described by an infinite continued fraction [q0; q1, q2, …].[133] Examples of infinite continued fractions are the golden ratio φ = [1; 1, 1, ...] and the square root of two, √2 = [1; 2, 2, ...].[134] The algorithm is unlikely to stop, since almost all ratios a/b of two real numbers are irrational.[135] An infinite continued fraction may be truncated at a step k [q0; q1, q2, ..., qk] to yield an approximation to a/b that improves as k is increased. The approximation is described by convergents mk/nk; the numerator and denominators are coprime and obey the recurrence relation ${\begin{aligned}m_{k}&=q_{k}m_{k-1}+m_{k-2}\\n_{k}&=q_{k}n_{k-1}+n_{k-2},\end{aligned}}$ where m−1 = n−2 = 1 and m−2 = n−1 = 0 are the initial values of the recursion. The convergent mk/nk is the best rational number approximation to a/b with denominator nk:[136] $\left\|{\frac {a}{b}}-{\frac {m_{k}}{n_{k}}}\right\|<{\frac {1}{n_{k}^{2}}}.$ Polynomials Main article: Polynomial greatest common divisor Polynomials in a single variable x can be added, multiplied and factored into irreducible polynomials, which are the analogs of the prime numbers for integers. The greatest common divisor polynomial g(x) of two polynomials a(x) and b(x) is defined as the product of their shared irreducible polynomials, which can be identified using the Euclidean algorithm.[128] The basic procedure is similar to that for integers. At each step k, a quotient polynomial qk(x) and a remainder polynomial rk(x) are identified to satisfy the recursive equation $r_{k-2}(x)=q_{k}(x)r_{k-1}(x)+r_{k}(x),$ where r−2(x) = a(x) and r−1(x) = b(x). Each quotient polynomial is chosen such that each remainder is either zero or has a degree that is smaller than the degree of its predecessor: deg[rk(x)] < deg[rk−1(x)]. Since the degree is a nonnegative integer, and since it decreases with every step, the Euclidean algorithm concludes in a finite number of steps. The last nonzero remainder is the greatest common divisor of the original two polynomials, a(x) and b(x).[137] For example, consider the following two quartic polynomials, which each factor into two quadratic polynomials ${\begin{aligned}a(x)&=x^{4}-4x^{3}+4x^{2}-3x+14=(x^{2}-5x+7)(x^{2}+x+2)\qquad {\text{and}}\\b(x)&=x^{4}+8x^{3}+12x^{2}+17x+6=(x^{2}+7x+3)(x^{2}+x+2).\end{aligned}}$ Dividing a(x) by b(x) yields a remainder r0(x) = x3 + (2/3)x2 + (5/3)x − (2/3). In the next step, b(x) is divided by r0(x) yielding a remainder r1(x) = x2 + x + 2. Finally, dividing r0(x) by r1(x) yields a zero remainder, indicating that r1(x) is the greatest common divisor polynomial of a(x) and b(x), consistent with their factorization. Many of the applications described above for integers carry over to polynomials.[138] The Euclidean algorithm can be used to solve linear Diophantine equations and Chinese remainder problems for polynomials; continued fractions of polynomials can also be defined. The polynomial Euclidean algorithm has other applications, such as Sturm chains, a method for counting the zeros of a polynomial that lie inside a given real interval.[139] This in turn has applications in several areas, such as the Routh–Hurwitz stability criterion in control theory.[140] Finally, the coefficients of the polynomials need not be drawn from integers, real numbers or even the complex numbers. For example, the coefficients may be drawn from a general field, such as the finite fields GF(p) described above. The corresponding conclusions about the Euclidean algorithm and its applications hold even for such polynomials.[128] Gaussian integers The Gaussian integers are complex numbers of the form α = u + vi, where u and v are ordinary integers[note 2] and i is the square root of negative one.[141] By defining an analog of the Euclidean algorithm, Gaussian integers can be shown to be uniquely factorizable, by the argument above.[42] This unique factorization is helpful in many applications, such as deriving all Pythagorean triples or proving Fermat's theorem on sums of two squares.[141] In general, the Euclidean algorithm is convenient in such applications, but not essential; for example, the theorems can often be proven by other arguments. The Euclidean algorithm developed for two Gaussian integers α and β is nearly the same as that for ordinary integers,[142] but differs in two respects. As before, we set r−2 = α and r−1 = β, and the task at each step k is to identify a quotient qk and a remainder rk such that $r_{k}=r_{k-2}-q_{k}r_{k-1},$ where every remainder is strictly smaller than its predecessor: \|rk\| < \|rk−1\|. The first difference is that the quotients and remainders are themselves Gaussian integers, and thus are complex numbers. The quotients qk are generally found by rounding the real and complex parts of the exact ratio (such as the complex number α/β) to the nearest integers.[142] The second difference lies in the necessity of defining how one complex remainder can be "smaller" than another. To do this, a norm function f(u + vi) = u2 + v2 is defined, which converts every Gaussian integer u + vi into an ordinary integer. After each step k of the Euclidean algorithm, the norm of the remainder f(rk) is smaller than the norm of the preceding remainder, f(rk−1). Since the norm is a nonnegative integer and decreases with every step, the Euclidean algorithm for Gaussian integers ends in a finite number of steps.[143] The final nonzero remainder is gcd(α, β), the Gaussian integer of largest norm that divides both α and β; it is unique up to multiplication by a unit, ±1 or ±i.[144] Many of the other applications of the Euclidean algorithm carry over to Gaussian integers. For example, it can be used to solve linear Diophantine equations and Chinese remainder problems for Gaussian integers;[145] continued fractions of Gaussian integers can also be defined.[142] Euclidean domains A set of elements under two binary operations, denoted as addition and multiplication, is called a Euclidean domain if it forms a commutative ring R and, roughly speaking, if a generalized Euclidean algorithm can be performed on them.[146][147] The two operations of such a ring need not be the addition and multiplication of ordinary arithmetic; rather, they can be more general, such as the operations of a mathematical group or monoid. Nevertheless, these general operations should respect many of the laws governing ordinary arithmetic, such as commutativity, associativity and distributivity. The generalized Euclidean algorithm requires a Euclidean function, i.e., a mapping f from R into the set of nonnegative integers such that, for any two nonzero elements a and b in R, there exist q and r in R such that a = qb + r and f(r) < f(b).[148] Examples of such mappings are the absolute value for integers, the degree for univariate polynomials, and the norm for Gaussian integers above.[149][150] The basic principle is that each step of the algorithm reduces f inexorably; hence, if f can be reduced only a finite number of times, the algorithm must stop in a finite number of steps. This principle relies on the well-ordering property of the non-negative integers, which asserts that every non-empty set of non-negative integers has a smallest member.[151] The fundamental theorem of arithmetic applies to any Euclidean domain: Any number from a Euclidean domain can be factored uniquely into irreducible elements. Any Euclidean domain is a unique factorization domain (UFD), although the converse is not true.[151] The Euclidean domains and the UFD's are subclasses of the GCD domains, domains in which a greatest common divisor of two numbers always exists.[152] In other words, a greatest common divisor may exist (for all pairs of elements in a domain), although it may not be possible to find it using a Euclidean algorithm. A Euclidean domain is always a principal ideal domain (PID), an integral domain in which every ideal is a principal ideal.[153] Again, the converse is not true: not every PID is a Euclidean domain. The unique factorization of Euclidean domains is useful in many applications. For example, the unique factorization of the Gaussian integers is convenient in deriving formulae for all Pythagorean triples and in proving Fermat's theorem on sums of two squares.[141] Unique factorization was also a key element in an attempted proof of Fermat's Last Theorem published in 1847 by Gabriel Lamé, the same mathematician who analyzed the efficiency of Euclid's algorithm, based on a suggestion of Joseph Liouville.[154] Lamé's approach required the unique factorization of numbers of the form x + ωy, where x and y are integers, and ω = e2iπ/n is an nth root of 1, that is, ωn = 1. Although this approach succeeds for some values of n (such as n = 3, the Eisenstein integers), in general such numbers do not factor uniquely. This failure of unique factorization in some cyclotomic fields led Ernst Kummer to the concept of ideal numbers and, later, Richard Dedekind to ideals.[155] Unique factorization of quadratic integers The quadratic integer rings are helpful to illustrate Euclidean domains. Quadratic integers are generalizations of the Gaussian integers in which the imaginary unit i is replaced by a number ω. Thus, they have the form u + vω, where u and v are integers and ω has one of two forms, depending on a parameter D. If D does not equal a multiple of four plus one, then $\omega ={\sqrt {D}}.$ If, however, D does equal a multiple of four plus one, then $\omega ={\frac {1+{\sqrt {D}}}{2}}.$ If the function f corresponds to a norm function, such as that used to order the Gaussian integers above, then the domain is known as norm-Euclidean. The norm-Euclidean rings of quadratic integers are exactly those where D is one of the values −11, −7, −3, −2, −1, 2, 3, 5, 6, 7, 11, 13, 17, 19, 21, 29, 33, 37, 41, 57, or 73.[156][157] The cases D = −1 and D = −3 yield the Gaussian integers and Eisenstein integers, respectively. If f is allowed to be any Euclidean function, then the list of possible values of D for which the domain is Euclidean is not yet known.[158] The first example of a Euclidean domain that was not norm-Euclidean (with D = 69) was published in 1994.[158] In 1973, Weinberger proved that a quadratic integer ring with D > 0 is Euclidean if, and only if, it is a principal ideal domain, provided that the generalized Riemann hypothesis holds.[129] Noncommutative rings The Euclidean algorithm may be applied to some noncommutative rings such as the set of Hurwitz quaternions.[130] Let α and β represent two elements from such a ring. They have a common right divisor δ if α = ξδ and β = ηδ for some choice of ξ and η in the ring. Similarly, they have a common left divisor if α = dξ and β = dη for some choice of ξ and η in the ring. Since multiplication is not commutative, there are two versions of the Euclidean algorithm, one for right divisors and one for left divisors.[130] Choosing the right divisors, the first step in finding the gcd(α, β) by the Euclidean algorithm can be written $\rho _{0}=\alpha -\psi _{0}\beta =(\xi -\psi _{0}\eta )\delta ,$ where ψ0 represents the quotient and ρ0 the remainder. This equation shows that any common right divisor of α and β is likewise a common divisor of the remainder ρ0. The analogous equation for the left divisors would be $\rho _{0}=\alpha -\beta \psi _{0}=\delta (\xi -\eta \psi _{0}).$ With either choice, the process is repeated as above until the greatest common right or left divisor is identified. As in the Euclidean domain, the "size" of the remainder ρ0 (formally, its norm) must be strictly smaller than β, and there must be only a finite number of possible sizes for ρ0, so that the algorithm is guaranteed to terminate.[159] Most of the results for the GCD carry over to noncommutative numbers. For example, Bézout's identity states that the right gcd(α, β) can be expressed as a linear combination of α and β.[160] In other words, there are numbers σ and τ such that $\Gamma _{\text{right}}=\sigma \alpha +\tau \beta .$ The analogous identity for the left GCD is nearly the same: $\Gamma _{\text{left}}=\alpha \sigma +\beta \tau .$ Bézout's identity can be used to solve Diophantine equations. For instance, one of the standard proofs of Lagrange's four-square theorem, that every positive integer can be represented as a sum of four squares, is based on quaternion GCDs in this way.[159] See also • Euclidean rhythm, a method for using the Euclidean algorithm to generate musical rhythms Notes 1. Some widely used textbooks, such as I. N. Herstein's Topics in Algebra and Serge Lang's Algebra, use the term "Euclidean algorithm" to refer to Euclidean division 2. The phrase "ordinary integer" is commonly used for distinguishing usual integers from Gaussian integers, and more generally from algebraic integers. References 1. Lamé, Gabriel (1844). "Note sur la limite du nombre des divisions dans la recherche du plus grand commun diviseur entre deux nombres entiers". Comptes Rendus des Séances de l'Académie des Sciences (in French). 19: 867–870. 2. Shallit, Jeffrey (1994-11-01). "Origins of the analysis of the Euclidean algorithm". 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Exercise 24, p. 205. ISBN 9780134912820. State and prove an analogue of the Chinese remainder theorem for the Gaussian integers. 146. Stark 1978, p. 290 147. Cohn 1962, pp. 104–105 148. Lauritzen, Niels (2003). Concrete Abstract Algebra: From Numbers to Gröbner Bases. Cambridge University Press. p. 130. ISBN 9780521534109. 149. Lauritzen (2003), p. 132 150. Lauritzen (2003), p. 161 151. Sharpe, David (1987). Rings and Factorization. Cambridge University Press. p. 55. ISBN 9780521337182. 152. Sharpe (1987), p. 52 153. Lauritzen (2003), p. 131 154. Lamé, G. (1847). "Mémoire sur la résolution, en nombres complexes, de l'équation An + Bn + Cn = 0". J. Math. Pures Appl. (in French). 12: 172–184. 155. Edwards, H. (2000). Fermat's last theorem: a genetic introduction to algebraic number theory. Springer. p. 76. 156. Cohn 1962, pp. 104–110 157. LeVeque, W. J. (2002) [1956]. Topics in Number Theory, Volumes I and II. New York: Dover Publications. pp. II:57, 81. ISBN 978-0-486-42539-9. Zbl 1009.11001. 158. Clark, D. A. (1994). "A quadratic field which is Euclidean but not norm-Euclidean". Manuscripta Mathematica. 83: 327–330. doi:10.1007/BF02567617. S2CID 895185. Zbl 0817.11047. 159. Davidoff, Giuliana; Sarnak, Peter; Valette, Alain (2003). "2.6 The Arithmetic of Integer Quaternions". Elementary Number Theory, Group Theory and Ramanujan Graphs. London Mathematical Society Student Texts. Vol. 55. Cambridge University Press. pp. 59–70. ISBN 9780521531436. 160. Ribenboim, Paulo (2001). Classical Theory of Algebraic Numbers. Universitext. Springer-Verlag. p. 104. ISBN 9780387950709. Bibliography • Cohen, H. (1993). A Course in Computational Algebraic Number Theory. New York: Springer-Verlag. ISBN 0-387-55640-0. • Cohn, H. (1962). Advanced Number Theory. New York: Dover. ISBN 0-486-64023-X. • Cox, D.; Little, J.; O'Shea, D. (1997). Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra (2nd ed.). Springer-Verlag. ISBN 0-387-94680-2. • Crandall, R.; Pomerance, C. (2001). Prime Numbers: A Computational Perspective (1st ed.). New York: Springer-Verlag. ISBN 0-387-94777-9. • Lejeune Dirichlet, P. G. (1894). Dedekind, Richard (ed.). Vorlesungen über Zahlentheorie (Lectures on Number Theory) (in German). Braunschweig: Vieweg. LCCN 03005859. OCLC 490186017.. See also Vorlesungen über Zahlentheorie • Knuth, D. E. (1997). The Art of Computer Programming, Volume 2: Seminumerical Algorithms (3rd ed.). Addison–Wesley. ISBN 0-201-89684-2. • LeVeque, W. J. (1996) [1977]. Fundamentals of Number Theory. New York: Dover. ISBN 0-486-68906-9. • Mollin, R. A. (2008). Fundamental Number Theory with Applications (2nd ed.). Boca Raton: Chapman & Hall/CRC. ISBN 978-1-4200-6659-3. • Ore, O. (1948). Number Theory and Its History. New York: McGraw–Hill. • Rosen, K. H. (2000). Elementary Number Theory and its Applications (4th ed.). Reading, MA: Addison–Wesley. ISBN 0-201-87073-8. • Schroeder, M. (2005). Number Theory in Science and Communication (4th ed.). Springer-Verlag. ISBN 0-387-15800-6. • Stark, H. (1978). An Introduction to Number Theory. MIT Press. ISBN 0-262-69060-8. • Stillwell, J. (1997). Numbers and Geometry. New York: Springer-Verlag. ISBN 0-387-98289-2. • Stillwell, J. (2003). Elements of Number Theory. New York: Springer-Verlag. ISBN 0-387-95587-9. • Tattersall, J. J. (2005). Elementary Number Theory in Nine Chapters. Cambridge: Cambridge University Press. ISBN 978-0-521-85014-8. External links Wikimedia Commons has media related to Euclidean algorithm. • Demonstrations of Euclid's algorithm • Weisstein, Eric W. "Euclidean Algorithm". MathWorld. • Euclid's Algorithm at cut-the-knot • Euclid's algorithm at PlanetMath. • The Euclidean Algorithm at MathPages • Euclid's Game at cut-the-knot • Music and Euclid's algorithm Number-theoretic algorithms Primality tests • AKS • APR • Baillie–PSW • Elliptic curve • Pocklington • Fermat • Lucas • Lucas–Lehmer • Lucas–Lehmer–Riesel • Proth's theorem • Pépin's • Quadratic Frobenius • Solovay–Strassen • Miller–Rabin Prime-generating • Sieve of Atkin • Sieve of Eratosthenes • Sieve of Pritchard • Sieve of Sundaram • Wheel factorization Integer factorization • Continued fraction (CFRAC) • Dixon's • Lenstra elliptic curve (ECM) • Euler's • Pollard's rho • p − 1 • p + 1 • Quadratic sieve (QS) • General number field sieve (GNFS) • Special number field sieve (SNFS) • Rational sieve • Fermat's • Shanks's square forms • Trial division • Shor's Multiplication • Ancient Egyptian • Long • Karatsuba • Toom–Cook • Schönhage–Strassen • Fürer's Euclidean division • Binary • Chunking • Fourier • Goldschmidt • Newton-Raphson • Long • Short • SRT Discrete logarithm • Baby-step giant-step • Pollard rho • Pollard kangaroo • Pohlig–Hellman • Index calculus • Function field sieve Greatest common divisor • Binary • Euclidean • Extended Euclidean • Lehmer's Modular square root • Cipolla • Pocklington's • Tonelli–Shanks • Berlekamp • Kunerth Other algorithms • Chakravala • Cornacchia • Exponentiation by squaring • Integer square root • Integer relation (LLL; KZ) • Modular exponentiation • Montgomery reduction • Schoof • Trachtenberg system • Italics indicate that algorithm is for numbers of special forms Ancient Greek mathematics Mathematicians (timeline) • Anaxagoras • Anthemius • Archytas • Aristaeus the Elder • Aristarchus • Aristotle • Apollonius • Archimedes • Autolycus • Bion • Bryson • Callippus • Carpus • Chrysippus • Cleomedes • Conon • Ctesibius • Democritus • Dicaearchus • Diocles • Diophantus • Dinostratus • Dionysodorus • Domninus • Eratosthenes • Eudemus • Euclid • Eudoxus • Eutocius • Geminus • Heliodorus • Heron • Hipparchus • Hippasus • Hippias • Hippocrates • Hypatia • Hypsicles • Isidore of Miletus • Leon • Marinus • Menaechmus • Menelaus • Metrodorus • Nicomachus • Nicomedes • Nicoteles • Oenopides • Pappus • Perseus • Philolaus • Philon • Philonides • Plato • Porphyry • Posidonius • Proclus • Ptolemy • Pythagoras • Serenus • Simplicius • Sosigenes • Sporus • Thales • Theaetetus • Theano • Theodorus • Theodosius • Theon of Alexandria • Theon of Smyrna • Thymaridas • Xenocrates • Zeno of Elea • Zeno of Sidon • Zenodorus Treatises • Almagest • Archimedes Palimpsest • Arithmetica • Conics (Apollonius) • Catoptrics • Data (Euclid) • Elements (Euclid) • Measurement of a Circle • On Conoids and Spheroids • On the Sizes and Distances (Aristarchus) • On Sizes and Distances (Hipparchus) • On the Moving Sphere (Autolycus) • Optics (Euclid) • On Spirals • On the Sphere and Cylinder • Ostomachion • Planisphaerium • Sphaerics • The Quadrature of the Parabola • The Sand Reckoner Problems • Constructible numbers • Angle trisection • Doubling the cube • Squaring the circle • Problem of Apollonius Concepts and definitions • Angle • Central • Inscribed • Axiomatic system • Axiom • Chord • Circles of Apollonius • Apollonian circles • Apollonian gasket • Circumscribed circle • Commensurability • Diophantine equation • Doctrine of proportionality • Euclidean geometry • Golden ratio • Greek numerals • Incircle and excircles of a triangle • Method of exhaustion • Parallel postulate • Platonic solid • Lune of Hippocrates • Quadratrix of Hippias • Regular polygon • Straightedge and compass construction • Triangle center Results In Elements • Angle bisector theorem • Exterior angle theorem • Euclidean algorithm • Euclid's theorem • Geometric mean theorem • Greek geometric algebra • Hinge theorem • Inscribed angle theorem • Intercept theorem • Intersecting chords theorem • Intersecting secants theorem • Law of cosines • Pons asinorum • Pythagorean theorem • Tangent-secant theorem • Thales's theorem • Theorem of the gnomon Apollonius • Apollonius's theorem Other • Aristarchus's inequality • Crossbar theorem • Heron's formula • Irrational numbers • Law of sines • Menelaus's theorem • Pappus's area theorem • Problem II.8 of Arithmetica • Ptolemy's inequality • Ptolemy's table of chords • Ptolemy's theorem • Spiral of Theodorus Centers • Cyrene • Mouseion of Alexandria • Platonic Academy Related • Ancient Greek astronomy • Attic numerals • Greek numerals • Latin translations of the 12th century • Non-Euclidean geometry • Philosophy of mathematics • Neusis construction History of • A History of Greek Mathematics • by Thomas Heath • algebra • timeline • arithmetic • timeline • calculus • timeline • geometry • timeline • logic • timeline • mathematics • timeline • numbers • prehistoric counting • numeral systems • list Other cultures • Arabian/Islamic • Babylonian • Chinese • Egyptian • Incan • Indian • Japanese Ancient Greece portal • Mathematics portal Authority control: National • Germany	Wikipedia
Evdokimov's algorithm In computational number theory, Evdokimov's algorithm, named after Sergei Evdokimov, is the asymptotically fastest known algorithm for factorization of polynomials (until 2019). It can factorize a one-variable polynomial of degree $n$ over an explicitly given finite field of cardinality $q$. Assuming the generalized Riemann hypothesis the algorithm runs in deterministic time $(n^{\log n}\log q)^{{\mathcal {O}}(1)}$ [1] (see Big O notation). This is an improvement of both Berlekamp's algorithm and Rónyai's algorithm[2] in the sense that the first algorithm is polynomial for small characteristic of the field, whearas the second one is polynomial for small $n$; however, both of them are exponential if no restriction is made. The factorization of a polynomial $f$ over a ground field $k$ is reduced to the case when $f$ has no multiple roots and is completely splitting over $k$ (i.e. $f$ has $n$ distinct roots in $k$). In order to find a root of $f$ in this case, the algorithm deals with polynomials not only over the ground field $k$ but also over a completely splitting semisimple algebra over $k$ (an example of such an algebra is given by $k[X]/(f)=k[A]$, where $A=X{\bmod {f}}$). The main problem here is to find efficiently a nonzero zero-divisor in the algebra. The GRH is used only to take roots in finite fields in polynomial time. Thus the Evdokimov algorithm, in fact, solves a polynomial equation over a finite field "by radicals" in quasipolynomial time, see Time complexity. The analyses of Evdokimov's algorithm is closely related with some problems in the association scheme theory. With the help of this approach, it was proved [3] that if $n$ is a prime and $n-1$ has a ‘large’ $r$-smooth divisor $s$, then a modification of the Evdokimov algorithm finds a nontrivial factor of the polynomial $f$ in deterministic $\operatorname {poly} (n^{r},\log q)$ time, assuming GRH and that $s=\Omega \left({\sqrt {n/2^{r}}}\,\right)$. References 1. Evdokimov, Sergei (1994), "Factorization of polynomials over finite fields in subexponential time under GRH", Algorithmic Number Theory, Lecture Notes in Computer Science, vol. 877, pp. 209–219, doi:10.1007/3-540-58691-1_58, ISBN 978-3-540-58691-3 2. Rónyai, Lajos (1988), "Factoring polynomials over finite fields", Journal of Algorithms, 9 (3): 391–400, doi:10.1016/0196-6774(88)90029-6, S2CID 16360930 3. Arora, Manuel; Ivanyos, Gabor; Karpinski, Marek; Saxena, Nitin (2014), "Deterministic polynomial factoring and association schemes", LMS Journal of Computation and Mathematics, 17: 123–140, arXiv:1205.5653, doi:10.1112/S1461157013000296, S2CID 31522031 Further reading • Shparlinski, I. (1999). Finite Fields: Theory and Computation. The Meeting Point of Number Theory, Computer Science, Coding Theory and Cryptography. Mathematics and Its Applications. Vol. 477. Springer Verlag.	Wikipedia
FEE method In mathematics, the FEE method, or fast E-function evaluation method, is the method of fast summation of series of a special form. It was constructed in 1990 by Ekaterina Karatsuba[1][2] and is so-named because it makes fast computations of the Siegel E-functions possible, in particular of $e^{x}$. A class of functions, which are "similar to the exponential function," was given the name "E-functions" by Carl Ludwig Siegel.[3] Among these functions are such special functions as the hypergeometric function, cylinder, spherical functions and so on. Using the FEE, it is possible to prove the following theorem: Theorem: Let $y=f(x)$ be an elementary transcendental function, that is the exponential function, or a trigonometric function, or an elementary algebraic function, or their superposition, or their inverse, or a superposition of the inverses. Then $s_{f}(n)=O(M(n)\log ^{2}n).\,$ Here $s_{f}(n)$ is the complexity of computation (bit) of the function $f(x)$ with accuracy up to $n$ digits, $M(n)$ is the complexity of multiplication of two $n$-digit integers. The algorithms based on the method FEE include the algorithms for fast calculation of any elementary transcendental function for any value of the argument, the classical constants e, $\pi ,$ the Euler constant $\gamma ,$ the Catalan and the Apéry constants,[4] such higher transcendental functions as the Euler gamma function and its derivatives, the hypergeometric,[5] spherical, cylinder (including the Bessel)[6] functions and some other functions for algebraic values of the argument and parameters, the Riemann zeta function for integer values of the argument[7][8] and the Hurwitz zeta function for integer argument and algebraic values of the parameter,[9] and also such special integrals as the integral of probability, the Fresnel integrals, the integral exponential function, the trigonometric integrals, and some other integrals[10] for algebraic values of the argument with the complexity bound which is close to the optimal one, namely $s_{f}(n)=O(M(n)\log ^{2}n).\,$ At present, only the FEE makes it possible to calculate fast the values of the functions from the class of higher transcendental functions,[11] certain special integrals of mathematical physics and such classical constants as Euler's, Catalan's[12] and Apéry's constants. An additional advantage of the method FEE is the possibility of parallelizing the algorithms based on the FEE. FEE computation of classical constants For fast evaluation of the constant $\pi ,$ one can use the Euler formula ${\frac {\pi }{4}}=\arctan {\frac {1}{2}}+\arctan {\frac {1}{3}},$ and apply the FEE to sum the Taylor series for $\arctan {\frac {1}{2}}={\frac {1}{1\cdot 2}}-{\frac {1}{3\cdot 2^{3}}}+\cdots +{\frac {(-1)^{r-1}}{(2r-1)2^{2r-1}}}+R_{1},$ $\arctan {\frac {1}{3}}={\frac {1}{1\cdot 3}}-{\frac {1}{3\cdot 3^{3}}}+\cdots +{\frac {(-1)^{r-1}}{(2r-1)3^{2r-1}}}+R_{2},$ with the remainder terms $R_{1},$ $R_{2},$ which satisfy the bounds $\|R_{1}\|\leq {\frac {4}{5}}{\frac {1}{2r+1}}{\frac {1}{2^{2r+1}}};$ $\|R_{2}\|\leq {\frac {9}{10}}{\frac {1}{2r+1}}{\frac {1}{3^{2r+1}}};$ and for $r=n,\,$ $4(\|R_{1}\|+\|R_{2}\|)\ <\ 2^{-n}.$ To calculate $\pi $ by the FEE it is possible to use also other approximations[13] In all cases the complexity is $s_{\pi }=O(M(n)\log ^{2}n).\,$ To compute the Euler constant gamma with accuracy up to $n$ digits, it is necessary to sum by the FEE two series. Namely, for $m=6n,\quad k=n,\quad k\geq 1,\,$ $\gamma =-\log n\sum _{r=0}^{12n}{\frac {(-1)^{r}n^{r+1}}{(r+1)!}}+\sum _{r=0}^{12n}{\frac {(-1)^{r}n^{r+1}}{(r+1)!(r+1)}}+O(2^{-n}).$ The complexity is $s_{\gamma }=O(M(n)\log ^{2}n).\,$ To evaluate fast the constant $\gamma $ it is possible to apply the FEE to other approximations.[14] FEE computation of certain power series By the FEE the two following series are calculated fast: $f_{1}=f_{1}(z)=\sum _{j=0}^{\infty }{\frac {a(j)}{b(j)}}z^{j},$ $f_{2}=f_{2}(z)=\sum _{j=0}^{\infty }{\frac {a(j)}{b(j)}}{\frac {z^{j}}{j!}},$ under the assumption that $a(j),\quad b(j)$ are integers, $\|a(j)\|+\|b(j)\|\leq (Cj)^{K};\quad \|z\|\ <\ 1;\quad K$ and $C$ are constants, and $z$ is an algebraic number. The complexity of the evaluation of the series is $s_{f_{1}}(n)=O\left(M(n)\log ^{2}n\right),\,$ $s_{f_{2}}(n)=O\left(M(n)\log n\right).$ FEE calculation of the classical constant e For the evaluation of the constant $e$ take $m=2^{k},\quad k\geq 1$, terms of the Taylor series for $e,$ $e=1+{\frac {1}{1!}}+{\frac {1}{2!}}+\cdots +{\frac {1}{(m-1)!}}+R_{m}.$ Here we choose $m$, requiring that for the remainder $R_{m}$ the inequality $R_{m}\leq 2^{-n-1}$ is fulfilled. This is the case, for example, when $m\geq {\frac {4n}{\log n}}.$ Thus, we take $m=2^{k}$ such that the natural number $k$ is determined by the inequalities: $2^{k}\geq {\frac {4n}{\log n}}>2^{k-1}.$ We calculate the sum $S=1+{\frac {1}{1!}}+{\frac {1}{2!}}+\cdots +{\frac {1}{(m-1)!}}=\sum _{j=0}^{m-1}{\frac {1}{(m-1-j)!}},$ in $k$ steps of the following process. Step 1. Combining in $S$ the summands sequentially in pairs we carry out of the brackets the "obvious" common factor and obtain ${\begin{aligned}S&=\left({\frac {1}{(m-1)!}}+{\frac {1}{(m-2)!}}\right)+\left({\frac {1}{(m-3)!}}+{\frac {1}{(m-4)!}}\right)+\cdots \\&={\frac {1}{(m-1)!}}(1+m-1)+{\frac {1}{(m-3)!}}(1+m-3)+\cdots .\end{aligned}}$ We shall compute only integer values of the expressions in the parentheses, that is the values $m,m-2,m-4,\dots .\,$ Thus, at the first step the sum $S$ is into $S=S(1)=\sum _{j=0}^{m_{1}-1}{\frac {1}{(m-1-2j)!}}\alpha _{m_{1}-j}(1),$ $m_{1}={\frac {m}{2}},m=2^{k},k\geq 1.$ At the first step ${\frac {m}{2}}$ integers of the form $\alpha _{m_{1}-j}(1)=m-2j,\quad j=0,1,\dots ,m_{1}-1,$ are calculated. After that we act in a similar way: combining on each step the summands of the sum $S$ sequentially in pairs, we take out of the brackets the 'obvious' common factor and compute only the integer values of the expressions in the brackets. Assume that the first $i$ steps of this process are completed. Step $i+1$ ($i+1\leq k$). $S=S(i+1)=\sum _{j=0}^{m_{i+1}-1}{\frac {1}{(m-1-2^{i+1}j)!}}\alpha _{m_{i+1}-j}(i+1),$ $m_{i+1}={\frac {m_{i}}{2}}={\frac {m}{2^{i+1}}},$ we compute only ${\frac {m}{2^{i+1}}}$ integers of the form $\alpha _{m_{i+1}-j}(i+1)=\alpha _{m_{i}-2j}(i)+\alpha _{m_{i}-(2j+1)}(i){\frac {(m-1-2^{i+1}j)!}{(m-1-2^{i}-2^{i+1}j)!}},$ $j=0,1,\dots ,\quad m_{i+1}-1,\quad m=2^{k},\quad k\geq i+1.$ Here ${\frac {(m-1-2^{i+1}j)!}{(m-1-2^{i}-2^{i+1}j)!}}$ is the product of $2^{i}$ integers. Etc. Step $k$, the last one. We compute one integer value $\alpha _{1}(k),$ we compute, using the fast algorithm described above the value $(m-1)!,$ and make one division of the integer $\alpha _{1}(k)$ by the integer $(m-1)!,$ with accuracy up to $n$ digits. The obtained result is the sum $S,$ or the constant $e$ up to $n$ digits. The complexity of all computations is $O\left(M(m)\log ^{2}m\right)=O\left(M(n)\log n\right).\,$ See also • Fast algorithms • AGM method • Computational complexity References 1. E. A. Karatsuba, Fast evaluations of transcendental functions. Probl. Peredachi Informat., Vol. 27, No. 4, (1991) 2. D. W. Lozier and F. W. J. Olver, Numerical Evaluation of Special Functions. Mathematics of Computation 1943–1993: A Half-Century of Computational Mathematics, W. Gautschi, eds., Proc. Sympos. Applied Mathematics, AMS, Vol. 48 (1994). 3. C. L. Siegel, Transcendental numbers. Princeton University Press, Princeton (1949). 4. Karatsuba E. A., Fast evaluation of $\zeta (3)$, Probl. Peredachi Informat., Vol. 29, No. 1 (1993) 5. Ekatharine A. Karatsuba, Fast evaluation of hypergeometric function by FEE. Computational Methods and Function Theory (CMFT'97), N. Papamichael, St. Ruscheweyh and E. B. Saff, eds., World Sc. Pub. (1999) 6. Catherine A. Karatsuba, Fast evaluation of Bessel functions. Integral Transforms and Special Functions, Vol. 1, No. 4 (1993) 7. E. A. Karatsuba, Fast Evaluation of Riemann zeta-function $\zeta (s)$ for integer values of argument $s$. Probl. Peredachi Informat., Vol. 31, No. 4 (1995). 8. J. M. Borwein, D. M. Bradley and R. E. Crandall, Computational strategies for the Riemann zeta function. J. of Comput. Appl. Math., Vol. 121, No. 1–2 (2000). 9. E. A. Karatsuba, Fast evaluation of Hurwitz zeta function and Dirichlet $L$-series, Problem. Peredachi Informat., Vol. 34, No. 4, pp. 342–353, (1998). 10. E. A. Karatsuba, Fast computation of some special integrals of mathematical physics. Scientific Computing, Validated Numerics, Interval Methods, W. Kramer, J. W. von Gudenberg, eds.(2001). 11. E. Bach, The complexity of number-theoretic constants. Info. Proc. Letters, No. 62 (1997). 12. E. A. Karatsuba, Fast computation of $\zeta(3)$ and of some special integrals, using the polylogarithms, the Ramanujan formula and its generalization. J. of Numerical Mathematics BIT, Vol. 41, No. 4 (2001). 13. D. H. Bailey, P. B. Borwein and S. Plouffe, On the rapid computation of various polylogarithmic constants. Math. Comp., Vol. 66 (1997). 14. R. P. Brent and E. M. McMillan, Some new algorithms for high-precision computation of Euler's constant. Math. Comp., Vol. 34 (1980). External links • http://www.ccas.ru/personal/karatsuba/divcen.htm • http://www.ccas.ru/personal/karatsuba/algen.htm	Wikipedia
The Fibonacci Association The Fibonacci Association is a mathematical organization that specializes in the Fibonacci number sequence and a wide variety of related subjects, generalizations, and applications, including recurrence relations, combinatorial identities, binomial coefficients, prime numbers, pseudoprimes, continued fractions, the golden ratio, linear algebra, geometry, real analysis, and complex analysis. History The organization was founded in 1963 by Brother Alfred Brousseau, F.S.C. of St. Mary's College (Moraga, California) and Verner E. Hoggatt Jr. of San Jose State College (now San Jose State University). Details regarding the early history of The Fibonacci Association are given Marjorie Bicknell-Johnson's "A Short History of The Fibonacci Quarterly", published in The Fibonacci Quarterly 25:1 (February 1987) 2-5, during the Twenty-Fifth Anniversary year of the journal. Publications Since the year of its founding, the Fibonacci Association has published an international mathematical journal, The Fibonacci Quarterly . The Fibonacci Association also publishes Proceedings for its international conferences, held every two years since 1984. The 2008 conference, formally entitled the Thirteenth International Conference on Fibonacci Numbers and Their Applications, took place at the University of Patras (Greece), preceded by conferences at San Francisco State University (USA, 2006), Technische Universität Braunschweig (Germany, 2004), Northern Arizona University (USA, 2002), and Institut Supérieur de Technologie (Luxemburg, 2000). The 2010 Conference was held at the Instituto de Matemáticas de la UNAM, Morelia, Mexico, as announced at the Fibonacci Association website: . The 2012 Conference will take place during June 25–30 at the Institute of Mathematics and Informatics, Eszerházy Károly College, Eger, Hungary, with keynote speaker Neil Sloane, founder of the Encyclopedia of Integer Sequences. External links • The Official website of the Fibonacci Association • The Fibonacci Quarterly • Up-to-date list of issues of The Fibonacci Quarterly Fibonacci Books • Liber Abaci (1202) • The Book of Squares (1225) Theories • Fibonacci sequence • Greedy algorithm for Egyptian fractions Related • Fibonacci numbers in popular culture • List of things named after Fibonacci • Generalizations of Fibonacci numbers • The Fibonacci Association • Fibonacci Quarterly	Wikipedia
The Geometry of Musical Rhythm The Geometry of Musical Rhythm: What Makes a "Good" Rhythm Good? is a book on the mathematics of rhythms and drum beats. It was written by Godfried Toussaint, and published by Chapman & Hall/CRC in 2013 and in an expanded second edition in 2020. The Basic Library List Committee of the Mathematical Association of America has suggested its inclusion in undergraduate mathematics libraries.[1] Author Godfried Toussaint (1944–2019) was a Belgian–Canadian computer scientist who worked as a professor of computer science for McGill University and New York University. His main professional expertise was in computational geometry,[2] but he was also a jazz drummer,[3] held a long-term interest in the mathematics of music and musical rhythm, and since 2005 held an affiliation as a researcher in the Centre for Interdisciplinary Research in Music Media and Technology in the Schulich School of Music at McGill.[2] In 2009 he visited Harvard University as a Radcliffe Fellow in advancement of his research in musical rhythm.[2][3] Topics In order to study rhythms mathematically, Toussaint abstracts away many of their features that are important musically, involving the sounds or strengths of the individual beats, the phasing of the beats, hierarchically-structured rhythms, or the possibility of music that changes from one rhythm to another. The information that remains describes the beats of each bar (an evenly-spaced cyclic sequence of times) as being either on-beats (times at which a beat is emphasized in the musical performance) or off-beats (times at which it is skipped or performed only weakly). This can be represented combinatorially as a necklace, an equivalence class of binary sequences under rotations, with true binary values representing on-beats and false representing off-beats. Alternatively, Toussaint uses a geometric representation as a convex polygon, the convex hull of a subset of the vertices of a regular polygon, where the vertices of the hull represent times when a beat is performed; two rhythms are considered the same if the corresponding polygons are congruent.[4][5] As an example, reviewer William Sethares (himself a music theorist and engineer) presents a representation of this type for the tresillo rhythm, in which three beats are hit out of an eight-beat bar, with two long gaps and one short gap between each beat. The tresillo may be represented geometrically as an isosceles triangle, formed from three vertices of a regular octahedron, with the two long sides and one short side of the triangle corresponding to the gaps between beats. In the figure, the conventional start to a tresillo bar, the beat before the first of its two longer gaps, is at the top vertex, and the chronological progression of beats corresponds to the clockwise ordering of vertices around the polygon.[5] The book uses this method to study and classify existing rhythms from world music, to analyze their mathematical properties (for instance, the fact that many of these rhythms have a spacing between their beats that, like the tresillo, is near-uniform but not exactly uniform), to devise algorithms that can generate similar nearly uniformly spaced beat patterns for arbitrary numbers of beats in the rhythm and in the bar, to measure the similarity between rhythms, to cluster rhythms into related groups using their similarities, and ultimately to try to capture the suitability of a rhythm for use in music by a mathematical formula.[5][6] Audience and reception Toussaint has used this book as auxiliary material in introductory computer programming courses, to provide programming tasks for the students.[5] It is accessible to readers without much background in mathematics or music theory,[4][7] and Setheres writes that it "would make a great introduction to ideas from mathematics and computer science for the musically inspired student".[5] Reviewer Russell Jay Hendel suggests that, as well as being read for pleasure, it could be a textbook for an advanced elective for a mathematics student, or a general education course in mathematics for non-mathematicians.[1] Professionals in ethnomusicology, music history, the psychology of music, music theory, and musical composition may also find it of interest.[7] Despite concerns with some misused terminology, with "naïveté towards core music theory", and with a mismatch between the visual representation of rhythm and its aural perception, music theorist Mark Gotham calls the book "a substantial contribution to a field that still lags behind the more developed theoretical literature on pitch".[7] And although reviewer Juan G. Escudero complains that the mathematical abstractions of the book misses many important aspects of music and musical rhythm, and that many rhythmic features of contemporary classical music have been overlooked, he concludes that "transdisciplinary efforts of this kind are necessary".[4] Reviewer Ilhand Izmirli calls the book "delightful, informative, and innovative".[6] Hendel adds that the book's presentation of its material as speculative and exploratory, rather than as definitive and completed, is "exactly what [mathematics] students need".[1] References 1. Hendel, Russell Jay (May 2013), "Review of The Geometry of Musical Rhythm", MAA Reviews, Mathematical Association of America 2. Toussaint, Godfried, Biography, McGill University, retrieved 2020-05-24 3. Ireland, Corydon (October 19, 2009), "Hunting for rhythm's DNA: Computational geometry unlocks a musical phylogeny", Harvard Gazette 4. Escudero, Juan G., "Review of The Geometry of Musical Rhythm", zbMATH, Zbl 1275.00024 5. Sethares, William A. (April 2014), "Review of The Geometry of Musical Rhythm", Journal of Mathematics and the Arts, 8 (3–4): 135–137, doi:10.1080/17513472.2014.906116 6. Izmirli, Ilhan M., "Review of The Geometry of Musical Rhythm", Mathematical Reviews, MR 3012379 7. Gotham, Mark (June 2013), "Review of The Geometry of Musical Rhythm", Music Theory Online, 19 (2)	Wikipedia
The Geometry of Numbers The Geometry of Numbers is a book on the geometry of numbers, an area of mathematics in which the geometry of lattices, repeating sets of points in the plane or higher dimensions, is used to derive results in number theory. It was written by Carl D. Olds, Anneli Cahn Lax, and Giuliana Davidoff, and published by the Mathematical Association of America in 2000 as volume 41 of their Anneli Lax New Mathematical Library book series. Authorship and publication history The Geometry of Numbers is based on a book manuscript that Carl D. Olds, a New Zealand-born mathematician working in California at San Jose State University, was still writing when he died in 1979. Anneli Cahn Lax, the editor of the New Mathematical Library of the Mathematical Association of America, took up the task of editing it, but it remained unfinished when she died in 1999. Finally, Giuliana Davidoff took over the project, and saw it through to publication in 2000.[1][2] Topics The Geometry of Numbers is relatively short,[3][4] and is divided into two parts. The first part applies number theory to the geometry of lattices, and the second applies results on lattices to number theory.[1] Topics in the first part include the relation between the maximum distance between parallel lines that are not separated by any point of a lattice and the slope of the lines,[5] Pick's theorem relating the area of a lattice polygon to the number of lattice points it contains,[4] and the Gauss circle problem of counting lattice points in a circle centered at the origin of the plane.[1] The second part begins with Minkowski's theorem, that centrally symmetric convex sets of large enough area (or volume in higher dimensions) necessarily contain a nonzero lattice point. It applies this to Diophantine approximation, the problem of accurately approximating one or more irrational numbers by rational numbers. After another chapter on the linear transformations of lattices, the book studies the problem of finding the smallest nonzero values of quadratic forms, and Lagrange's four-square theorem, the theorem that every non-negative integer can be represented as a sum of four squares of integers. The final two chapters concern Blichfeldt's theorem, that bounded planar regions with area $A$ can be translated to cover at least $\lceil A\rceil $ lattice points, and additional results in Diophantine approximation.[1] The chapters on Minkowski's theorem and Blichfeldt's theorem, particularly, have been called the "foundation stones" of the book by reviewer Philip J. Davis.[2] An appendix by Peter Lax concerns the Gaussian integers.[6] A second appendix concerns lattice-based methods for packing problems including circle packing and, in higher dimensions, sphere packing.[4][6] The book closes with biographies of Hermann Minkowski and Hans Frederick Blichfeldt.[6] Audience and reception The Geometry of Numbers is intended for secondary-school and undergraduate mathematics students, although it may be too advanced for the secondary-school students; it contains exercises making it suitable for classroom use.[3] It has been described as "expository",[4] "self-contained",[1][3][4] and "readable".[6] However, reviewer Henry Cohn notes several copyediting oversights, complains about its selection of topics, in which "curiosities are placed on an equal footing with deep results", and misses certain well-known examples which were not included. Despite this, he recommends the book to readers who are not yet ready for more advanced treatments of this material and wish to see "some beautiful mathematics".[5] References 1. Hoare, Graham (July 2002), "Review of The Geometry of Numbers", The Mathematical Gazette, 86 (506): 368–369, doi:10.2307/3621910, JSTOR 3621910 2. Davis, Philip J. (October 2001), "From spots and dots to deep stuff (review of The Geometry of Numbers)", SIAM News, vol. 34, no. 8 3. Giesbrecht, Edwin C. (February 2002), "Review of The Geometry of Numbers", The Mathematics Teacher, 95 (2): 156, 158, JSTOR 20870960 4. Wills, Jörg M., "Review of The Geometry of Numbers", zbMATH, Zbl 0967.11023 5. Cohn, Henry (December 2002), "Review of The Geometry of Numbers", MAA Reviews, Mathematical Association of America 6. Burger, Edward B. (2002), "Review of The Geometry of Numbers", MathSciNet, MR 1817689	Wikipedia
The Geometry of an Art The Geometry of an Art: The History of the Mathematical Theory of Perspective from Alberti to Monge is a book in the history of mathematics, on the mathematics of graphical perspective. It was written by Kirsti Andersen, and published in 2007 by Springer-Verlag in their book series Sources and Studies in the History of Mathematics and Physical Sciences. The Geometry of an Art: The History of the Mathematical Theory of Perspective from Alberti to Monge AuthorKirsti Andersen LanguageEnglish SeriesSources and Studies in the History of Mathematics and Physical Sciences SubjectHistory of the mathematics of graphical perspective PublisherSpringer-Verlag Publication date 2007 Pages812 ISBN978-0-387-25961-1 Topics This book covers a wide span of mathematical history, from 1435 to 1800, and a wide field of "around 250 publications by more than 200 authors".[1] After three introductory chapters on the beginnings of perspective with the works of Leon Battista Alberti, Piero della Francesca, Leonardo da Vinci, and others from their time, the remainder of the book is organized geographically rather than chronologically, in order to set the works it discusses into their local context.[2] Thus, Chapter 4 covers the spread of perspective among the artists and artisans of 15th-century Italy, including the works of Luca Pacioli and Daniele Barbaro, while Chapter 5 concerns developments in Northern Europe in the same timeframe by Albrecht Dürer, Wenzel Jamnitzer, and Paul Vredeman de Vries, among others.[1] In what reviewer Riccardo Bellé calls "the core of the book", chapters 6 through 12 cover the developments of the theory by Guidobaldo del Monte, Simon Stevin, Willem 's Gravesande, and Brook Taylor.[2] Again, after an initial chapter on del Monte's discovery of the vanishing point and Stevin's mathematical explication of del Monte's work, these chapters are divided geographically. Chapter 7 concerns the Netherlands, including the Dutch painters of the 17th century, the book on perspective by Samuel Marolois, and the work of 's Gravesande. Chapter 8 returns to Italy, and the work of architects and stage designers there, including Andrea Pozzo among the Jesuits. Chapter 9 covers over 40 works from France and Belgium, including the anonymously-published work of Jean Du Breuil, who brought the Jesuit knowledge of architecture from Italy to France, and the work on anamorphosis by Jean François Niceron. This chapter also covers Girard Desargues, although it disagrees with the widely-held opinion that Desargues was the inventor of projective geometry. Chapter 10, the longest of the book, concerns Britain, including Taylor, and his followers. Chapters 11 and 12 both concern the German-speaking countries, with Chapter 12 focusing on Johann Heinrich Lambert, who "concluded the process of understanding the geometry behind perspective by creating perspectival geometry".[1] A penultimate chapter concerns Gaspard Monge, the development of descriptive geometry, and its relation to the earlier perspective geometry and projective geometry. After a final summary chapter, the book includes four appendices and two bibliographies. The book is illustrated with over 600 black and white images, some from the works described and others new-created visualizations of their mathematical concepts,[1] with older diagrams consistently relabeled to make their common features more apparent.[2] From this history, reviewer Jeremy Gray draws several interesting conclusions: that, after their initial joint formulation, the mathematical and artistic aspects of the subject remained more or less separate, with later developments in mathematics having little influence on artistic practice, that (despite frequent accounts of their being directly connected) the earlier work on perspective geometry had little influence on the creation of projective geometry, and that despite covering so many contributors to this history, Andersen could find no women among them.[3] Audience and reception Reviewer Christa Binder describes this book as Kirsti Andersen's life work and the "definitive reference work on perspective, a classic in its field". Riccardo Bellé recommends the book to "a wide range of scholars, especially historians of mathematics, historians of art, historians of architecture", but also to practitioners of architecture, engineering, or perspective art, and to art teachers.[2] Philip J. Davis recommends it to anyone who wishes to understand the roots of contemporary computer graphics.[4] Gray calls it "a remarkable piece of historical research" that "will surely become the definitive text on the subject".[3] However, although finding the book clearly written and comprehensive as a history of perspective, reviewer Greg St. George warns against trying to use this book as an introduction to the mathematics of perspective, for which a more focused text would be more appropriate.[5] Similarly, Judith V. Field finds that the book's attempts to make the mathematics more clear, by unifying its notation and terminology and basing its explanations on modern mathematical treatments, tend to muddle its treatment of the history and historical sources of the subject. Field also takes fault with the book's superficial and dismissive treatment of Desargues, with its uncritical reliance on modern sources that Field considers dubious such as the work of Morris Kline, with its "coy refusal" to draw conclusions from the story it tells, and with its publisher's poor copyediting.[6] References 1. Binder, Christa (February 2012), Annals of Science, 69 (2): 291–294, doi:10.1080/00033790902730636{{citation}}: CS1 maint: untitled periodical (link) 2. Bellé, Riccardo (March 2009), Isis, 100 (1): 132–133, doi:10.1086/599638, JSTOR 10.1086/599638{{citation}}: CS1 maint: untitled periodical (link) 3. Gray, Jeremy (May 2009), Historia Mathematica, 36 (2): 182–183, doi:10.1016/j.hm.2008.08.007{{citation}}: CS1 maint: untitled periodical (link) 4. Davis, Philip J. (October 2008), Centaurus, 50 (4): 332–334, doi:10.1111/j.1600-0498.2008.00111.x{{citation}}: CS1 maint: untitled periodical (link) 5. St. George, Greg (July 2007), Zentralblatt für Didaktik der Mathematik, 39 (5–6): 553–554, doi:10.1007/s11858-007-0046-z{{citation}}: CS1 maint: untitled periodical (link) 6. Field, J. V. (September 2008), "Review", MAA Reviews, Mathematical Association of America	Wikipedia
The Geometry of the Octonions The Geometry of the Octonions is a mathematics book on the octonions, a system of numbers generalizing the complex numbers and quaternions, presenting its material at a level suitable for undergraduate mathematics students. It was written by Tevian Dray and Corinne Manogue, and published in 2015 by World Scientific. The Basic Library List Committee of the Mathematical Association of America has suggested its inclusion in undergraduate mathematics libraries.[1] Topics The book is subdivided into three parts, with the second part being the most significant.[2] Its contents combine both a survey of past work in this area, and much of its authors' own researches.[3] The first part explains the Cayley–Dickson construction,[1][3] which constructs the complex numbers from the real numbers, the quaternions from the complex numbers, and the octonions from the quaternions. Related algebras are also discussed, including the sedenions (a 16-dimensional real algebra formed in the same way by taking one more step past the octonions) and the split real unital composition algebras (also called Hurwitz algebras).[2] A particular focus here is on interpreting the multiplication operation of these algebras in a geometric way.[4] Reviewer Danail Brezov notes with disappointment that Clifford algebras, although very relevant to this material, are not covered.[3] The second part of the book uses the octonions and the other division algebras associated with it to provide concrete descriptions of the Lie groups of geometric symmetries. These include rotation groups, spin groups, symplectic groups, and the exceptional Lie groups, which the book interprets as octonionic variants of classical Lie groups.[2][4] The third part applies the octonions in geometric constructions including the Hopf fibration and its generalizations, the Cayley plane, and the E8 lattice. It also connects them to problems in physics involving the four-dimensional Dirac equation, the quantum mechanics of relativistic fermions, spinors, and the formulation of quantum mechanics using Jordan algebras.[2][3][4] It also includes material on octonionic number theory,[3][4] and concludes with a chapter on the Freudenthal magic square and related constructions.[2] Audience and reception Although presented at an undergraduate level, The Geometry of the Octonions is not a textbook: its material is likely too specialized for an undergraduate course, and it lacks exercises or similar material that would be needed to use it as a textbook.[1] Readers should be familiar with linear algebra, and some experience with Lie groups would also be helpful.[2] The later chapters on applications in physics are heavier going, and require familiarity with quantum mechanics.[1] The book avoids a proof-heavy formal style of mathematical writing,[2] so much so that reviewer Danail Brezov writes that at points it "seems to lack mathematical rigor".[3] Related reading Multiple reviewers suggest that this work would make a good introduction to the octonions, as a stepping stone to the more advanced material presented in other works on the same topic.[2][3][4] Their suggestions include the following: • Baez, John C. (2002), "The octonions", Bulletin of the American Mathematical Society, New Series, 39 (2): 145–205, doi:10.1090/S0273-0979-01-00934-X, MR 1886087, S2CID 586512 • Conway, John H.; Smith, Derek A. (2003), On Quaternions and Octonions: Their Geometry, Arithmetic, and Symmetry, Natick, Massachusetts: A K Peters, ISBN 1-56881-134-9, MR 1957212 • Kantor, I. L.; Solodovnikov, A. S. (1989), Hypercomplex Numbers: An Elementary Introduction to Algebras, New York: Springer-Verlag, doi:10.1007/978-1-4612-3650-4, ISBN 0-387-96980-2, MR 0996029 • Salzmann, Helmut; Betten, Dieter; Grundhöfer, Theo; Hähl, Hermann; Löwen, Rainer; Stroppel, Markus (1995), Compact Projective Planes: With an Introduction to Octonion Geometry, De Gruyter Expositions in Mathematics, vol. 21, Walter de Gruyter & Co., Berlin, doi:10.1515/9783110876833, ISBN 3-11-011480-1, MR 1384300 • Springer, Tonny A.; Veldkamp, Ferdinand D. (2000), Octonions, Jordan Algebras and Exceptional Groups, Springer Monographs in Mathematics, Berlin: Springer-Verlag, doi:10.1007/978-3-662-12622-6, ISBN 3-540-66337-1, MR 1763974 • Ward, J. P. (1997), Quaternions and Cayley Numbers: Algebra and Applications, Mathematics and its Applications, vol. 403, Dordrecht: Kluwer Academic Publishers, doi:10.1007/978-94-011-5768-1, ISBN 0-7923-4513-4, MR 1458894 References 1. Hunacek, Mark (June 2015), "Review of The Geometry of the Octonions", MAA Reviews 2. Elduque, Alberto, "Review of The Geometry of the Octonions", MathSciNet, MR 3361898 3. Brezov, Danail (2015), "Review of The Geometry of the Octonions", Journal of Geometry and Symmetry in Physics, 39: 99–101, Zbl 1417.00016 4. Knarr, Norbert, "Review of The Geometry of the Octonions", zbMATH, Zbl 1333.17004	Wikipedia
George Green (mathematician) George Green (14 July 1793 – 31 May 1841) was a British mathematical physicist who wrote An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism in 1828.[2][3] The essay introduced several important concepts, among them a theorem similar to the modern Green's theorem, the idea of potential functions as currently used in physics, and the concept of what are now called Green's functions. Green was the first person to create a mathematical theory of electricity and magnetism and his theory formed the foundation for the work of other scientists such as James Clerk Maxwell, William Thomson, and others. His work on potential theory ran parallel to that of Carl Friedrich Gauss. George Green Born(1793-07-14)14 July 1793 Sneinton, Nottinghamshire, England Died31 May 1841(1841-05-31) (aged 47) Nottingham, Nottinghamshire, England Alma materGonville and Caius College, Cambridge (BA, 1838) Known forGreen measure Green's deformation tensor Green's function Green's identities Green's law Green's matrix Green's theorem Liouville–Green method Scientific career FieldsMathematics InstitutionsGonville and Caius College, Cambridge[1] InfluencedLord Kelvin Julian Schwinger Green's life story is remarkable in that he was almost entirely self-taught. He received only about one year of formal schooling as a child, between the ages of 8 and 9. Early life Green was born and lived for most of his life in the English town of Sneinton, Nottinghamshire, now part of the city of Nottingham. His father, also named George, was a baker who had built and owned a brick windmill used to grind grain.[1] In his youth, Green was described as having a frail constitution and a dislike for doing work in his father's bakery. He had no choice in the matter, however, and as was common for the time he likely began working daily to earn his living at the age of five. Robert Goodacre's Academy During this era it was common for only 25–50% of children in Nottingham to receive any schooling. The majority of schools were Sunday schools, run by the Church, and children would typically attend for one or two years only. Recognizing the young Green's above average intellect, and being in a strong financial situation due to his successful bakery, his father enrolled him in March 1801 at Robert Goodacre's Academy in Upper Parliament Street. Robert Goodacre was a well-known science populariser and educator of the time. He published Essay on the Education of Youth, in which he wrote that he did not "study the interest of the boy but the embryo Man". To a non-specialist, he would have seemed deeply knowledgeable in science and mathematics, but a close inspection of his essay and curriculum revealed that the extent of his mathematical teachings was limited to algebra, trigonometry and logarithms. Thus, Green's later mathematical contributions, which exhibited knowledge of very modern developments in mathematics, could not have resulted from his tenure at the Robert Goodacre Academy. He stayed for only four terms (one school year), and it was speculated by his contemporaries that he had exhausted all they had to teach him. Move from Nottingham to Sneinton In 1773 George's father moved to Nottingham, which at the time had a reputation for being a pleasant town with open spaces and wide roads. By 1831, however, the population had increased nearly five times, in part due to the budding industrial revolution, and the city became known as one of the worst slums in England. There were frequent riots by starving workers, often associated with special hostility towards bakers and millers on the suspicion that they were hiding grain to drive up food prices. For these reasons, in 1807, George Green senior bought a plot of land in Sneinton. On this plot of land he built a "brick wind corn mill", now referred to as Green's Windmill. It was technologically impressive for its time, but required nearly twenty-four-hour maintenance, which was to become Green's burden for the next twenty years. Adult life Miller Just as with baking, Green found the responsibilities of operating the mill annoying and tedious. Grain from the fields was arriving continuously at the mill's doorstep, and the sails of the windmill had to be constantly adjusted to the windspeed, both to prevent damage in high winds, and to maximise rotational speed in low winds. The millstones that would continuously grind against each other, could wear down or cause a fire if they ran out of grain to grind. Every month the stones, which weighed over a ton, would have to be replaced or repaired. Family life In 1823 Green formed a relationship with Jane Smith, the daughter of William Smith, hired by Green Senior as mill manager. Although Green and Jane Smith never married, Jane eventually became known as Jane Green and the couple had seven children together; all but the first had Green as a baptismal name. The youngest child was born 13 months before Green's death. Green provided for his common-law wife and children in his will.[4] Nottingham Subscription Library When Green was thirty, he became a member of the Nottingham Subscription Library. This library exists today, and was likely the main source of Green's advanced mathematical knowledge. Unlike more conventional libraries, the subscription library was exclusive to a hundred or so subscribers, and the first on the list of subscribers was the Duke of Newcastle. This library catered to requests for specialised books and journals that satisfied the particular interests of their subscribers. 1828 essay In 1828, Green published An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism, which is the essay he is most famous for today. It was published privately at the author's expense, because he thought it would be presumptuous for a person like himself, with no formal education in mathematics, to submit the paper to an established journal. When Green published his Essay, it was sold on a subscription basis to 51 people, most of whom were friends who probably could not understand it. The wealthy landowner and mathematician Sir Edward Bromhead bought a copy and encouraged Green to do further work in mathematics. Not believing the offer was sincere, Green did not contact Bromhead for two years. Mathematician By the time Green's father died in 1829, the senior Green had become one of the gentry due to his considerable accumulated wealth and land owned, roughly half of which he left to his son and the other half to his daughter. The young Green, now thirty-six years old, consequently was able to use this wealth to abandon his miller duties and pursue mathematical studies. Cambridge Members of the Nottingham Subscription Library who knew Green repeatedly insisted that he obtain a proper University education. In particular, one of the library's most prestigious subscribers was Sir Edward Bromhead, with whom Green shared many correspondences; he insisted that Green go to Cambridge. In 1832, aged nearly forty, Green was admitted as an undergraduate at Gonville and Caius College, Cambridge.[5] He was particularly insecure about his lack of knowledge of Greek and Latin, which were prerequisites, but it turned out not to be as hard for him to learn these as he had envisaged, as the degree of mastery required was not as high as he had expected. In the mathematics examinations, he won the first-year mathematical prize. He graduated with a BA in 1838 as a 4th Wrangler (the 4th highest scoring student in his graduating class, coming after James Joseph Sylvester who scored 2nd).[5] College fellow Following his graduation, Green was elected a fellow of the Cambridge Philosophical Society. Even without his stellar academic standing, the Society had already read and made note of his Essay and three other publications, so Green was welcomed. The next two years provided an unparalleled opportunity for Green to read, write, and discuss his scientific ideas. In this short time he published an additional six publications with applications to hydrodynamics, sound, and optics. Final years and posthumous fame In his final years at Cambridge, Green became rather ill, and in 1840 he returned to Sneinton, only to die a year later. There are rumours that at Cambridge, Green had "succumbed to alcohol", and some of his earlier supporters, such as Sir Edward Bromhead, tried to distance themselves from him. Green's work was not well known in the mathematical community during his lifetime. Besides Green himself, the first mathematician to quote his 1828 work was the Briton Robert Murphy (1806–1843) in his 1833 work.[6] In 1845, four years after Green's death, Green's work was rediscovered by the young William Thomson (then aged 21), later known as Lord Kelvin, who popularised it for future mathematicians. According to the book "George Green" by D.M. Cannell, William Thomson noticed Murphy's citation of Green's 1828 essay but found it difficult to locate Green's 1828 work; he finally got some copies of Green's 1828 work from William Hopkins in 1845. In 1871 N. M. Ferrers assembled The Mathematical Papers of the late George Green for publication.[7] Green's work on the motion of waves in a canal (resulting in what is known as Green's law) anticipates the WKB approximation of quantum mechanics, while his research on light-waves and the properties of the Aether produced what is now known as the Cauchy-Green tensor. Green's theorem and functions were important tools in classical mechanics, and were revised by Schwinger's 1948 work on electrodynamics that led to his 1965 Nobel prize (shared with Feynman and Tomonaga). Green's functions later also proved useful in analysing superconductivity. On a visit to Nottingham in 1930, Albert Einstein commented that Green had been 20 years ahead of his time. The theoretical physicist Julian Schwinger, who used Green's functions in his ground-breaking works, published a tribute entitled "The Greening of Quantum Field Theory: George and I" in 1993.[8] The George Green Library at the University of Nottingham is named after him, and houses the majority of the university's science and engineering Collection. The George Green Institute for Electromagnetics Research, a research group in the University of Nottingham engineering department, is also named after him.[9] In 1986, Green's Windmill was restored to working order. It now serves both as a working example of a 19th-century windmill and as a museum and science centre dedicated to Green. Westminster Abbey has a memorial stone for Green in the nave adjoining the graves of Sir Isaac Newton and Lord Kelvin.[10] His work and influence on 19th-century applied physics had been largely forgotten until the publication of his biography by Mary Cannell in 1993. Source of knowledge It is unclear to historians exactly where Green obtained information on current developments in mathematics, as Nottingham had little in the way of intellectual resources. What is even more mysterious is that Green had used "the Mathematical Analysis", a form of calculus derived from Leibniz that was virtually unheard of, or even actively discouraged, in England at the time (due to Leibniz being a contemporary of Newton, who had his own methods that were championed in England). This form of calculus, and the developments of mathematicians such as the French mathematicians Laplace, Lacroix and Poisson, were not taught even at Cambridge, let alone Nottingham, and yet Green not only had heard of these developments, but improved upon them.[11] It is speculated that only one person educated in mathematics, John Toplis, headmaster of Nottingham High School 1806–1819, graduate from Cambridge as 11th Wrangler and an enthusiast of French mathematics, lived in Nottingham at the time. List of publications • An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism. By George Green, Nottingham. Printed for the Author by T. Wheelhouse, Nottingham. 1828. (Quarto, vii + 72 pages.) • Green, George (1835). "Mathematical investigations concerning the laws of the equilibrium of fluids analogous to the electric fluid, with other similar researches". Transactions of the Cambridge Philosophical Society. 5 (part i): 1–63. Presented 12 November 1832. • Green, George (1835). "On the determination of the exterior and interior attractions of ellipsoids of variable densities". Transactions of the Cambridge Philosophical Society. 5 (part iii): 395–429. Bibcode:1835TCaPS...5..395G. Presented 6 May 1833. • Green, George (1836). "Researches on the vibration of pendulums in fluid media". Transactions of the Royal Society of Edinburgh. 13 (1): 54–62. doi:10.1017/S0080456800022183. S2CID 124762445. Presented 16 December 1833. • Green, George (1838). "On the reflexion and refraction of sound". Transactions of the Cambridge Philosophical Society. 6 (part iii): 403–413. Presented 11 December 1837. • Green, George (1838). "On the motion of waves in a variable canal of small depth and width". Transactions of the Cambridge Philosophical Society. 6 (part iii): 457–462. Bibcode:1838TCaPS...6..457G. Presented 15 May 1837. • Green, George (1842). "On the laws of the reflexion and refraction of light at the common surface of two non-crystallized media". Transactions of the Cambridge Philosophical Society. 7 (part i): 1–24. Presented 11 December 1837. • Green, George (1842). "Note on the motion of waves in canals". Transactions of the Cambridge Philosophical Society. 7 (part i): 87–95. Presented 18 February 1839. • Green, George (1842). "Supplement to a memoir on the reflection and refraction of light". Transactions of the Cambridge Philosophical Society. 7 (part i): 113–120. Presented 6 May 1839. • Green, George (1842). "On the propagation of light in crystallized media". Transactions of the Cambridge Philosophical Society. 7 (part ii): 121–140. Presented 20 May 1839. Notes 1. O'Connor, John J.; Robertson, Edmund F., "George Green (mathematician)", MacTutor History of Mathematics Archive, University of St Andrews 2. This 1828 essay can be found in Mathematical papers of the late George Green, edited by N. M. Ferrers. The website for this is given below. 3. Cannell, D.M. (1999). "George Green: An Enigmatic Mathematician". American Mathematical Monthly. 106 (2): 136–151. doi:10.2307/2589050. JSTOR 2589050. 4. Cannel, D. M.; Lord, N. J.; Lord, N. J (1993). "George Green, mathematician and physicist 1793–1841". The Mathematical Gazette. 77 (478): 26–51. doi:10.2307/3619259. JSTOR 3619259. S2CID 238490315. 5. "Green, George (GRN832G)". A Cambridge Alumni Database. University of Cambridge. 6. Murphy, R. (1833). "On the inverse method of definite integrals, with physical applications". Transactions of the Cambridge Philosophical Society. 4: 353–408. Green is mentioned in a footnote on p. 357. 7. N. M. Ferrers editor (1871) The Mathematical Papers of the late George Green, Macmillan Publishers, link from University of Michigan Historical Math Collection 8. Schwinger, Julian (1993). "The Greening of quantum Field Theory: George and I": 10283. arXiv:hep-ph/9310283. Bibcode:1993hep.ph...10283S. {{cite journal}}: Cite journal requires \|journal= (help) 9. "George Green Institute for Electromagnetics Research". Archived from the original on 17 January 2014. Retrieved 17 February 2014. 10. George Green from Westminster Abbey 11. Cannell, D.M. (1999). "George Green: An Enigmatic Mathematician". The American Mathematical Monthly. 106 (2): 137, 140. CiteSeerX 10.1.1.383.6824. doi:10.1080/00029890.1999.12005020. References • Ivor Grattan-Guinness, 'Green, George (1793–1841)', Oxford Dictionary of National Biography, Oxford University Press, 2004 accessed 26 May 2009 • D. M. Cannell, "George Green mathematician and physicist 1793–1841", The Athlone Press, London, 1993. • Murphy, Robert (1833). "On the inverse method of definite integrals". Transactions of the Cambridge Philosophical Society. 4: 353–408. (Note: This was the first quotation of Green's 1828 work by somebody other than Green himself.) • O'Connor, John J.; Robertson, Edmund F., "George Green (mathematician)", MacTutor History of Mathematics Archive, University of St Andrews • "George Green". Archived from the original on 26 December 2010. – An excellent on-line source of George Green information • Green, George (1828). "An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism". arXiv:0807.0088 [physics.hist-ph]. • Cannel, D. M. and Lord, N. J.; Lord, N. J. (March 1993). "George Green, mathematician and physicist 1793–1841". The Mathematical Gazette. The Mathematical Gazette, Vol. 77, No. 478. 77 (478): 26–51. doi:10.2307/3619259. JSTOR 3619259. S2CID 238490315.{{cite journal}}: CS1 maint: multiple names: authors list (link) • Challis, L. and Sheard, F.; Sheard, Fred (December 2003). "The Green of Green Functions". Physics Today. 56 (12): 41–46. Bibcode:2003PhT....56l..41C. doi:10.1063/1.1650227. S2CID 17977976.{{cite journal}}: CS1 maint: multiple names: authors list (link) • "Green's Mill and Science Centre" (Web page). Retrieved 22 November 2005. External links • List of References for George Green • Quotations related to George Green at Wikiquote • Bowley, Roger. "George Green & Green's Functions". Sixty Symbols. Brady Haran for the University of Nottingham. Authority control International • FAST • ISNI • VIAF National • France • BnF data • Catalonia • Germany • Israel • United States • Japan • Czech Republic • Australia • Netherlands Academics • CiNii • MathSciNet • zbMATH People • Deutsche Biographie • Trove Other • SNAC • IdRef	Wikipedia
Gibbs lemma In game theory and in particular the study of Blotto games and operational research, the Gibbs lemma is a result that is useful in maximization problems.[1] It is named for Josiah Willard Gibbs. Consider $\phi =\sum _{i=1}^{n}f_{i}(x_{i})$. Suppose $\phi $ is maximized, subject to $\sum x_{i}=X$ and $x_{i}\geq 0$, at $x^{0}=(x_{1}^{0},\ldots ,x_{n}^{0})$. If the $f_{i}$ are differentiable, then the Gibbs lemma states that there exists a $\lambda $ such that ${\begin{aligned}f'_{i}(x_{i}^{0})&=\lambda {\mbox{ if }}x_{i}^{0}>0\\&\leq \lambda {\mbox{ if }}x_{i}^{0}=0.\end{aligned}}$ Notes 1. J. M. Danskin (6 December 2012). The Theory of Max-Min and its Application to Weapons Allocation Problems. Springer Science & Business Media. ISBN 978-3-642-46092-0. ... problems in which one side must make his move knowing that the other side will then learn what the move is and optimally counter. They are fundamental in particular to military weapons-selection problems involving large systems... References	Wikipedia
Gradient discretisation method In numerical mathematics, the gradient discretisation method (GDM) is a framework which contains classical and recent numerical schemes for diffusion problems of various kinds: linear or non-linear, steady-state or time-dependent. The schemes may be conforming or non-conforming, and may rely on very general polygonal or polyhedral meshes (or may even be meshless). Differential equations Scope Fields • Natural sciences • Engineering • Astronomy • Physics • Chemistry • Biology • Geology Applied mathematics • Continuum mechanics • Chaos theory • Dynamical systems Social sciences • Economics • Population dynamics List of named differential equations Classification Types • Ordinary • Partial • Differential-algebraic • Integro-differential • Fractional • Linear • Non-linear By variable type • Dependent and independent variables • Autonomous • Coupled / Decoupled • Exact • Homogeneous / Nonhomogeneous Features • Order • Operator • Notation Relation to processes • Difference (discrete analogue) • Stochastic • Stochastic partial • Delay Solution Existence and uniqueness • Picard–Lindelöf theorem • Peano existence theorem • Carathéodory's existence theorem • Cauchy–Kowalevski theorem General topics • Initial conditions • Boundary values • Dirichlet • Neumann • Robin • Cauchy problem • Wronskian • Phase portrait • Lyapunov / Asymptotic / Exponential stability • Rate of convergence • Series / Integral solutions • Numerical integration • Dirac delta function Solution methods • Inspection • Method of characteristics • Euler • Exponential response formula • Finite difference (Crank–Nicolson) • Finite element • Infinite element • Finite volume • Galerkin • Petrov–Galerkin • Green's function • Integrating factor • Integral transforms • Perturbation theory • Runge–Kutta • Separation of variables • Undetermined coefficients • Variation of parameters People List • Isaac Newton • Gottfried Leibniz • Jacob Bernoulli • Leonhard Euler • Józef Maria Hoene-Wroński • Joseph Fourier • Augustin-Louis Cauchy • George Green • Carl David Tolmé Runge • Martin Kutta • Rudolf Lipschitz • Ernst Lindelöf • Émile Picard • Phyllis Nicolson • John Crank Some core properties are required to prove the convergence of a GDM. These core properties enable complete proofs of convergence of the GDM for elliptic and parabolic problems, linear or non-linear. For linear problems, stationary or transient, error estimates can be established based on three indicators specific to the GDM [1] (the quantities $C_{D}$, $S_{D}$ and $W_{D}$, see below). For non-linear problems, the proofs are based on compactness techniques and do not require any non-physical strong regularity assumption on the solution or the model data.[2] Non-linear models for which such convergence proof of the GDM have been carried out comprise: the Stefan problem which is modelling a melting material, two-phase flows in porous media, the Richards equation of underground water flow, the fully non-linear Leray—Lions equations.[3] Any scheme entering the GDM framework is then known to converge on all these problems. This applies in particular to conforming Finite Elements, Mixed Finite Elements, nonconforming Finite Elements, and, in the case of more recent schemes, the Discontinuous Galerkin method, Hybrid Mixed Mimetic method, the Nodal Mimetic Finite Difference method, some Discrete Duality Finite Volume schemes, and some Multi-Point Flux Approximation schemes The example of a linear diffusion problem Consider Poisson's equation in a bounded open domain $\Omega \subset \mathbb {R} ^{d}$, with homogeneous Dirichlet boundary condition $-\Delta {\overline {u}}=f,$ (1) where $f\in L^{2}(\Omega )$. The usual sense of weak solution [4] to this model is: ${\mbox{Find }}{\overline {u}}\in H_{0}^{1}(\Omega ){\mbox{ such that, for all }}{\overline {v}}\in H_{0}^{1}(\Omega ),\quad \int _{\Omega }\nabla {\overline {u}}(x)\cdot \nabla {\overline {v}}(x)dx=\int _{\Omega }f(x){\overline {v}}(x)dx.$ (2) In a nutshell, the GDM for such a model consists in selecting a finite-dimensional space and two reconstruction operators (one for the functions, one for the gradients) and to substitute these discrete elements in lieu of the continuous elements in (2). More precisely, the GDM starts by defining a Gradient Discretization (GD), which is a triplet $D=(X_{D,0},\Pi _{D},\nabla _{D})$, where: • the set of discrete unknowns $X_{D,0}$ is a finite dimensional real vector space, • the function reconstruction $\Pi _{D}~:~X_{D,0}\to L^{2}(\Omega )$ is a linear mapping that reconstructs, from an element of $X_{D,0}$, a function over $\Omega $, • the gradient reconstruction $\nabla _{D}~:~X_{D,0}\to L^{2}(\Omega )^{d}$ is a linear mapping which reconstructs, from an element of $X_{D,0}$, a "gradient" (vector-valued function) over $\Omega $. This gradient reconstruction must be chosen such that $\Vert \nabla _{D}\cdot \Vert _{L^{2}(\Omega )^{d}}$ is a norm on $X_{D,0}$. The related Gradient Scheme for the approximation of (2) is given by: find $u\in X_{D,0}$ such that $\forall v\in X_{D,0},\qquad \int _{\Omega }\nabla _{D}u(x)\cdot \nabla _{D}v(x)dx=\int _{\Omega }f(x)\Pi _{D}v(x)dx.$ (3) The GDM is then in this case a nonconforming method for the approximation of (2), which includes the nonconforming finite element method. Note that the reciprocal is not true, in the sense that the GDM framework includes methods such that the function $\nabla _{D}u$ cannot be computed from the function $\Pi _{D}u$. The following error estimate, inspired by G. Strang's second lemma,[5] holds $W_{D}(\nabla {\overline {u}})\leq \Vert \nabla {\overline {u}}-\nabla _{D}u_{D}\Vert _{L^{2}(\Omega )^{d}}\leq W_{D}(\nabla {\overline {u}})+2S_{D}({\overline {u}}),$ (4) and $\Vert {\overline {u}}-\Pi _{D}u_{D}\Vert _{L^{2}(\Omega )}\leq C_{D}W_{D}(\nabla {\overline {u}})+(C_{D}+1)S_{D}({\overline {u}}),$ (5) defining: $C_{D}=\max _{v\in X_{D,0}\setminus \{0\}}{\frac {\Vert \Pi _{D}v\Vert _{L^{2}(\Omega )}}{\Vert \nabla _{D}v\Vert _{L^{2}(\Omega )^{d}}}},$ (6) which measures the coercivity (discrete Poincaré constant), $\forall \varphi \in H_{0}^{1}(\Omega ),\,S_{D}(\varphi )=\min _{v\in X_{D,0}}\left(\Vert \Pi _{D}v-\varphi \Vert _{L^{2}(\Omega )}+\Vert \nabla _{D}v-\nabla \varphi \Vert _{L^{2}(\Omega )^{d}}\right),$ (7) which measures the interpolation error, $\forall \varphi \in H_{\operatorname {div} }(\Omega ),\,W_{D}(\varphi )=\max _{v\in X_{D,0}\setminus \{0\}}{\frac {\left\|\int _{\Omega }\left(\nabla _{D}v(x)\cdot \varphi (x)+\Pi _{D}v(x)\operatorname {div} \varphi (x)\right)\,dx\right\|}{\Vert \nabla _{D}v\Vert _{L^{2}(\Omega )^{d}}}},$ (8) which measures the defect of conformity. Note that the following upper and lower bounds of the approximation error can be derived: ${\begin{aligned}&&{\frac {1}{2}}[S_{D}({\overline {u}})+W_{D}(\nabla {\overline {u}})]\\&\leq &\Vert {\overline {u}}-\Pi _{D}u_{D}\Vert _{L^{2}(\Omega )}+\Vert \nabla {\overline {u}}-\nabla _{D}u_{D}\Vert _{L^{2}(\Omega )^{d}}\\&\leq &(C_{D}+2)[S_{D}({\overline {u}})+W_{D}(\nabla {\overline {u}})].\end{aligned}}$ (9) Then the core properties which are necessary and sufficient for the convergence of the method are, for a family of GDs, the coercivity, the GD-consistency and the limit-conformity properties, as defined in the next section. More generally, these three core properties are sufficient to prove the convergence of the GDM for linear problems and for some nonlinear problems like the $p$-Laplace problem. For nonlinear problems such as nonlinear diffusion, degenerate parabolic problems..., we add in the next section two other core properties which may be required. The core properties allowing for the convergence of a GDM Let $(D_{m})_{m\in \mathbb {N} }$ be a family of GDs, defined as above (generally associated with a sequence of regular meshes whose size tends to 0). Coercivity The sequence $(C_{D_{m}})_{m\in \mathbb {N} }$ (defined by (6)) remains bounded. GD-consistency For all $\varphi \in H_{0}^{1}(\Omega )$, $\lim _{m\to \infty }S_{D_{m}}(\varphi )=0$ (defined by (7)). Limit-conformity For all $\varphi \in H_{\operatorname {div} }(\Omega )$, $\lim _{m\to \infty }W_{D_{m}}(\varphi )=0$ (defined by (8)). This property implies the coercivity property. Compactness (needed for some nonlinear problems) For all sequence $(u_{m})_{m\in \mathbb {N} }$ such that $u_{m}\in X_{D_{m},0}$ for all $m\in \mathbb {N} $ and $(\Vert u_{m}\Vert _{D_{m}})_{m\in \mathbb {N} }$ is bounded, then the sequence $(\Pi _{D_{m}}u_{m})_{m\in \mathbb {N} }$ is relatively compact in $L^{2}(\Omega )$ (this property implies the coercivity property). Piecewise constant reconstruction (needed for some nonlinear problems) Let $D=(X_{D,0},\Pi _{D},\nabla _{D})$ be a gradient discretisation as defined above. The operator $\Pi _{D}$ is a piecewise constant reconstruction if there exists a basis $(e_{i})_{i\in B}$ of $X_{D,0}$ and a family of disjoint subsets $(\Omega _{i})_{i\in B}$ of $\Omega $ such that $ \Pi _{D}u=\sum _{i\in B}u_{i}\chi _{\Omega _{i}}$ for all $ u=\sum _{i\in B}u_{i}e_{i}\in X_{D,0}$, where $\chi _{\Omega _{i}}$ is the characteristic function of $\Omega _{i}$. Some non-linear problems with complete convergence proofs of the GDM We review some problems for which the GDM can be proved to converge when the above core properties are satisfied. Nonlinear stationary diffusion problems $-\operatorname {div} (\Lambda ({\overline {u}})\nabla {\overline {u}})=f$ In this case, the GDM converges under the coercivity, GD-consistency, limit-conformity and compactness properties. p-Laplace problem for p > 1 $-\operatorname {div} \left(\|\nabla {\overline {u}}\|^{p-2}\nabla {\overline {u}}\right)=f$ In this case, the core properties must be written, replacing $L^{2}(\Omega )$ by $L^{p}(\Omega )$, $H_{0}^{1}(\Omega )$ by $W_{0}^{1,p}(\Omega )$ and $H_{\operatorname {div} }(\Omega )$ by $W_{\operatorname {div} }^{p'}(\Omega )$ with $ {\frac {1}{p}}+{\frac {1}{p'}}=1$, and the GDM converges only under the coercivity, GD-consistency and limit-conformity properties. Linear and nonlinear heat equation $\partial _{t}{\overline {u}}-\operatorname {div} (\Lambda ({\overline {u}})\nabla {\overline {u}})=f$ In this case, the GDM converges under the coercivity, GD-consistency (adapted to space-time problems), limit-conformity and compactness (for the nonlinear case) properties. Degenerate parabolic problems Assume that $\beta $ and $\zeta $ are nondecreasing Lipschitz continuous functions: $\partial _{t}\beta ({\overline {u}})-\Delta \zeta ({\overline {u}})=f$ Note that, for this problem, the piecewise constant reconstruction property is needed, in addition to the coercivity, GD-consistency (adapted to space-time problems), limit-conformity and compactness properties. Review of some numerical methods which are GDM All the methods below satisfy the first four core properties of GDM (coercivity, GD-consistency, limit-conformity, compactness), and in some cases the fifth one (piecewise constant reconstruction). Galerkin methods and conforming finite element methods Let $V_{h}\subset H_{0}^{1}(\Omega )$ be spanned by the finite basis $(\psi _{i})_{i\in I}$. The Galerkin method in $V_{h}$ is identical to the GDM where one defines • $X_{D,0}=\{u=(u_{i})_{i\in I}\}=\mathbb {R} ^{I},$ • $\Pi _{D}u=\sum _{i\in I}u_{i}\psi _{i}$ • $\nabla _{D}u=\sum _{i\in I}u_{i}\nabla \psi _{i}.$ In this case, $C_{D}$ is the constant involved in the continuous Poincaré inequality, and, for all $\varphi \in H_{\operatorname {div} }(\Omega )$, $W_{D}(\varphi )=0$ (defined by (8)). Then (4) and (5) are implied by Céa's lemma. The "mass-lumped" $P^{1}$ finite element case enters the framework of the GDM, replacing $\Pi _{D}u$ by $ {\widetilde {\Pi }}_{D}u=\sum _{i\in I}u_{i}\chi _{\Omega _{i}}$, where $\Omega _{i}$ is a dual cell centred on the vertex indexed by $i\in I$. Using mass lumping allows to get the piecewise constant reconstruction property. Nonconforming finite element On a mesh $T$ which is a conforming set of simplices of $\mathbb {R} ^{d}$, the nonconforming $P^{1}$ finite elements are defined by the basis $(\psi _{i})_{i\in I}$ of the functions which are affine in any $K\in T$, and whose value at the centre of gravity of one given face of the mesh is 1 and 0 at all the others (these finite elements are used in [Crouzeix et al][6] for the approximation of the Stokes and Navier-Stokes equations). Then the method enters the GDM framework with the same definition as in the case of the Galerkin method, except for the fact that $\nabla \psi _{i}$ must be understood as the "broken gradient" of $\psi _{i}$, in the sense that it is the piecewise constant function equal in each simplex to the gradient of the affine function in the simplex. Mixed finite element The mixed finite element method consists in defining two discrete spaces, one for the approximation of $\nabla {\overline {u}}$ and another one for ${\overline {u}}$.[7] It suffices to use the discrete relations between these approximations to define a GDM. Using the low degree Raviart–Thomas basis functions allows to get the piecewise constant reconstruction property. Discontinuous Galerkin method The Discontinuous Galerkin method consists in approximating problems by a piecewise polynomial function, without requirements on the jumps from an element to the other.[8] It is plugged in the GDM framework by including in the discrete gradient a jump term, acting as the regularization of the gradient in the distribution sense. Mimetic finite difference method and nodal mimetic finite difference method This family of methods is introduced by [Brezzi et al][9] and completed in [Lipnikov et al].[10] It allows the approximation of elliptic problems using a large class of polyhedral meshes. The proof that it enters the GDM framework is done in [Droniou et al].[2] See also • Finite element method References 1. R. Eymard, C. Guichard, and R. Herbin. Small-stencil 3d schemes for diffusive flows in porous media. M2AN, 46:265–290, 2012. 2. J. Droniou, R. Eymard, T. Gallouët, and R. Herbin. Gradient schemes: a generic framework for the discretisation of linear, nonlinear and nonlocal elliptic and parabolic equations. Math. Models Methods Appl. Sci. (M3AS), 23(13):2395–2432, 2013. 3. J. Leray and J. Lions. Quelques résultats de Višik sur les problèmes elliptiques non linéaires par les méthodes de Minty-Browder. Bull. Soc. Math. France, 93:97–107, 1965. 4. H. Brezis. Functional analysis, Sobolev spaces and partial differential equations. Universitext. Springer, New York, 2011. 5. G. Strang. Variational crimes in the finite element method. In The mathematical foundations of the finite element method with applications to partial differential equations (Proc. Sympos., Univ. Maryland, Baltimore, Md., 1972), pages 689–710. Academic Press, New York, 1972. 6. M. Crouzeix and P.-A. Raviart. Conforming and nonconforming finite element methods for solving the stationary Stokes equations. I. Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge, 7(R-3):33–75, 1973. 7. P.-A. Raviart and J. M. Thomas. A mixed finite element method for 2nd order elliptic problems. In Mathematical aspects of finite element methods (Proc. Conf., Consiglio Naz. delle Ricerche (C.N.R.), Rome, 1975), pages 292–315. Lecture Notes in Math., Vol. 606. Springer, Berlin, 1977. 8. D. A. Di Pietro and A. Ern. Mathematical aspects of discontinuous Galerkin methods, volume 69 of Mathématiques & Applications (Berlin) [Mathematics & Applications]. Springer, Heidelberg, 2012. 9. F. Brezzi, K. Lipnikov, and M. Shashkov. Convergence of the mimetic finite difference method for diffusion problems on polyhedral meshes. SIAM J. Numer. Anal., 43(5):1872–1896, 2005. 10. K. Lipnikov, G. Manzini, and M. Shashkov. Mimetic finite difference method. J. Comput. Phys., 257-Part B:1163–1227, 2014. External links • The Gradient Discretisation Method by Jérôme Droniou, Robert Eymard, Thierry Gallouët, Cindy Guichard and Raphaèle Herbin Numerical methods for partial differential equations Finite difference Parabolic • Forward-time central-space (FTCS) • Crank–Nicolson Hyperbolic • Lax–Friedrichs • Lax–Wendroff • MacCormack • Upwind • Method of characteristics Others • Alternating direction-implicit (ADI) • Finite-difference time-domain (FDTD) Finite volume • Godunov • High-resolution • Monotonic upstream-centered (MUSCL) • Advection upstream-splitting (AUSM) • Riemann solver • Essentially non-oscillatory (ENO) • Weighted essentially non-oscillatory (WENO) Finite element • hp-FEM • Extended (XFEM) • Discontinuous Galerkin (DG) • Spectral element (SEM) • Mortar • Gradient discretisation (GDM) • Loubignac iteration • Smoothed (S-FEM) Meshless/Meshfree • Smoothed-particle hydrodynamics (SPH) • Peridynamics (PD) • Moving particle semi-implicit method (MPS) • Material point method (MPM) • Particle-in-cell (PIC) Domain decomposition • Schur complement • Fictitious domain • Schwarz alternating • additive • abstract additive • Neumann–Dirichlet • Neumann–Neumann • Poincaré–Steklov operator • Balancing (BDD) • Balancing by constraints (BDDC) • Tearing and interconnect (FETI) • FETI-DP Others • Spectral • Pseudospectral (DVR) • Method of lines • Multigrid • Collocation • Level-set • Boundary element • Method of moments • Immersed boundary • Analytic element • Isogeometric analysis • Infinite difference method • Infinite element method • Galerkin method • Petrov–Galerkin method • Validated numerics • Computer-assisted proof • Integrable algorithm • Method of fundamental solutions	Wikipedia
Fischer's inequality In mathematics, Fischer's inequality gives an upper bound for the determinant of a positive-semidefinite matrix whose entries are complex numbers in terms of the determinants of its principal diagonal blocks. Suppose A, C are respectively p×p, q×q positive-semidefinite complex matrices and B is a p×q complex matrix. Let $M:=\left[{\begin{matrix}A&B\\B^{}&C\end{matrix}}\right]$ so that M is a (p+q)×(p+q) matrix. Then Fischer's inequality states that $\det(M)\leq \det(A)\det(C).$ If M is positive-definite, equality is achieved in Fischer's inequality if and only if all the entries of B are 0. Inductively one may conclude that a similar inequality holds for a block decomposition of M with multiple principal diagonal blocks. Considering 1×1 blocks, a corollary is Hadamard's inequality. On the other hand, Fischer's inequality can also be proved by using Hadamard's inequality, see the proof of Theorem 7.8.5 in Horn and Johnson's Matrix Analysis. Proof Assume that A and C are positive-definite. We have $A^{-1}$ and $C^{-1}$ are positive-definite. Let $D:=\left[{\begin{matrix}A&0\\0&C\end{matrix}}\right].$ We note that $D^{-{\frac {1}{2}}}MD^{-{\frac {1}{2}}}=\left[{\begin{matrix}A^{-{\frac {1}{2}}}&0\\0&C^{-{\frac {1}{2}}}\end{matrix}}\right]\left[{\begin{matrix}A&B\\B^{}&C\end{matrix}}\right]\left[{\begin{matrix}A^{-{\frac {1}{2}}}&0\\0&C^{-{\frac {1}{2}}}\end{matrix}}\right]=\left[{\begin{matrix}I_{p}&A^{-{\frac {1}{2}}}BC^{-{\frac {1}{2}}}\\C^{-{\frac {1}{2}}}B^{}A^{-{\frac {1}{2}}}&I_{q}\end{matrix}}\right]$ Applying the AM-GM inequality to the eigenvalues of $D^{-{\frac {1}{2}}}MD^{-{\frac {1}{2}}}$, we see $\det(D^{-{\frac {1}{2}}}MD^{-{\frac {1}{2}}})\leq \left({1 \over p+q}\mathrm {tr} (D^{-{\frac {1}{2}}}MD^{-{\frac {1}{2}}})\right)^{p+q}=1^{p+q}=1.$ By multiplicativity of determinant, we have ${\begin{aligned}\det(D^{-{\frac {1}{2}}})\det(M)\det(D^{-{\frac {1}{2}}})\leq 1\\\Longrightarrow \det(M)\leq \det(D)=\det(A)\det(C).\end{aligned}}$ In this case, equality holds if and only if M = D that is, all entries of B are 0. For $\varepsilon >0$, as $A+\varepsilon I_{p}$ and $C+\varepsilon I_{q}$ are positive-definite, we have $\det(M+\varepsilon I_{p+q})\leq \det(A+\varepsilon I_{p})\det(C+\varepsilon I_{q}).$ Taking the limit as $\varepsilon \rightarrow 0$ proves the inequality. From the inequality we note that if M is invertible, then both A and C are invertible and we get the desired equality condition. Improvements If M can be partitioned in square blocks Mij, then the following inequality by Thompson is valid:[1] $\det(M)\leq \det([\det(M_{ij})])$ where [det(Mij)] is the matrix whose (i,j) entry is det(Mij). In particular, if the block matrices B and C are also square matrices, then the following inequality by Everett is valid:[2] $\det(M)\leq \det {\begin{bmatrix}\det(A)&&\det(B)\\\det(B^{})&&\det(C)\end{bmatrix}}$ Thompson's inequality can also be generalized by an inequality in terms of the coefficients of the characteristic polynomial of the block matrices. Expressing the characteristic polynomial of the matrix A as $p_{A}(t)=\sum _{k=0}^{n}t^{n-k}(-1)^{k}\operatorname {tr} (\Lambda ^{k}A)$ and supposing that the blocks Mij are m x m matrices, the following inequality by Lin and Zhang is valid:[3] $\det(M)\leq \left({\frac {\det([\operatorname {tr} (\Lambda ^{r}M_{ij}]))}{\binom {m}{r}}}\right)^{\frac {m}{r}},\quad r=1,\ldots ,m$ Note that if r = m, then this inequality is identical to Thompson's inequality. See also • Hadamard's inequality Notes 1. Thompson, R. C. (1961). "A determinantal inequality for positive definite matrices". Canadian Mathematical Bulletin. 4: 57–62. doi:10.4153/cmb-1961-010-9. 2. Everitt, W. N. (1958). "A note on positive definite matrices". Glasgow Mathematical Journal. 3 (4): 173–175. doi:10.1017/S2040618500033670. ISSN 2051-2104. 3. Lin, Minghua; Zhang, Pingping (2017). "Unifying a result of Thompson and a result of Fiedler and Markham on block positive definite matrices". Linear Algebra and Its Applications. 533: 380–385. doi:10.1016/j.laa.2017.07.032. References • Fischer, Ernst (1907), "Über den Hadamardschen Determinentsatz", Arch. Math. U. Phys. (3), 13: 32–40. • Horn, Roger A.; Johnson, Charles R. (2012), Matrix Analysis, doi:10.1017/cbo9781139020411, ISBN 9781139020411.	Wikipedia
Cycloid In geometry, a cycloid is the curve traced by a point on a circle as it rolls along a straight line without slipping. A cycloid is a specific form of trochoid and is an example of a roulette, a curve generated by a curve rolling on another curve. The cycloid, with the cusps pointing upward, is the curve of fastest descent under uniform gravity (the brachistochrone curve). It is also the form of a curve for which the period of an object in simple harmonic motion (rolling up and down repetitively) along the curve does not depend on the object's starting position (the tautochrone curve). In physics, when a charged particle at rest is put under a uniform electric and magnetic field perpendicular to one another, the particle’s trajectory draws out a cycloid. History It was in the left hand try-pot of the Pequod, with the soapstone diligently circling round me, that I was first indirectly struck by the remarkable fact, that in geometry all bodies gliding along the cycloid, my soapstone for example, will descend from any point in precisely the same time. Moby Dick by Herman Melville, 1851 The cycloid has been called "The Helen of Geometers" as, like Helen of Troy, it caused frequent quarrels among 17th-century mathematicians, while Sarah Hart sees it named as such "because the properties of this curve are so beautiful".[1][2] Historians of mathematics have proposed several candidates for the discoverer of the cycloid. Mathematical historian Paul Tannery speculated that such a simple curve must have been known to the ancients, citing similar work by Carpus of Antioch described by Iamblichus.[3] English mathematician John Wallis writing in 1679 attributed the discovery to Nicholas of Cusa,[4] but subsequent scholarship indicates that either Wallis was mistaken or the evidence he used is now lost.[5] Galileo Galilei's name was put forward at the end of the 19th century[6] and at least one author reports credit being given to Marin Mersenne.[7] Beginning with the work of Moritz Cantor[8] and Siegmund Günther,[9] scholars now assign priority to French mathematician Charles de Bovelles[10][11][12] based on his description of the cycloid in his Introductio in geometriam, published in 1503.[13] In this work, Bovelles mistakes the arch traced by a rolling wheel as part of a larger circle with a radius 120% larger than the smaller wheel.[5] Galileo originated the term cycloid and was the first to make a serious study of the curve.[5] According to his student Evangelista Torricelli,[14] in 1599 Galileo attempted the quadrature of the cycloid (determining the area under the cycloid) with an unusually empirical approach that involved tracing both the generating circle and the resulting cycloid on sheet metal, cutting them out and weighing them. He discovered the ratio was roughly 3:1, which is the true value, but he incorrectly concluded the ratio was an irrational fraction, which would have made quadrature impossible.[7] Around 1628, Gilles Persone de Roberval likely learned of the quadrature problem from Père Marin Mersenne and effected the quadrature in 1634 by using Cavalieri's Theorem.[5] However, this work was not published until 1693 (in his Traité des Indivisibles).[15] Constructing the tangent of the cycloid dates to August 1638 when Mersenne received unique methods from Roberval, Pierre de Fermat and René Descartes. Mersenne passed these results along to Galileo, who gave them to his students Torricelli and Viviani, who were able to produce a quadrature. This result and others were published by Torricelli in 1644,[14] which is also the first printed work on the cycloid. This led to Roberval charging Torricelli with plagiarism, with the controversy cut short by Torricelli's early death in 1647.[15] In 1658, Blaise Pascal had given up mathematics for theology but, while suffering from a toothache, began considering several problems concerning the cycloid. His toothache disappeared, and he took this as a heavenly sign to proceed with his research. Eight days later he had completed his essay and, to publicize the results, proposed a contest. Pascal proposed three questions relating to the center of gravity, area and volume of the cycloid, with the winner or winners to receive prizes of 20 and 40 Spanish doubloons. Pascal, Roberval and Senator Carcavy were the judges, and neither of the two submissions (by John Wallis and Antoine de Lalouvère) was judged to be adequate.[16]: 198 While the contest was ongoing, Christopher Wren sent Pascal a proposal for a proof of the rectification of the cycloid; Roberval claimed promptly that he had known of the proof for years. Wallis published Wren's proof (crediting Wren) in Wallis's Tractatus Duo, giving Wren priority for the first published proof.[15] Fifteen years later, Christiaan Huygens had deployed the cycloidal pendulum to improve chronometers and had discovered that a particle would traverse a segment of an inverted cycloidal arch in the same amount of time, regardless of its starting point. In 1686, Gottfried Wilhelm Leibniz used analytic geometry to describe the curve with a single equation. In 1696, Johann Bernoulli posed the brachistochrone problem, the solution of which is a cycloid.[15] Equations The cycloid through the origin, generated by a circle of radius r rolling over the x-axis on the positive side (y ≥ 0), consists of the points (x, y), with ${\begin{aligned}x&=r(t-\sin t)\\y&=r(1-\cos t),\end{aligned}}$ where t is a real parameter corresponding to the angle through which the rolling circle has rotated. For given t, the circle's centre lies at (x, y) = (rt, r). The Cartesian equation is obtained by solving the y-equation for t and substituting into the x-equation: $x=r\cos ^{-1}\left(1-{\frac {y}{r}}\right)-{\sqrt {y(2r-y)}},$ or, eliminating the multiple-valued inverse cosine: $r\cos \!\left({\frac {x+{\sqrt {y(2r-y)}}}{r}}\right)+y=r.$ When y is viewed as a function of x, the cycloid is differentiable everywhere except at the cusps on the x-axis, with the derivative tending toward $\infty $ or $-\infty $ near a cusp. The map from t to (x, y) is differentiable, in fact of class C∞, with derivative 0 at the cusps. The slope of the tangent to the cycloid at the point $(x,y)$ is given by $ {\frac {dy}{dx}}=\cot({\frac {t}{2}})$. A cycloid segment from one cusp to the next is called an arch of the cycloid, for example the points with $0\leq t\leq 2\pi $ and $0\leq x\leq 2\pi $. Considering the cycloid as the graph of a function $y=f(x)$, it satisfies the differential equation:[17] $\left({\frac {dy}{dx}}\right)^{2}={\frac {2r}{y}}-1.$ Involute The involute of the cycloid has exactly the same shape as the cycloid it originates from. This can be visualized as the path traced by the tip of a wire initially lying on a half arch of the cycloid: as it unrolls while remaining tangent to the original cycloid, it describes a new cycloid (see also cycloidal pendulum and arc length). Demonstration This demonstration uses the rolling-wheel definition of cycloid, as well as the instantaneous velocity vector of a moving point, tangent to its trajectory. In the adjacent picture, $P_{1}$ and $P_{2}$ are two points belonging to two rolling circles, with the base of the first just above the top of the second. Initially, $P_{1}$ and $P_{2}$ coincide at the intersection point of the two circles. When the circles roll horizontally with the same speed, $P_{1}$ and $P_{2}$ traverse two cycloid curves. Considering the red line connecting $P_{1}$ and $P_{2}$ at a given time, one proves the line is always tangent to the lower arc at $P_{2}$ and orthogonal to the upper arc at $P_{1}$. Let $Q$ be the point in common between the upper and lower circles at the given time. Then: • $P_{1},Q,P_{2}$ are colinear: indeed the equal rolling speed gives equal angles ${\widehat {P_{1}O_{1}Q}}={\widehat {P_{2}O_{2}Q}}$, and thus ${\widehat {O_{1}QP_{1}}}={\widehat {O_{2}QP_{2}}}$ . The point $Q$ lies on the line $O_{1}O_{2}$ therefore ${\widehat {P_{1}QO_{1}}}+{\widehat {P_{1}QO_{2}}}=\pi $ and analogously ${\widehat {P_{2}QO_{2}}}+{\widehat {P_{2}QO_{1}}}=\pi $. From the equality of ${\widehat {O_{1}QP_{1}}}$ and ${\widehat {O_{2}QP_{2}}}$ one has that also ${\widehat {P_{1}QO_{2}}}={\widehat {P_{2}QO_{1}}}$. It follows ${\widehat {P_{1}QO_{1}}}+{\widehat {P_{2}QO_{1}}}=\pi $ . • If $A$ is the meeting point between the perpendicular from $P_{1}$ to the line segment $O_{1}O_{2}$ and the tangent to the circle at $P_{2}$ , then the triangle $P_{1}AP_{2}$ is isosceles, as is easily seen from the construction: ${\widehat {QP_{2}A}}={\tfrac {1}{2}}{\widehat {P_{2}O_{2}Q}}$ and ${\widehat {QP_{1}A}}={\tfrac {1}{2}}{\widehat {QO_{1}R}}=$${\tfrac {1}{2}}{\widehat {QO_{1}P_{1}}}$ . For the previous noted equality between ${\widehat {P_{1}O_{1}Q}}$ and ${\widehat {QO_{2}P_{2}}}$ then ${\widehat {QP_{1}A}}={\widehat {QP_{2}A}}$ and $P_{1}AP_{2}$ is isosceles. • Drawing from $P_{2}$ the orthogonal segment to $O_{1}O_{2}$, from $P_{1}$ the straight line tangent to the upper circle, and calling $B$ the meeting point, one sees that $P_{1}AP_{2}B$ is a rhombus using the theorems on angles between parallel lines • Now consider the velocity $V_{2}$ of $P_{2}$ . It can be seen as the sum of two components, the rolling velocity $V_{a}$ and the drifting velocity $V_{d}$, which are equal in modulus because the circles roll without skidding. $V_{d}$ is parallel to $P_{1}A$, while $V_{a}$ is tangent to the lower circle at $P_{2}$ and therefore is parallel to $P_{2}A$. The rhombus constituted from the components $V_{d}$ and $V_{a}$ is therefore similar (same angles) to the rhombus $BP_{1}AP_{2}$ because they have parallel sides. Then $V_{2}$, the total velocity of $P_{2}$, is parallel to $P_{2}P_{1}$ because both are diagonals of two rhombuses with parallel sides and has in common with $P_{1}P_{2}$ the contact point $P_{2}$. Thus the velocity vector $V_{2}$ lies on the prolongation of $P_{1}P_{2}$ . Because $V_{2}$ is tangent to the cycloid at $P_{2}$, it follows that also $P_{1}P_{2}$ coincides with the tangent to the lower cycloid at $P_{2}$. • Analogously, it can be easily demonstrated that $P_{1}P_{2}$ is orthogonal to $V_{1}$ (the other diagonal of the rhombus). • This proves that the tip of a wire initially stretched on a half arch of the lower cycloid and fixed to the upper circle at $P_{1}$ will follow the point along its path without changing its length because the speed of the tip is at each moment orthogonal to the wire (no stretching or compression). The wire will be at the same time tangent at $P_{2}$ to the lower arc because of the tension and the facts demonstrated above. (If it were not tangent there would be a discontinuity at $P_{2}$ and consequently unbalanced tension forces.) Area Using the above parameterization $ x=r(t-\sin t),\ y=r(1-\cos t)$, the area under one arch, $0\leq t\leq 2\pi ,$ is given by: $A=\int _{x=0}^{2\pi r}y\,dx=\int _{t=0}^{2\pi }r^{2}(1-\cos t)^{2}dt=3\pi r^{2}.$ This is three times the area of the rolling circle. Arc length The arc length S of one arch is given by ${\begin{aligned}S&=\int _{0}^{2\pi }{\sqrt {\left({\frac {dx}{dt}}\right)^{2}+\left({\frac {dy}{dt}}\right)^{2}}}dt\\&=\int _{0}^{2\pi }r{\sqrt {2-2\cos t}}\,dt\\&=2r\int _{0}^{2\pi }\sin {\frac {t}{2}}\,dt\\&=8r.\end{aligned}}$ Another geometric way to calculate the length of the cycloid is to notice that when a wire describing an involute has been completely unwrapped from half an arch, it extends itself along two diameters, a length of 4r. This is thus equal to half the length of arch, and that of a complete arch is 8r. Cycloidal pendulum If a simple pendulum is suspended from the cusp of an inverted cycloid, such that the string is constrained to be tangent to one of its arches, and the pendulum's length L is equal to that of half the arc length of the cycloid (i.e., twice the diameter of the generating circle, L = 4r), the bob of the pendulum also traces a cycloid path. Such a pendulum is isochronous, with equal-time swings regardless of amplitude. Introducing a coordinate system centred in the position of the cusp, the equation of motion is given by: ${\begin{aligned}x&=r[2\theta (t)+\sin 2\theta (t)]\\y&=r[-3-\cos 2\theta (t)],\end{aligned}}$ where $\theta $ is the angle that the straight part of the string makes with the vertical axis, and is given by $\sin \theta (t)=A\cos(\omega t),\qquad \omega ^{2}={\frac {g}{L}}={\frac {g}{4r}},$ where A < 1 is the "amplitude", $\omega $ is the radian frequency of the pendulum and g the gravitational acceleration. The 17th-century Dutch mathematician Christiaan Huygens discovered and proved these properties of the cycloid while searching for more accurate pendulum clock designs to be used in navigation.[18] Related curves Several curves are related to the cycloid. • Trochoid: generalization of a cycloid in which the point tracing the curve may be inside the rolling circle (curtate) or outside (prolate). • Hypocycloid: variant of a cycloid in which a circle rolls on the inside of another circle instead of a line. • Epicycloid: variant of a cycloid in which a circle rolls on the outside of another circle instead of a line. • Hypotrochoid: generalization of a hypocycloid where the generating point may not be on the edge of the rolling circle. • Epitrochoid: generalization of an epicycloid where the generating point may not be on the edge of the rolling circle. All these curves are roulettes with a circle rolled along another curve of uniform curvature. The cycloid, epicycloids, and hypocycloids have the property that each is similar to its evolute. If q is the product of that curvature with the circle's radius, signed positive for epi- and negative for hypo-, then the similitude ratio of curve to evolute is 1 + 2q. The classic Spirograph toy traces out hypotrochoid and epitrochoid curves. Other uses The cycloidal arch was used by architect Louis Kahn in his design for the Kimbell Art Museum in Fort Worth, Texas. It was also used by Wallace K. Harrison in the design of the Hopkins Center at Dartmouth College in Hanover, New Hampshire.[19] Early research indicated that some transverse arching curves of the plates of golden age violins are closely modeled by curtate cycloid curves.[20] Later work indicates that curtate cycloids do not serve as general models for these curves,[21] which vary considerably. See also • Cyclogon • Cycloid gear • List of periodic functions • Tautochrone curve References 1. Cajori, Florian (1999). A History of Mathematics. New York: Chelsea. p. 177. ISBN 978-0-8218-2102-2. 2. Hart, Sarah (7 April 2023). "The Wondrous Connections Between Mathematics and Literature". New York Times. Retrieved 7 April 2023. 3. Tannery, Paul (1883), "Pour l'histoire des lignes et surfaces courbes dans l'antiquité", Mélanges, Bulletin des sciences mathématiques et astronomiques, Ser. 2, 7: 278–291, p. 284: Avant de quitter la citation de Jamblique, j'ajouterai que, dans la courbe de double mouvement de Carpos, il est difficile de ne pas reconnaître la cycloïde dont la génération si simple n'a pas dû échapper aux anciens. [Before leaving the citation of Iamblichus, I will add that, in the curve of double movement of Carpus, it is difficult not to recognize the cycloid, whose so-simple generation couldn't have escaped the ancients.] (cited in Whitman 1943); 4. Wallis, D. (1695). "An Extract of a Letter from Dr. Wallis, of May 4. 1697, Concerning the Cycloeid Known to Cardinal Cusanus, about the Year 1450; and to Carolus Bovillus about the Year 1500". Philosophical Transactions of the Royal Society of London. 19 (215–235): 561–566. doi:10.1098/rstl.1695.0098. (Cited in Günther, p. 5) 5. Whitman, E. A. (May 1943), "Some historical notes on the cycloid", The American Mathematical Monthly, 50 (5): 309–315, doi:10.2307/2302830, JSTOR 2302830 (subscription required) 6. Cajori, Florian (1999), A History of Mathematics (5th ed.), p. 162, ISBN 0-8218-2102-4(Note: The first (1893) edition and its reprints state that Galileo invented the cycloid. According to Phillips, this was corrected in the second (1919) edition and has remained through the most recent (fifth) edition.) 7. Roidt, Tom (2011). Cycloids and Paths (PDF) (MS). Portland State University. p. 4. Archived (PDF) from the original on 2022-10-09. 8. Cantor, Moritz (1892), Vorlesungen über Geschichte der Mathematik, Bd. 2, Leipzig: B. G. Teubner, OCLC 25376971 9. Günther, Siegmund (1876), Vermischte untersuchungen zur geschichte der mathematischen wissenschaften, Leipzig: Druck und Verlag Von B. G. Teubner, p. 352, OCLC 2060559 10. Phillips, J. P. (May 1967), "Brachistochrone, Tautochrone, Cycloid—Apple of Discord", The Mathematics Teacher, 60 (5): 506–508, doi:10.5951/MT.60.5.0506, JSTOR 27957609(subscription required) 11. Victor, Joseph M. (1978), Charles de Bovelles, 1479-1553: An Intellectual Biography, p. 42, ISBN 978-2-600-03073-1 12. Martin, J. (2010). "The Helen of Geometry". The College Mathematics Journal. 41: 17–28. doi:10.4169/074683410X475083. S2CID 55099463. 13. de Bouelles, Charles (1503), Introductio in geometriam ... Liber de quadratura circuli. Liber de cubicatione sphere. Perspectiva introductio., OCLC 660960655 14. Torricelli, Evangelista (1644), Opera geometrica, OCLC 55541940 15. Walker, Evelyn (1932), A Study of Roberval's Traité des Indivisibles, Columbia University (cited in Whitman 1943); 16. Conner, James A. (2006), Pascal's Wager: The Man Who Played Dice with God (1st ed.), HarperCollins, pp. 224, ISBN 9780060766917 17. Roberts, Charles (2018). Elementary Differential Equations: Applications, Models, and Computing (2nd illustrated ed.). CRC Press. p. 141. ISBN 978-1-4987-7609-7. Extract of page 141, equation (f) with their K=2r 18. C. Huygens, "The Pendulum Clock or Geometrical Demonstrations Concerning the Motion of Pendula (sic) as Applied to Clocks," Translated by R. J. Blackwell, Iowa State University Press (Ames, Iowa, USA, 1986). 19. 101 Reasons to Love Dartmouth, Dartmouth Alumni Magazine, 2016 20. Playfair, Q. "Curtate Cycloid Arching in Golden Age Cremonese Violin Family Instruments". Catgut Acoustical Society Journal. II. 4 (7): 48–58. 21. Mottola, RM (2011). "Comparison of Arching Profiles of Golden Age Cremonese Violins and Some Mathematically Generated Curves". Savart Journal. 1 (1). Further reading • An application from physics: Ghatak, A. & Mahadevan, L. Crack street: the cycloidal wake of a cylinder tearing through a sheet. Physical Review Letters, 91, (2003). link.aps.org • Edward Kasner & James Newman (1940) Mathematics and the Imagination, pp 196–200, Simon & Schuster. • Wells D (1991). The Penguin Dictionary of Curious and Interesting Geometry. New York: Penguin Books. pp. 445–47. ISBN 0-14-011813-6. External links • O'Connor, John J.; Robertson, Edmund F., "Cycloid", MacTutor History of Mathematics Archive, University of St Andrews • Weisstein, Eric W. "Cycloid". MathWorld. Retrieved April 27, 2007. • Cycloids at cut-the-knot • A Treatise on The Cycloid and all forms of Cycloidal Curves, monograph by Richard A. Proctor, B.A. posted by Cornell University Library. • Cycloid Curves by Sean Madsen with contributions by David von Seggern, Wolfram Demonstrations Project. • Cycloid on PlanetPTC (Mathcad) • A VISUAL Approach to CALCULUS problems by Tom Apostol Authority control: National • Israel • United States • Latvia	Wikipedia
The Higher Infinite The Higher Infinite: Large Cardinals in Set Theory from their Beginnings is a monograph in set theory by Akihiro Kanamori, concerning the history and theory of large cardinals, infinite sets characterized by such strong properties that their existence cannot be proven in Zermelo–Fraenkel set theory (ZFC).[1] This book was published in 1994 by Springer-Verlag in their series Perspectives in Mathematical Logic, with a second edition in 2003 in their Springer Monographs in Mathematics series,[2] and a paperback reprint of the second edition in 2009 (ISBN 978-3-540-88866-6).[3] Topics Not counting introductory material and appendices, there are six chapters in The Higher Infinite, arranged roughly in chronological order by the history of the development of the subject. The author writes that he chose this ordering "both because it provides the most coherent exposition of the mathematics and because it holds the key to any epistemological concerns".[1][4] In the first chapter, "Beginnings",[4] the material includes inaccessible cardinals, Mahlo cardinals, measurable cardinals, compact cardinals and indescribable cardinals. The chapter covers the constructible universe and inner models, elementary embeddings and ultrapowers, and a result of Dana Scott that measurable cardinals are inconsistent with the axiom of constructibility.[5][6] The second chapter, "Partition properties",[4] includes the partition calculus of Paul Erdős and Richard Rado, trees and Aronszajn trees, the model-theoretic study of large cardinals, and the existence of the set 0# of true formulae about indiscernibles. It also includes Jónsson cardinals and Rowbottom cardinals.[5][6] Next are two chapters on "Forcing and sets of reals" and "Aspects of measurability".[4] The main topic of the first of these chapters is forcing, a technique introduced by Paul Cohen for proving consistency and inconsistency results in set theory; it also includes material in descriptive set theory. The second of these chapters covers the application of forcing by Robert M. Solovay to prove the consistency of measurable cardinals, and related results using stronger notions of forcing.[5] Chapter five is "Strong hypotheses".[4] It includes material on supercompact cardinals and their reflection properties, on huge cardinals, on Vopěnka's principle,[5] on extendible cardinals, on strong cardinals, and on Woodin cardinals.[6] The book concludes with the chapter "Determinacy",[4] involving the axiom of determinacy and the theory of infinite games.[5] Reviewer Frank R. Drake views this chapter, and the proof in it by Donald A. Martin of the Borel determinacy theorem, as central for Kanamori, "a triumph for the theory he presents".[7] Although quotations expressing the philosophical positions of researchers in this area appear throughout the book,[1] more detailed coverage of issues in the philosophy of mathematics regarding the foundations of mathematics are deferred to an appendix.[8] Audience and reception Reviewer Pierre Matet writes that this book "will no doubt serve for many years to come as the main reference for large cardinals",[4] and reviewers Joel David Hamkins, Azriel Lévy and Philip Welch express similar sentiments.[1][6][8] Hamkins writes that the book is "full of historical insight, clear writing, interesting theorems, and elegant proofs".[1] Because this topic uses many of the important tools of set theory more generally, Lévy recommends the book "to anybody who wants to start doing research in set theory",[6] and Welch recommends it to all university libraries.[8] References 1. Hamkins, Joel David (August 2000), "Review of The Higher Infinite", Studia Logica, 65 (3): 443–446, JSTOR 20016207 2. MR1994835; Zbl 1022.03033 3. MR2731169; Zbl 1154.03033 4. Matet, Pierre (1996), "Review of The Higher Infinite", Mathematical Reviews, MR 1321144 5. Weese, M., "Review of The Higher Infinite", zbMATH, Zbl 0813.03034 6. Lévy, Azriel (March 1996), "Review of The Higher Infinite", Journal of Symbolic Logic, 61 (1): 334–336, doi:10.2307/2275615, JSTOR 2275615, S2CID 119055819 7. Drake, F. R. (1997), "Review of The Higher Infinite", Bulletin of the London Mathematical Society, 29 (1): 111–113, doi:10.1112/S0024609396221678 8. Welch, P. D. (February 1998), "Review of The Higher Infinite", Proceedings of the Edinburgh Mathematical Society, 41 (1): 208–209, doi:10.1017/s0013091500019532 External links • The Higher Infinite(registration required) (1st edition) at the Internet Archive	Wikipedia
The History of Mathematical Tables The History of Mathematical Tables: from Sumer to Spreadsheets is an edited volume in the history of mathematics on mathematical tables. It was edited by Martin Campbell-Kelly, Mary Croarken, Raymond Flood, and Eleanor Robson, developed out of the presentations at a conference on the subject organised in 2001 by the British Society for the History of Mathematics,[1][2] and published in 2003 by the Oxford University Press. Topics An introductory chapter classifies tables broadly according to whether they are intended as aids to calculation (based on mathematical formulas) or as analyses and records of data, and further subdivides them according to how they were compiled.[2] Following this, the contributions to the book include articles on the following topics:[1][2][3] • Tables of data in Babylonian mathematics, administration, and astronomy, by Eleanor Robson • Early tables of logarithms, by Graham Jagger • Life tables in actuarial science, by Christopher Lewin and Margaret de Valois • The work of Gaspard de Prony constructing mathematical tables in revolutionary France, by Ivor Grattan-Guinness • Difference engines, by Michael Williams • The uses and advantages of machines in table-making, and error correction in mechanical tables, by Doron Swade • Astronomical tables, by Arthur Norberg • The data processing and statistical analyses used to produce tables of census data from punched cards, by Edward Higgs • British table-making committees, and the transition from calculators to computers, by Mary Croarken • The Mathematical Tables Project of the Works Progress Administration, in New York during the Great Depression of the 1930s and early 1940s, by David Alan Grier • The work of the British Nautical Almanac Office, by George A. Wilkins • Spreadsheets, by Martin Campbell-Kelly. The work is presented on VIII + 361 pages in a unified format with illustrations throughout, and with the historical and biographical context of the material set aside in separate text boxes.[1] Audience and reception Reviewer Paul J. Campbell finds it ironic that, unlike the works it discusses, "there are no tables in the back of the book".[4] Reviewer Sandy L. Zabell calls the book "interesting and highly readable".[2] Both Peggy A. Kidwell and Fernando Q. Gouvêa note several topics that would have been worthwhile to include, including tables in mathematics in medieval Islam or other non-Western cultures, the book printing industry that provided inexpensive books of tables in the 19th century, and the development of mathematical tables in Germany. As Kidwell writes, "like most good books, this one not only tells good stories, but leaves the reader hoping to learn more". Gouvêa evaluates the book as being useful in its coverage of a topic often missed in broader surveys of the history of mathematics, of interest both to historians of mathematics and to a more general audience interested in the development of these topics, and "a must-have for libraries".[5] References 1. Kidwell, Peggy Aldrich (July 2004), "Review of The History of Mathematical Tables", Technology and Culture, 45 (3): 662–664, doi:10.1353/tech.2004.0136, JSTOR 40060668 2. Zabell, S. L. (June 2005), "Review of The History of Mathematical Tables", Isis, 96 (2): 258, doi:10.1086/491481, JSTOR 10.1086/491481 3. Gray, Jeremy (September 2004), "Table mountain (review of The History of Mathematical Tables)", Notes and Records of the Royal Society of London, 58 (3): 311–312, JSTOR 4142071 4. Campbell, Paul J. (April 2004), "Review of The History of Mathematical Tables", Mathematics Magazine, 77 (2): 163, doi:10.1080/0025570X.2004.11953245, JSTOR 3219109 5. Gouvêa, Fernando Q. (May 2004), "Review of The History of Mathematical Tables", MAA Reviews, Mathematical Association of America External links • The History of Mathematical Tables on the Internet Archive	Wikipedia
The History of Mathematics: A Very Short Introduction The History of Mathematics: A Very Short Introduction is a book on the history of mathematics. Rather than giving a systematic overview of the historical development of mathematics, it provides an introduction to how the discipline of the history of mathematics is studied and researched, through a sequence of case studies in historical topics. It was written by British historian of mathematics Jackie Stedall (1950–2014), and published in 2012 as part of the Oxford University Press Very Short Introductions series of books. It has been listed as essential for mathematics libraries, and won the Neumann Prize for books on the history of mathematics. Topics The History of Mathematics consists of seven chapters,[1] featuring many case studies.[2][3] Its first, "Mathematics: myth and history", gives a case study of the history of Fermat's Last Theorem and of Wiles's proof of Fermat's Last Theorem,[4] making a case that the proper understanding of this history should go beyond a chronicle of individual mathematicians and their accomplishments,[5] and that mathematics has always been done as part of a cultural milieu rather than as an isolated activity in an ivory tower.[2][6] The second chapter takes a wider view, using the Chinese Book on Numbers and Computation as one of its case studies;[2] it asks which cultural accomplishments should be counted as mathematics, and who counted as mathematicians, making the point that these two questions are not the same.[4] The third chapter covers the way that ancient mathematics has been passed down to present historians, the ways that documentary evidence has been destroyed over the years, and the ways that, when it has been preserved, it has been altered in transmission.[4] A case study here is Euclid's Elements and its transmission through Mathematics in medieval Islam to Europe.[7] Chapter 4 concerns the ways in which mathematics has been taught and learned, beginning with the scribal schools of Babylon and including also the historical role of women in mathematics, and chapter 5 concerns the ways in which its practitioners have supported themselves.[2][4] Chapter 6 provides another case study, of the Pythagorean theorem, the different ways in which it has been reinterpreted, or other pieces of mathematics reinterpreted in its light, and the ways in which historians have shifted their views on questions of who was first for results like this that have been found across multiple ancient cultures.[4] Another point of both chapters 3 and 6 is the importance of understanding mathematical works within the view of their subject at the time, rather than reinterpreting them anachronistically into modern concepts of mathematics.[1][2][6] A final chapter discusses the history of the history of mathematics, as a discipline.[4] Audience and reception The History of Mathematics is a quick read,[6] largely avoids the use of formulas or other technical material,[7] and is accessible to readers without significant background knowledge in this area.[1][3] As well as being of general interest, it is suitable as reading material for courses in the history of mathematics,[5][7] and informative for professional mathematicians and historians of mathematics as well as for students.[2][3][5] A rare negative opinion is provided by reviewer Franz Lemmermeyer, a German historian of number theory, who strongly disagrees with the book's philosophy of studying everyday work in mathematics instead of the achievements of great men, calling it "rewriting the history of mathematics in a politically corrected way". Nevertheless, Lemmermeyer writes that it "should be read by everyone interested in the history of mathematics".[8] The Basic Library List Committee of the Mathematical Association of America has listed The History of Mathematics as an essential book for all undergraduate mathematics libraries,[5] and it won the 2013 Neumann Prize of the British Society for the History of Mathematics (named for Peter M. Neumann) for the best general-audience English-language book on the history of mathematics.[9] References 1. Ferguson, Wallace A. (June 2013), "Review of The History of Mathematics: A Very Short Introduction", IMA Reviews, Institute of Mathematics and its Applications 2. Blanco, Mònica (2015), "Review of The History of Mathematics: A Very Short Introduction", Actes d'História de la Ciència i de la Técnica (in Catalan), 8: 161–163 3. Harkleroad, Leon, "Review of The History of Mathematics: A Very Short Introduction", MathSciNet, MR 3137003 4. Sonar, Thomas (September 2014), "Review of The History of Mathematics: A Very Short Introduction", BSHM Bulletin: Journal of the British Society for the History of Mathematics, 29 (3): 217–219, doi:10.1080/17498430.2014.920217 5. Gouvêa, Fernando Q. (21 December 2012), "Review of The History of Mathematics: A Very Short Introduction", MAA Reviews, Mathematical Association of America, retrieved 2021-07-30 6. Leversha, Gerry (March 2014), "Review of The History of Mathematics: A Very Short Introduction", The Mathematical Gazette, 98 (541): 155–156, doi:10.1017/s0025557200000917, JSTOR 24496613 7. Schneebeli, H. R. (2013), "Review of The History of Mathematics: A Very Short Introduction", Elemente der Mathematik (in German), 68 (3): 136, doi:10.4171/EM/231 8. Lemmermeyer, Franz, "Review of The History of Mathematics: A Very Short Introduction", zbMATH, Zbl 1244.00001 9. Neumann Prize, British Society for the History of Mathematics, retrieved 2021-07-30	Wikipedia
Scott Flansburg Scott Flansburg (born December 28, 1963) is an American dubbed "The Human Calculator" and listed in the Guinness Book of World Records for speed of mental calculation. He is the annual host and ambassador for The National Counting Bee, a math educator, and media personality. He has published the books Math Magic and Math Magic for Your Kids.[1] Scott Flansburg Flansburg in 2011 Born (1963-12-28) December 28, 1963 Herkimer, New York Pen nameThe Human Calculator OccupationMental calculator LanguageEnglish NationalityAmerican Notable worksMath Magic Math Magic for Kids Notable awardsGuinness World Record 2001 Website www.scottflansburg.com Biography Early life Flansburg was born December 28, 1963, in Herkimer, New York. He has stated that he was nine years old when he discovered his mental calculation abilities. He says he wasn't paying attention in math class when his teacher asked him to add 4 numbers on the blackboard. Instead of adding the columns from right to left, as they'd been taught, he added them from left to right, and solved the problem. Intrigued by math, he began keeping a running tally of his family's groceries at the store so his father could give the cashier an exact check before the total was rung up.[2] In his youth he also began noticing that the shapes and numbers of angles in numbers were clues to their values, and began counting from 0 to 9 on his fingers instead of 1 to 10.[3] Early career Flansburg can mentally add, subtract, multiply, divide, and find square and cube roots almost instantly, with calculator accuracy. Around 1990 he began using his abilities in an entertainment and educational context.[4] In 1991, he was involved in the creation of "The Human Calculator System", a product designed for direct-marketing salespeople, consisting of a study guide and four cassette recordings teaching his method. In the spring of that year, he appeared on the infomercial show "Amazing Discoveries", hosted by Mike Levey, to promote his product. At the beginning of the program, he was introduced as "The Human Calculator" for the first time in national media.[5] Flansburg was dubbed "The Human Calculator" by Regis Philbin after appearing on Live with Regis and Kathy Lee.[2] The Guinness Book of World Records listed him as "Fastest Human Calculator"[4] in 2001 and 2003,[6] after he broke the record for adding the same number to itself more times in 15 seconds than someone could do with a calculator.[7] In 1999 he invented a 13-month calendar using zero as a day, month, and year, which he called "The Human Calculator Calendar".[6] In 1998 he published the Harper Paperbacks book Math Magic for Your Kids: Hundreds of Games and Exercises from the Human Calculator to Make Math Fun and Easy.[8] A revised edition of his book Math Magic: How to Master Everyday Math Problems was published in 2004.[3] The Counting Bee Flansburg is the creator of the Counting Bee, an annual, fast-paced competition to find the fastest human counters in different age categories around the world. Based on a 15-second countdown, "mathletes" race to compute as many numeric answers in each level. For each level, they are given random starting numbers and told to count by a fixed number. The starting numbers and counting numbers increase in difficulty with each level. The top mathletes are those who successfully complete the most levels. State-wide and country-wide competitions determine Counting Bee World Champions by age groups. The Human Calculator Scott Flansburg developed a mental math program called 'The Human Calculator.[9]' Scott's course provides a step-by-step guide to becoming 'A Human Calculator.' Scott created this course to help people master mental mathematics. The training covers the unknown and hidden patterns behind numbers and how they work in our daily life. As an educator Since about 1990[3] Flansburg has regularly given lectures and presentations at schools.[7] He has been a presenter at organizations such as NASA, IBM, The Smithsonian Institution, the National Council of Teachers of Mathematics,[4] and the Mental Calculation World Cup. The latter described Flansburg as "more an auditory than a visual [mental] calculator".[10] One of Flansburg's "personal missions" is to use education to elevate math confidence and self-esteem in adults and children. "Why has it become so socially acceptable to be bad at math?," he stated. "If you were illiterate, you wouldn’t say that on TV, but you can say that you are bad at math. We have to change the attitude." He believes students should become proficient with calculation methods rather than relying on table memorization.[3] Flansburg is the annual host and ambassador for World Maths Day,[11] and an official promoter of the American Math Challenge, a competition for students preparing for World Math Day.[7] Media appearances Flansburg has appeared on television shows such as The Oprah Winfrey Show, The Ellen DeGeneres Show, The Tonight Show with Jay Leno, and Larry King Live. On April 26, 2009, on the Japanese primetime show Asahi's Otona no Sonata, he broke his own world record with 37 answers in 15 seconds.[11] He was featured as The Human Calculator in the first episode of Stan Lee's Superhumans, which aired on The History Channel on August 5, 2010. Part of the episode analyzed his brain activity.[12] An fMRI scan while he was doing complex calculations revealed that activity in the Brodmann area 44 region of the frontal cortex was absent; instead, there was activity somewhat higher from area 44 and closer to the motor cortex.[13] In January 2016, Flansburg hosted the TV show The Human Calculator on H2.[14] Personal life Flansburg resides in Herkimer, New York. Publications • Math Magic for Your Kids (1998)[8] • Math Magic (2004)[15] References 1. "Scott Flansburg". HarperCollins Publishers. Retrieved 2011-09-09. 2. Smith, Jacqueline (August 17, 2009). "'Human calculator' looking for Kiwi mathletes". New Zealand Herald. Retrieved 2011-09-09. 3. "Scott Flansburg: The Math King". Children's Literature Network. March 21, 2011. Retrieved 2011-09-09. 4. "Meet Scott Flansburg". ScottFlansburg.com. Retrieved 2011-09-09. 5. "Amazing Discoveries Infomercial: Human Calculator". YouTube. Retrieved 7 June 2018. 6. Noory, George (June 16, 2002). "Guests: Scott Flansburg". Coast to Coast AM. Retrieved 2011-09-09. 7. Reiter, Angela (October 22, 2010). "Woodlands Academy Mesmerized By The Human Calculator". Trib Local. Retrieved 2011-09-09. 8. Scott, Flansburg (1998). Math Magic for Your Kids. Harper Paperbacks. ISBN 978-0-06-097731-3. 9. Flansburg, Scott. "The Human Calculator \| Mental Math Trainer - Scott Flansburg". scottflansburg.com/. Retrieved 2021-09-19. 10. Brain, Mr. (July 12, 2010). "Mental Calculation World Cup 2010". Mental Calculation World Cup. Retrieved 2011-09-09. 11. "Scott Flansburg: The Human Calculator". ScottFlansburg.com. Archived from the original on 2011-09-24. Retrieved 2011-09-09. 12. "Featured Superhumans: The Human Calculator". History.com. Retrieved 2011-09-09. 13. "Electro Man: Episode 101". Stan Lee's Superhumans. August 5, 2011. 14. The Human Calculator Full Episodes, Video & More HISTORY 15. Flansburg, Scott (1993). Math Magic. Harper Paperbacks. ISBN 978-0-06-072635-5. External links • Official website Authority control International • FAST • ISNI • VIAF National • Israel • United States	Wikipedia
The Indispensability of Mathematics The Indispensability of Mathematics is a 2001 book by Mark Colyvan in which he examines the Quine–Putnam indispensability argument in the philosophy of mathematics. This thesis is based on the premise that mathematical entities are placed on the same ontological foundation as other theoretical entities indispensable to our best scientific theories.[1][2][3] The Indispensability of Mathematics AuthorMark Colyvan SubjectPhilosophy of mathematics Published2001 PublisherOxford University Press Pages192 pp. ISBN9780195137545 References 1. Melia, Joseph (March 2003). "Mark Colyvan, The Indispensability of Mathematics". Metascience. 12 (1): 55–58. doi:10.1023/A:1024411117330. S2CID 169768062. 2. Cole, J. (1 March 2003). "Review: The Indispensability of Mathematics". Mind. 112 (446): 331–336. doi:10.1093/mind/112.446.331. ISSN 0026-4423. Retrieved 26 October 2018. 3. Cheyne, C. (September 2002). "The Indispensability of Mathematics". Australasian Journal of Philosophy. 80 (3): 378–379. doi:10.1080/713659474. ISSN 0004-8402. S2CID 170188260. External links • The Indispensability of Mathematics	Wikipedia
Johansen test In statistics, the Johansen test,[1] named after Søren Johansen, is a procedure for testing cointegration of several, say k, I(1) time series.[2] This test permits more than one cointegrating relationship so is more generally applicable than the Engle–Granger test which is based on the Dickey–Fuller (or the augmented) test for unit roots in the residuals from a single (estimated) cointegrating relationship.[3] There are two types of Johansen test, either with trace or with eigenvalue, and the inferences might be a little bit different.[4] The null hypothesis for the trace test is that the number of cointegration vectors is r = r* < k, vs. the alternative that r = k. Testing proceeds sequentially for r* = 1,2, etc. and the first non-rejection of the null is taken as an estimate of r. The null hypothesis for the "maximum eigenvalue" test is as for the trace test but the alternative is r = r* + 1 and, again, testing proceeds sequentially for r* = 1,2,etc., with the first non-rejection used as an estimator for r. Just like a unit root test, there can be a constant term, a trend term, both, or neither in the model. For a general VAR(p) model: $X_{t}=\mu +\Phi D_{t}+\Pi _{p}X_{t-p}+\cdots +\Pi _{1}X_{t-1}+e_{t},\quad t=1,\dots ,T$ There are two possible specifications for error correction: that is, two vector error correction models (VECM): 1. The longrun VECM: $\Delta X_{t}=\mu +\Phi D_{t}+\Pi X_{t-p}+\Gamma _{p-1}\Delta X_{t-p+1}+\cdots +\Gamma _{1}\Delta X_{t-1}+\varepsilon _{t},\quad t=1,\dots ,T$ where $\Gamma _{i}=\Pi _{1}+\cdots +\Pi _{i}-I,\quad i=1,\dots ,p-1.\,$ 2. The transitory VECM: $\Delta X_{t}=\mu +\Phi D_{t}+\Pi X_{t-1}-\sum _{j=1}^{p-1}\Gamma _{j}\Delta X_{t-j}+\varepsilon _{t},\quad t=1,\cdots ,T$ where $\Gamma _{i}=\left(\Pi _{i+1}+\cdots +\Pi _{p}\right),\quad i=1,\dots ,p-1.\,$ Be aware that the two are the same. In both VECM, $\Pi =\Pi _{1}+\cdots +\Pi _{p}-I.\,$ Inferences are drawn on Π, and they will be the same, so is the explanatory power. References 1. Johansen, Søren (1991). "Estimation and Hypothesis Testing of Cointegration Vectors in Gaussian Vector Autoregressive Models". Econometrica. 59 (6): 1551–1580. JSTOR 2938278. 2. For the presence of I(2) variables see Ch. 9 of Johansen, Søren (1995). Likelihood-based Inference in Cointegrated Vector Autoregressive Models. Oxford University Press. 3. Davidson, James (2000). Econometric Theory. Wiley. ISBN 0-631-21584-0. 4. Hänninen, R. (2012). "The Law of One Price in United Kingdom Soft Sawnwood Imports – A Cointegration Approach". Modern Time Series Analysis in Forest Products Markets. Springer. p. 66. ISBN 978-94-011-4772-9. Further reading • Banerjee, Anindya; et al. (1993). Co-Integration, Error Correction, and the Econometric Analysis of Non-Stationary Data. New York: Oxford University Press. pp. 266–268. ISBN 0-19-828810-7. • Favero, Carlo A. (2001). Applied Macroeconometrics. New York: Oxford University Press. pp. 56–71. ISBN 0-19-829685-1. • Hatanaka, Michio (1996). Time-Series-Based Econometrics: Unit Roots and Cointegration. New York: Oxford University Press. pp. 219–246. ISBN 0-19-877353-6. • Maddala, G. S.; Kim, In-Moo (1998). Unit Roots, Cointegration, and Structural Change. Cambridge University Press. pp. 198–248. ISBN 0-521-58782-4. Statistics • Outline • Index Descriptive statistics Continuous data Center • Mean • Arithmetic • Arithmetic-Geometric • Cubic • Generalized/power • Geometric • Harmonic • Heronian • Heinz • Lehmer • Median • Mode Dispersion • Average absolute deviation • Coefficient of variation • Interquartile range • Percentile • Range • Standard deviation • Variance Shape • Central limit theorem • Moments • Kurtosis • L-moments • Skewness Count data • Index of dispersion Summary tables • Contingency table • Frequency distribution • Grouped data Dependence • Partial correlation • Pearson product-moment correlation • Rank correlation • Kendall's τ • Spearman's ρ • Scatter plot Graphics • Bar chart • Biplot • Box plot • Control chart • Correlogram • Fan chart • Forest plot • Histogram • Pie chart • Q–Q plot • Radar chart • Run chart • Scatter plot • Stem-and-leaf display • Violin plot Data collection Study design • Effect size • Missing data • Optimal design • Population • Replication • Sample size 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London Mathematical Society The London Mathematical Society (LMS) is one of the United Kingdom's learned societies for mathematics (the others being the Royal Statistical Society (RSS), the Institute of Mathematics and its Applications (IMA), the Edinburgh Mathematical Society and the Operational Research Society (ORS). London Mathematical Society De Morgan House Formation1865 TypeLearned society HeadquartersLondon, WC1 United Kingdom President Ulrike Tillman Key people Catherine Hobbs Iain Gordon (Vice President) Websitewww.lms.ac.uk History The Society was established on 16 January 1865, the first president being Augustus De Morgan. The earliest meetings were held in University College, but the Society soon moved into Burlington House, Piccadilly. The initial activities of the Society included talks and publication of a journal. The LMS was used as a model for the establishment of the American Mathematical Society in 1888. Mary Cartwright was the first woman to be President of the LMS (in 1961–62).[1] The Society was granted a royal charter in 1965, a century after its foundation. In 1998 the Society moved from rooms in Burlington House into De Morgan House (named after the society's first president), at 57–58 Russell Square, Bloomsbury, to accommodate an expansion of its staff. In 2015 the Society celebrated its 150th Anniversary. During the year the anniversary was celebrated with a wide range of meetings, events, and other activities, highlighting the historical and continuing value and prevalence of mathematics in society, and in everyday life. Membership Membership is open to those who are interested in mathematics. Currently, there are four classes of membership, namely: (a) Ordinary, (b) Reciprocity, (c) Associate, and (d) Associate (undergraduate). In addition, Honorary Members of the Society are distinguished mathematicians who are not normally resident in the UK, who are proposed by the Society's Council for election to Membership at a Society Meeting.[2] LMS Activities The Society publishes books and periodicals; organises mathematical conferences; provides funding to promote mathematics research and education; and awards a number of prizes and fellowships for excellence in mathematical research. Grants The Society supports mathematics in the UK through its grant schemes. These schemes provide support for mathematicians at different stages in their careers. The Society’s grants include research grants for mathematicians, early career researchers and computer scientists working at the interface of mathematics and computer science; education grants for teachers and other educators; travel grants to attend conferences; and grants for those with caring responsibilities. Awarding grants is one of the primary mechanisms through which the Society achieves its central purpose, namely to 'promote and extend mathematical knowledge’. Fellowships The Society also offers a range of Fellowships: LMS Early Career Fellowships; LMS Atiyah-Lebanon UK Fellowships; LMS Emmy Noether Fellowships and Grace Chisholm Young Fellowships. Society lectures and meetings The Society organises an annual programme of events and meetings. The programme provides meetings of interest to undergraduates, through early career researchers to established mathematicians. These include LMS-Bath Mathematical Symposia, Lecture Series (Aitken/Forder, Hardy, Invited), Research Schools, LMS Prospects in Mathematics Meeting, Public Lectures, Society Meetings, LMS Undergraduate Summer Schools and Women in Mathematics Days. Publications The Society's periodical publications include five journals: • Bulletin of the London Mathematical Society (1969–present)[3] • Journal of the London Mathematical Society (1926–present)[4] • Proceedings of the London Mathematical Society (1865–present)[5] • Transactions of the London Mathematical Society (2014–present)[6] • Journal of Topology (2006 – present) It also publishes the journal Compositio Mathematica on behalf of its owning foundation, Mathematika on behalf of University College London and copublishes Nonlinearity with the Institute of Physics. It also co-publishes four series of translations: Russian Mathematical Surveys, Izvestiya: Mathematics and Sbornik: Mathematics (jointly with the Russian Academy of Sciences and Turpion), and Transactions of the Moscow Mathematical Society (jointly with the American Mathematical Society). Books The Society publishes two book series, the LMS Lecture Notes and LMS Student Texts. Previously it published a series of Monographs and (jointly with the American Mathematical Society) the History of Mathematics series. An electronic journal, the LMS Journal of Computation and Mathematics ceased publication at the end of 2017. Prizes The named prizes are: • De Morgan Medal (triennial) — the most prestigious • Pólya Prize (two years out of three) • Louis Bachelier Prize (biennial) • Senior Berwick Prize • Senior Whitehead Prize (biennial) • Naylor Prize and Lectureship • Forder Lectureship (biennial) • Berwick Prize • Anne Bennett Prize • Senior Anne Bennett Prize • Fröhlich Prize (biennial) • Shephard Prize • Whitehead Prize (annual) • Hirst Prize In addition, the Society jointly with the Institute of Mathematics and its Applications awards the David Crighton Medal and Christopher Zeeman Medal on alternating years.[7] The LMS also awards the Emmy Noether Fellowship. List of presidents Source: [8] • 1865–1866 Augustus De Morgan • 1866–1868 James Joseph Sylvester • 1868–1870 Arthur Cayley • 1870–1872 William Spottiswoode • 1872–1874 Thomas Archer Hirst • 1874–1876 Henry John Stephen Smith • 1876–1878 Lord Rayleigh • 1878–1880 Charles Watkins Merrifield • 1880–1882 Samuel Roberts • 1882–1884 Olaus Henrici • 1884–1886 James Whitbread Lee Glaisher • 1886–1888 James Cockle • 1888–1890 John James Walker • 1890–1892 Alfred George Greenhill • 1892–1894 Alfred Kempe • 1894–1896 Percy Alexander MacMahon • 1896–1898 Edwin Elliott • 1898–1900 William Thomson, 1st Baron Kelvin • 1900–1902 E. W. Hobson • 1902–1904 Horace Lamb • 1904–1906 Andrew Forsyth • 1906–1908 William Burnside • 1908–1910 William Davidson Niven • 1910–1912 H. F. Baker • 1912–1914 Augustus Edward Hough Love • 1914–1916 Joseph Larmor • 1916–1918 Hector Macdonald • 1918–1920 John Edward Campbell • 1920–1922 Herbert Richmond • 1922–1924 William Henry Young • 1924–1926 Arthur Lee Dixon • 1926–1928 G. H. Hardy • 1928–1929 E. T. Whittaker • 1929–1931 Sydney Chapman • 1931–1933 Alfred Cardew Dixon • 1933–1935 G. N. Watson • 1935–1937 George Barker Jeffery • 1937–1939 Edward Arthur Milne • 1939–1941 G. H. Hardy • 1941–1943 John Edensor Littlewood • 1943–1945 L. J. Mordell • 1945–1947 Edward Charles Titchmarsh • 1947–1949 W. V. D. Hodge • 1949–1951 Max Newman • 1951–1953 George Frederick James Temple • 1953–1955 J. H. C. Whitehead • 1955–1957 Philip Hall • 1957–1959 Harold Davenport • 1959–1961 Hans Heilbronn • 1961–1963 Mary Cartwright • 1963–1965 Arthur Geoffrey Walker • 1965–1967 Graham Higman • 1967–1969 J. A. Todd • 1969–1970 Edward Collingwood • 1970–1972 Claude Ambrose Rogers • 1972–1974 David George Kendall • 1974–1976 Michael Atiyah • 1976–1978 J. W. S. Cassels • 1978–1980 C. T. C. Wall • 1980–1982 Barry Johnson • 1982–1984 Paul Cohn • 1984–1986 Ioan James • 1986–1988 Erik Christopher Zeeman • 1988–1990 John H. Coates • 1990–1992 John Kingman • 1992–1994 John Ringrose • 1994–1996 Nigel Hitchin • 1996–1998 John M. Ball • 1998–2000 Martin J. Taylor • 2000–2002 Trevor Stuart • 2002–2003 Peter Goddard • 2003–2005 Frances Kirwan • 2005–2007 John Toland • 2007–2009 E. Brian Davies • 2009 (interim) John M. Ball • 2009–2011 Angus Macintyre • 2011–2013 Graeme Segal[9] • 2013–2015 Terry Lyons • 2015–2017 Simon Tavaré • 2017–2019 Caroline Series • 2019–2021 Jonathan Keating • 2022– Ulrike Tillman See also • American Mathematical Society • Edinburgh Mathematical Society • European Mathematical Society • List of Mathematical Societies • Council for the Mathematical Sciences • BCS-FACS Specialist Group References 1. O'Connor, J. J.; Robertson, E. F. "Dame Mary Lucy Cartwright". School of Mathematics and Statistics, University of St Andrews. Retrieved 3 April 2019. 2. "Membership classes of London Mathematical Society". 3. "Bulletin of the London Mathematical Society \| London Mathematical Society". 4. "Journal of the London Mathematical Society \| London Mathematical Society". 5. "Proceedings of the London Mathematical Society \| London Mathematical Society". 6. "Transactions of the London Mathematical Society \| London Mathematical Society". 7. "IMA-LMS Prizes". London Mathematical Society. Retrieved 10 February 2020. 8. "List of Presidents of the London Mathematical Society" (PDF). London Mathematical Society. Retrieved 4 October 2018. 9. "2011 LMS Election Results". London Mathematical Society. 18 November 2011. • Oakes, Susan Margaret; Pears, Alan Robson; Rice, Adrian Clifford (2005). The Book of Presidents 1865–1965. London Mathematical Society. ISBN 0-9502734-1-4. External links Wikimedia Commons has media related to London Mathematical Society. • London Mathematical Society website • A History of the London Mathematical Society • MacTutor: The London Mathematical Society Mathematics in the United Kingdom Organizations and Projects • International Centre for Mathematical Sciences • Advisory Committee on Mathematics Education • Association of Teachers of Mathematics • British Society for Research into Learning Mathematics • Council for the Mathematical Sciences • Count On • Edinburgh Mathematical Society • HoDoMS • Institute of Mathematics and its Applications • Isaac Newton Institute • United Kingdom Mathematics Trust • Joint Mathematical Council • Kent Mathematics Project • London Mathematical Society • Making Mathematics Count • Mathematical Association • Mathematics and Computing College • Mathematics in Education and Industry • Megamaths • Millennium Mathematics Project • More Maths Grads • National Centre for Excellence in the Teaching of Mathematics • National Numeracy • National Numeracy Strategy • El Nombre • Numbertime • Oxford University Invariant Society • School Mathematics Project • Science, Technology, Engineering and Mathematics Network • Sentinus Maths schools • Exeter Mathematics School • King's College London Mathematics School • Lancaster University School of Mathematics • University of Liverpool Mathematics School Journals • Compositio Mathematica • Eureka • Forum of Mathematics • Glasgow Mathematical Journal • The Mathematical Gazette • Philosophy of Mathematics Education Journal • Plus Magazine Competitions • British Mathematical Olympiad • British Mathematical Olympiad Subtrust • National Cipher Challenge Awards • Chartered Mathematician • Smith's Prize • Adams Prize • Thomas Bond Sprague Prize • Rollo Davidson Prize The European Mathematical Society International member societies • European Consortium for Mathematics in Industry • European Society for Mathematical and Theoretical Biology National member societies • Austria • Belarus • Belgium • Belgian Mathematical Society • Belgian Statistical Society • Bosnia and Herzegovina • Bulgaria • Croatia • Cyprus • Czech Republic • Denmark • Estonia • Finland • France • Mathematical Society of France • Society of Applied & Industrial Mathematics • Société Francaise de Statistique • Georgia • Germany • German Mathematical Society • Association of Applied Mathematics and Mechanics • Greece • Hungary • Iceland • Ireland • Israel • Italy • Italian Mathematical Union • Società Italiana di Matematica Applicata e Industriale • The Italian Association of Mathematics applied to Economic and Social Sciences • Latvia • Lithuania • Luxembourg • Macedonia • Malta • Montenegro • Netherlands • Norway • Norwegian Mathematical Society • Norwegian Statistical Association • Poland • Portugal • Romania • Romanian Mathematical Society • Romanian Society of Mathematicians • Russia • Moscow Mathematical Society • St. Petersburg Mathematical Society • Ural Mathematical Society • Slovakia • Slovak Mathematical Society • Union of Slovak Mathematicians and Physicists • Slovenia • Spain • Catalan Society of Mathematics • Royal Spanish Mathematical Society • Spanish Society of Statistics and Operations Research • The Spanish Society of Applied Mathematics • Sweden • Swedish Mathematical Society • Swedish Society of Statisticians • Switzerland • Turkey • Ukraine • United Kingdom • Edinburgh Mathematical Society • Institute of Mathematics and its Applications • London Mathematical Society Academic Institutional Members • Abdus Salam International Centre for Theoretical Physics • Academy of Sciences of Moldova • Bernoulli Center • Centre de Recerca Matemàtica • Centre International de Rencontres Mathématiques • Centrum voor Wiskunde en Informatica • Emmy Noether Research Institute for Mathematics • Erwin Schrödinger International Institute for Mathematical Physics • European Institute for Statistics, Probability and Operations Research • Institut des Hautes Études Scientifiques • Institut Henri Poincaré • Institut Mittag-Leffler • Institute for Mathematical Research • International Centre for Mathematical Sciences • Isaac Newton Institute for Mathematical Sciences • Mathematisches Forschungsinstitut Oberwolfach • Mathematical Research Institute • Max Planck Institute for Mathematics in the Sciences • Research Institute of Mathematics of the Voronezh State University • Serbian Academy of Science and Arts • Mathematical Society of Serbia • Stefan Banach International Mathematical Center • Thomas Stieltjes Institute for Mathematics Institutional Members • Central European University • Faculty of Mathematics at the University of Barcelona • Cellule MathDoc Authority control International • ISNI • VIAF National • Catalonia • Israel • United States • Czech Republic Academics • CiNii Other • IdRef 51.5212°N 0.1243°W / 51.5212; -0.1243	Wikipedia
Karatsuba algorithm The Karatsuba algorithm is a fast multiplication algorithm. It was discovered by Anatoly Karatsuba in 1960 and published in 1962.[1][2][3] It is a divide-and-conquer algorithm that reduces the multiplication of two n-digit numbers to three multiplications of n/2-digit numbers and, by repeating this reduction, to at most $n^{\log _{2}3}\approx n^{1.58}$ single-digit multiplications. It is therefore asymptotically faster than the traditional algorithm, which performs $n^{2}$ single-digit products. Karatsuba algorithm ClassMultiplication algorithm The Karatsuba algorithm was the first multiplication algorithm asymptotically faster than the quadratic "grade school" algorithm. The Toom–Cook algorithm (1963) is a faster generalization of Karatsuba's method, and the Schönhage–Strassen algorithm (1971) is even faster, for sufficiently large n. History The standard procedure for multiplication of two n-digit numbers requires a number of elementary operations proportional to $n^{2}\,\!$, or $O(n^{2})\,\!$ in big-O notation. Andrey Kolmogorov conjectured that the traditional algorithm was asymptotically optimal, meaning that any algorithm for that task would require $\Omega (n^{2})\,\!$ elementary operations. In 1960, Kolmogorov organized a seminar on mathematical problems in cybernetics at the Moscow State University, where he stated the $\Omega (n^{2})\,\!$ conjecture and other problems in the complexity of computation. Within a week, Karatsuba, then a 23-year-old student, found an algorithm that multiplies two n-digit numbers in $O(n^{\log _{2}3})$ elementary steps, thus disproving the conjecture. Kolmogorov was very excited about the discovery; he communicated it at the next meeting of the seminar, which was then terminated. Kolmogorov gave some lectures on the Karatsuba result at conferences all over the world (see, for example, "Proceedings of the International Congress of Mathematicians 1962", pp. 351–356, and also "6 Lectures delivered at the International Congress of Mathematicians in Stockholm, 1962") and published the method in 1962, in the Proceedings of the USSR Academy of Sciences. The article had been written by Kolmogorov and contained two results on multiplication, Karatsuba's algorithm and a separate result by Yuri Ofman; it listed "A. Karatsuba and Yu. Ofman" as the authors. Karatsuba only became aware of the paper when he received the reprints from the publisher.[2] Algorithm Basic step The basic principle of Karatsuba's algorithm is divide-and-conquer, using a formula that allows one to compute the product of two large numbers $x$ and $y$ using three multiplications of smaller numbers, each with about half as many digits as $x$ or $y$, plus some additions and digit shifts. This basic step is, in fact, a generalization of a similar complex multiplication algorithm, where the imaginary unit i is replaced by a power of the base. Let $x$ and $y$ be represented as $n$-digit strings in some base $B$. For any positive integer $m$ less than $n$, one can write the two given numbers as $x=x_{1}B^{m}+x_{0},$ $y=y_{1}B^{m}+y_{0},$ where $x_{0}$ and $y_{0}$ are less than $B^{m}$. The product is then ${\begin{aligned}xy&=(x_{1}B^{m}+x_{0})(y_{1}B^{m}+y_{0})\\&=x_{1}y_{1}B^{2m}+(x_{1}y_{0}+x_{0}y_{1})B^{m}+x_{0}y_{0}\\&=z_{2}B^{2m}+z_{1}B^{m}+z_{0},\\\end{aligned}}$ where $z_{2}=x_{1}y_{1},$ $z_{1}=x_{1}y_{0}+x_{0}y_{1},$ $z_{0}=x_{0}y_{0}.$ These formulae require four multiplications and were known to Charles Babbage.[4] Karatsuba observed that $xy$ can be computed in only three multiplications, at the cost of a few extra additions. With $z_{0}$ and $z_{2}$ as before one can observe that ${\begin{aligned}z_{1}&=x_{1}y_{0}+x_{0}y_{1}\\&=x_{1}y_{0}+x_{0}y_{1}+x_{1}y_{1}-x_{1}y_{1}+x_{0}y_{0}-x_{0}y_{0}\\&=x_{1}y_{0}+x_{0}y_{0}+x_{0}y_{1}+x_{1}y_{1}-x_{1}y_{1}-x_{0}y_{0}\\&=(x_{1}+x_{0})y_{0}+(x_{0}+x_{1})y_{1}-x_{1}y_{1}-x_{0}y_{0}\\&=(x_{1}+x_{0})(y_{0}+y_{1})-x_{1}y_{1}-x_{0}y_{0}\\&=(x_{1}+x_{0})(y_{1}+y_{0})-z_{2}-z_{0}.\\\end{aligned}}$ Example To compute the product of 12345 and 6789, where B = 10, choose m = 3. We use m right shifts for decomposing the input operands using the resulting base (Bm = 1000), as: 12345 = 12 · 1000 + 345 6789 = 6 · 1000 + 789 Only three multiplications, which operate on smaller integers, are used to compute three partial results: z2 = 12 × 6 = 72 z0 = 345 × 789 = 272205 z1 = (12 + 345) × (6 + 789) − z2 − z0 = 357 × 795 − 72 − 272205 = 283815 − 72 − 272205 = 11538 We get the result by just adding these three partial results, shifted accordingly (and then taking carries into account by decomposing these three inputs in base 1000 like for the input operands): result = z2 · (Bm)2 + z1 · (Bm)1 + z0 · (Bm)0, i.e. result = 72 · 10002 + 11538 · 1000 + 272205 = 83810205. Note that the intermediate third multiplication operates on an input domain which is less than two times larger than for the two first multiplications, its output domain is less than four times larger, and base-1000 carries computed from the first two multiplications must be taken into account when computing these two subtractions. Recursive application If n is four or more, the three multiplications in Karatsuba's basic step involve operands with fewer than n digits. Therefore, those products can be computed by recursive calls of the Karatsuba algorithm. The recursion can be applied until the numbers are so small that they can (or must) be computed directly. In a computer with a full 32-bit by 32-bit multiplier, for example, one could choose B = 231 and store each digit as a separate 32-bit binary word. Then the sums x1 + x0 and y1 + y0 will not need an extra binary word for storing the carry-over digit (as in carry-save adder), and the Karatsuba recursion can be applied until the numbers to multiply are only one digit long. Time complexity analysis Karatsuba's basic step works for any base B and any m, but the recursive algorithm is most efficient when m is equal to n/2, rounded up. In particular, if n is 2k, for some integer k, and the recursion stops only when n is 1, then the number of single-digit multiplications is 3k, which is nc where c = log23. Since one can extend any inputs with zero digits until their length is a power of two, it follows that the number of elementary multiplications, for any n, is at most $3^{\lceil \log _{2}n\rceil }\leq 3n^{\log _{2}3}\,\!$. Since the additions, subtractions, and digit shifts (multiplications by powers of B) in Karatsuba's basic step take time proportional to n, their cost becomes negligible as n increases. More precisely, if T(n) denotes the total number of elementary operations that the algorithm performs when multiplying two n-digit numbers, then $T(n)=3T(\lceil n/2\rceil )+cn+d$ for some constants c and d. For this recurrence relation, the master theorem for divide-and-conquer recurrences gives the asymptotic bound $T(n)=\Theta (n^{\log _{2}3})\,\!$. It follows that, for sufficiently large n, Karatsuba's algorithm will perform fewer shifts and single-digit additions than longhand multiplication, even though its basic step uses more additions and shifts than the straightforward formula. For small values of n, however, the extra shift and add operations may make it run slower than the longhand method. Implementation Here is the pseudocode for this algorithm, using numbers represented in base ten. For the binary representation of integers, it suffices to replace everywhere 10 by 2.[5] The second argument of the split_at function specifies the number of digits to extract from the right: for example, split_at("12345", 3) will extract the 3 final digits, giving: high="12", low="345". function karatsuba(num1, num2) if (num1 < 10 or num2 < 10) return num1 × num2 /* fall back to traditional multiplication / / Calculates the size of the numbers. / m = max(size_base10(num1), size_base10(num2)) m2 = floor(m / 2) / m2 = ceil (m / 2) will also work / / Split the digit sequences in the middle. / high1, low1 = split_at(num1, m2) high2, low2 = split_at(num2, m2) / 3 recursive calls made to numbers approximately half the size. */ z0 = karatsuba(low1, low2) z1 = karatsuba(low1 + high1, low2 + high2) z2 = karatsuba(high1, high2) return (z2 × 10 ^ (m2 × 2)) + ((z1 - z2 - z0) × 10 ^ m2) + z0 An issue that occurs when implementation is that the above computation of $(x_{1}+x_{0})$ and $(y_{1}+y_{0})$ for $z_{1}$ may result in overflow (will produce a result in the range $B^{m}\leq {\text{result}}<2B^{m}$), which require a multiplier having one extra bit. This can be avoided by noting that $z_{1}=(x_{0}-x_{1})(y_{1}-y_{0})+z_{2}+z_{0}.$ This computation of $(x_{0}-x_{1})$ and $(y_{1}-y_{0})$ will produce a result in the range of $-B^{m}<{\text{result}}<B^{m}$. This method may produce negative numbers, which require one extra bit to encode signedness, and would still require one extra bit for the multiplier. However, one way to avoid this is to record the sign and then use the absolute value of $(x_{0}-x_{1})$ and $(y_{1}-y_{0})$ to perform an unsigned multiplication, after which the result may be negated when both signs originally differed. Another advantage is that even though $(x_{0}-x_{1})(y_{1}-y_{0})$ may be negative, the final computation of $z_{1}$ only involves additions. References 1. A. Karatsuba and Yu. Ofman (1962). "Multiplication of Many-Digital Numbers by Automatic Computers". Proceedings of the USSR Academy of Sciences. 145: 293–294. Translation in the academic journal Physics-Doklady, 7 (1963), pp. 595–596{{cite journal}}: CS1 maint: postscript (link) 2. A. A. Karatsuba (1995). "The Complexity of Computations" (PDF). Proceedings of the Steklov Institute of Mathematics. 211: 169–183. Translation from Trudy Mat. Inst. Steklova, 211, 186–202 (1995){{cite journal}}: CS1 maint: postscript (link) 3. Knuth D.E. (1969) The Art of Computer Programming. v.2. Addison-Wesley Publ.Co., 724 pp. 4. Charles Babbage, Chapter VIII – Of the Analytical Engine, Larger Numbers Treated, Passages from the Life of a Philosopher, Longman Green, London, 1864; page 125. 5. Weiss, Mark A. (2005). Data Structures and Algorithm Analysis in C++. Addison-Wesley. p. 480. ISBN 0321375319. External links • Karatsuba's Algorithm for Polynomial Multiplication • Weisstein, Eric W. "Karatsuba Multiplication". MathWorld. • Bernstein, D. J., "Multidigit multiplication for mathematicians". Covers Karatsuba and many other multiplication algorithms. Number-theoretic algorithms Primality tests • AKS • APR • Baillie–PSW • Elliptic curve • Pocklington • Fermat • Lucas • Lucas–Lehmer • Lucas–Lehmer–Riesel • Proth's theorem • Pépin's • Quadratic Frobenius • Solovay–Strassen • Miller–Rabin Prime-generating • Sieve of Atkin • Sieve of Eratosthenes • Sieve of Pritchard • Sieve of Sundaram • Wheel factorization Integer factorization • Continued fraction (CFRAC) • Dixon's • Lenstra elliptic curve (ECM) • Euler's • Pollard's rho • p − 1 • p + 1 • Quadratic sieve (QS) • General number field sieve (GNFS) • Special number field sieve (SNFS) • Rational sieve • Fermat's • Shanks's square forms • Trial division • Shor's Multiplication • Ancient Egyptian • Long • Karatsuba • Toom–Cook • Schönhage–Strassen • Fürer's Euclidean division • Binary • Chunking • Fourier • Goldschmidt • Newton-Raphson • Long • Short • SRT Discrete logarithm • Baby-step giant-step • Pollard rho • Pollard kangaroo • Pohlig–Hellman • Index calculus • Function field sieve Greatest common divisor • Binary • Euclidean • Extended Euclidean • Lehmer's Modular square root • Cipolla • Pocklington's • Tonelli–Shanks • Berlekamp • Kunerth Other algorithms • Chakravala • Cornacchia • Exponentiation by squaring • Integer square root • Integer relation (LLL; KZ) • Modular exponentiation • Montgomery reduction • Schoof • Trachtenberg system • Italics indicate that algorithm is for numbers of special forms	Wikipedia
Kirby calculus In mathematics, the Kirby calculus in geometric topology, named after Robion Kirby, is a method for modifying framed links in the 3-sphere using a finite set of moves, the Kirby moves. Using four-dimensional Cerf theory, he proved that if M and N are 3-manifolds, resulting from Dehn surgery on framed links L and J respectively, then they are homeomorphic if and only if L and J are related by a sequence of Kirby moves. According to the Lickorish–Wallace theorem any closed orientable 3-manifold is obtained by such surgery on some link in the 3-sphere. Some ambiguity exists in the literature on the precise use of the term "Kirby moves". Different presentations of "Kirby calculus" have a different set of moves and these are sometimes called Kirby moves. Kirby's original formulation involved two kinds of move, the "blow-up" and the "handle slide"; Roger Fenn and Colin Rourke exhibited an equivalent construction in terms of a single move, the Fenn–Rourke move, that appears in many expositions and extensions of the Kirby calculus. Dale Rolfsen's book, Knots and Links, from which many topologists have learned the Kirby calculus, describes a set of two moves: 1) delete or add a component with surgery coefficient infinity 2) twist along an unknotted component and modify surgery coefficients appropriately (this is called the Rolfsen twist). This allows an extension of the Kirby calculus to rational surgeries. There are also various tricks to modify surgery diagrams. One such useful move is the slam-dunk. An extended set of diagrams and moves are used for describing 4-manifolds. A framed link in the 3-sphere encodes instructions for attaching 2-handles to the 4-ball. (The 3-dimensional boundary of this manifold is the 3-manifold interpretation of the link diagram mentioned above.) 1-handles are denoted by either 1. a pair of 3-balls (the attaching region of the 1-handle) or, more commonly, 2. unknotted circles with dots. The dot indicates that a neighborhood of a standard 2-disk with boundary the dotted circle is to be excised from the interior of the 4-ball.[1] Excising this 2-handle is equivalent to adding a 1-handle; 3-handles and 4-handles are usually not indicated in the diagram. Handle decomposition • A closed, smooth 4-manifold is usually described by a handle decomposition. • A 0-handle is just a ball, and the attaching map is disjoint union. • A 1-handle is attached along two disjoint 3-balls. • A 2-handle is attached along a solid torus; since this solid torus is embedded in a 3-manifold, there is a relation between handle decompositions on 4-manifolds, and knot theory in 3-manifolds. • A pair of handles with index differing by 1, whose cores link each other in a sufficiently simple way can be cancelled without changing the underlying manifold. Similarly, such a cancelling pair can be created. Two different smooth handlebody decompositions of a smooth 4-manifold are related by a finite sequence of isotopies of the attaching maps, and the creation/cancellation of handle pairs. See also • Exotic $\mathbb {R} ^{4}$ References • Kirby, Robion (1978). "A calculus for framed links in S3". Inventiones Mathematicae. 45 (1): 35–56. Bibcode:1978InMat..45...35K. doi:10.1007/BF01406222. MR 0467753. S2CID 120770295. • Fenn, Roger; Rourke, Colin (1979). "On Kirby's calculus of links". Topology. 18 (1): 1–15. doi:10.1016/0040-9383(79)90010-7. MR 0528232. • Gompf, Robert; Stipsicz, András (1999). 4-Manifolds and Kirby Calculus. Graduate Studies in Mathematics. Vol. 20. Providence, RI: American Mathematical Society. ISBN 0-8218-0994-6. MR 1707327. 1. "Archived copy" (PDF). Archived from the original (PDF) on 2012-05-14. Retrieved 2012-01-02.{{cite web}}: CS1 maint: archived copy as title (link)	Wikipedia
Knight's tour A knight's tour is a sequence of moves of a knight on a chessboard such that the knight visits every square exactly once. If the knight ends on a square that is one knight's move from the beginning square (so that it could tour the board again immediately, following the same path), the tour is closed (or re-entrant); otherwise, it is open.[1][2] The knight's tour problem is the mathematical problem of finding a knight's tour. Creating a program to find a knight's tour is a common problem given to computer science students.[3] Variations of the knight's tour problem involve chessboards of different sizes than the usual 8 × 8, as well as irregular (non-rectangular) boards. Theory The knight's tour problem is an instance of the more general Hamiltonian path problem in graph theory. The problem of finding a closed knight's tour is similarly an instance of the Hamiltonian cycle problem. Unlike the general Hamiltonian path problem, the knight's tour problem can be solved in linear time.[4] History The earliest known reference to the knight's tour problem dates back to the 9th century AD. In Rudrata's Kavyalankara[5] (5.15), a Sanskrit work on Poetics, the pattern of a knight's tour on a half-board has been presented as an elaborate poetic figure (citra-alaṅkāra) called the turagapadabandha or 'arrangement in the steps of a horse'. The same verse in four lines of eight syllables each can be read from left to right or by following the path of the knight on tour. Since the Indic writing systems used for Sanskrit are syllabic, each syllable can be thought of as representing a square on a chessboard. Rudrata's example is as follows: सेनालीलीलीनानाली लीनानानानालीलीली नलीनालीलेनालीना लीलीलीनानानानाली transliterated: senālīlīlīnānālī līnānānānālīlīlī nalīnālīlenālīnā līlīlīnānānānālī For example, the first line can be read from left to right or by moving from the first square to the second line, third syllable (2.3) and then to 1.5 to 2.7 to 4.8 to 3.6 to 4.4 to 3.2. The Sri Vaishnava poet and philosopher Vedanta Desika, during the 14th century, in his 1,008-verse magnum opus praising the deity Ranganatha's divine sandals of Srirangam, Paduka Sahasram (in chapter 30: Chitra Paddhati) has composed two consecutive Sanskrit verses containing 32 letters each (in Anushtubh meter) where the second verse can be derived from the first verse by performing a Knight's tour on a 4 × 8 board, starting from the top-left corner.[6] The transliterated 19th verse is as follows: sThi (1) rA (30) ga (9) sAm (20) sa (3) dhA (24) rA (11) dhyA (26) vi (16) ha (19) thA (2) ka (29) tha (10) thA (27) ma (4) thA (23) sa (31) thpA (8) dhu (17) kE (14) sa (21) rA (6) sA (25) mA (12) ran (18) ga (15) rA (32) ja (7) pa (28) dha (13) nna (22) ya (5) The 20th verse that can be obtained by performing Knight's tour on the above verse is as follows: sThi thA sa ma ya rA ja thpA ga tha rA mA dha kE ga vi \| dhu ran ha sAm sa nna thA dhA sA dhyA thA pa ka rA sa rA \|\| It is believed that Desika composed all 1,008 verses (including the special Chaturanga Turanga Padabandham mentioned above) in a single night as a challenge.[7] A tour reported in the fifth book of Bhagavantabaskaraby by Bhat Nilakantha, a cyclopedic work in Sanskrit on ritual, law and politics, written either about 1600 or about 1700 describes three knight's tours. The tours are not only reentrant but also symmetrical, and the verses are based on the same tour, starting from different squares.[8] Nilakantha's work is an extraordinary achievement being a fully symmetric closed tour, predating the work of Euler (1759) by at least 60 years. After Nilakantha, one of the first mathematicians to investigate the knight's tour was Leonhard Euler. The first procedure for completing the knight's tour was Warnsdorf's rule, first described in 1823 by H. C. von Warnsdorf. In the 20th century, the Oulipo group of writers used it, among many others. The most notable example is the 10 × 10 knight's tour which sets the order of the chapters in Georges Perec's novel Life a User's Manual. The sixth game of the World Chess Championship 2010 between Viswanathan Anand and Veselin Topalov saw Anand making 13 consecutive knight moves (albeit using both knights); online commentators jested that Anand was trying to solve the knight's tour problem during the game. Existence Schwenk[10] proved that for any m × n board with m ≤ n, a closed knight's tour is always possible unless one or more of these three conditions are met: 1. m and n are both odd 2. m = 1, 2, or 4 3. m = 3 and n = 4, 6, or 8. Cull et al. and Conrad et al. proved that on any rectangular board whose smaller dimension is at least 5, there is a (possibly open) knight's tour.[4][11] For any m × n board with m ≤ n, a knight's tour is always possible unless one or more of these three conditions are met: 1. m = 1 or 2 2. m = 3 and n = 3, 5, or 6[12] 3. m = 4 and n = 4.[13] Number of tours On an 8 × 8 board, there are exactly 26,534,728,821,064 directed closed tours (i.e. two tours along the same path that travel in opposite directions are counted separately, as are rotations and reflections).[14][15][16] The number of undirected closed tours is half this number, since every tour can be traced in reverse. There are 9,862 undirected closed tours on a 6 × 6 board.[17] nNumber of directed tours (open and closed) on an n × n board (sequence A165134 in the OEIS) 11 20 30 40 51,728 66,637,920 7165,575,218,320 819,591,828,170,979,904 Finding tours with computers There are several ways to find a knight's tour on a given board with a computer. Some of these methods are algorithms, while others are heuristics. Brute-force algorithms A brute-force search for a knight's tour is impractical on all but the smallest boards.[18] For example, there are approximately 4×1051 possible move sequences on an 8 × 8 board,[19] and it is well beyond the capacity of modern computers (or networks of computers) to perform operations on such a large set. However, the size of this number is not indicative of the difficulty of the problem, which can be solved "by using human insight and ingenuity ... without much difficulty."[18] Divide-and-conquer algorithms By dividing the board into smaller pieces, constructing tours on each piece, and patching the pieces together, one can construct tours on most rectangular boards in linear time – that is, in a time proportional to the number of squares on the board.[11][20] Warnsdorff's rule abcdefgh 8 8 77 66 55 44 33 22 11 abcdefgh A graphical representation of Warnsdorff's Rule. Each square contains an integer giving the number of moves that the knight could make from that square. In this case, the rule tells us to move to the square with the smallest integer in it, namely 2. Warnsdorff's rule is a heuristic for finding a single knight's tour. The knight is moved so that it always proceeds to the square from which the knight will have the fewest onward moves. When calculating the number of onward moves for each candidate square, we do not count moves that revisit any square already visited. It is possible to have two or more choices for which the number of onward moves is equal; there are various methods for breaking such ties, including one devised by Pohl[21] and another by Squirrel and Cull.[22] This rule may also more generally be applied to any graph. In graph-theoretic terms, each move is made to the adjacent vertex with the least degree.[23] Although the Hamiltonian path problem is NP-hard in general, on many graphs that occur in practice this heuristic is able to successfully locate a solution in linear time.[21] The knight's tour is such a special case.[24] The heuristic was first described in "Des Rösselsprungs einfachste und allgemeinste Lösung" by H. C. von Warnsdorff in 1823.[24] A computer program that finds a knight's tour for any starting position using Warnsdorff's rule was written by Gordon Horsington and published in 1984 in the book Century/Acorn User Book of Computer Puzzles.[25] Neural network solutions The knight's tour problem also lends itself to being solved by a neural network implementation.[26] The network is set up such that every legal knight's move is represented by a neuron, and each neuron is initialized randomly to be either "active" or "inactive" (output of 1 or 0), with 1 implying that the neuron is part of the solution. Each neuron also has a state function (described below) which is initialized to 0. When the network is allowed to run, each neuron can change its state and output based on the states and outputs of its neighbors (those exactly one knight's move away) according to the following transition rules: $U_{t+1}(N_{i,j})=U_{t}(N_{i,j})+2-\sum _{N\in G(N_{i,j})}V_{t}(N)$ $V_{t+1}(N_{i,j})=\left\{{\begin{array}{ll}1&{\mbox{if}}\,\,U_{t+1}(N_{i,j})>3\\0&{\mbox{if}}\,\,U_{t+1}(N_{i,j})<0\\V_{t}(N_{i,j})&{\mbox{otherwise}},\end{array}}\right.$ where $t$ represents discrete intervals of time, $U(N_{i,j})$ is the state of the neuron connecting square $i$ to square $j$, $V(N_{i,j})$ is the output of the neuron from $i$ to $j$, and $G(N_{i,j})$ is the set of neighbors of the neuron. Although divergent cases are possible, the network should eventually converge, which occurs when no neuron changes its state from time $t$ to $t+1$. When the network converges, either the network encodes a knight's tour or a series of two or more independent circuits within the same board. See also • Abu Bakr bin Yahya al-Suli • Eight queens puzzle • George Koltanowski • Longest uncrossed knight's path • Self-avoiding walk Notes 1. Brown, Alfred James (2017). Knight's Tours and Zeta Functions (MS thesis). San José State University. p. 3. doi:10.31979/etd.e7ra-46ny. 2. Hooper, David; Whyld, Kenneth (1996) [First pub. 1992]. "knight's tour". The Oxford Companion to Chess (2nd ed.). Oxford University Press. p. 204. ISBN 0-19-280049-3. 3. Deitel, H. M.; Deitel, P. J. (2003). Java How To Program Fifth Edition (5th ed.). Prentice Hall. pp. 326–328. ISBN 978-0131016217. 4. Conrad, A.; Hindrichs, T.; Morsy, H. & Wegener, I. (1994). "Solution of the Knight's Hamiltonian Path Problem on Chessboards". Discrete Applied Mathematics. 50 (2): 125–134. doi:10.1016/0166-218X(92)00170-Q. 5. Satyadev, Chaudhary. Kavyalankara of Rudrata (Sanskrit text, with Hindi translation);. Delhitraversal: Parimal Sanskrit Series No. 30. 6. "Indian Institute of Information Technology, Bangalore". www.iiitb.ac.in. Retrieved 2019-10-11. 7. Bridge-india (2011-08-05). "Bridge-India: Paduka Sahasram by Vedanta Desika". Bridge-India. Retrieved 2019-10-16. 8. A History of Chess by Murray 9. "MathWorld News: There Are No Magic Knight's Tours on the Chessboard". 10. Allen J. Schwenk (1991). "Which Rectangular Chessboards Have a Knight's Tour?" (PDF). Mathematics Magazine. 64 (5): 325–332. doi:10.1080/0025570X.1991.11977627. S2CID 28726833. Archived from the original (PDF) on 2019-05-26. 11. Cull, P.; De Curtins, J. (1978). "Knight's Tour Revisited" (PDF). Fibonacci Quarterly. 16: 276–285. Archived (PDF) from the original on 2022-10-09. 12. "Knight's Tours on 3 by N Boards". 13. "Knight's Tours on 4 by N Boards". 14. Martin Loebbing; Ingo Wegener (1996). "The Number of Knight's Tours Equals 33,439,123,484,294 — Counting with Binary Decision Diagrams". The Electronic Journal of Combinatorics. 3 (1): R5. doi:10.37236/1229. Remark: The authors later admitted that the announced number is incorrect. According to McKay's report, the correct number is 13,267,364,410,532 and this number is repeated in Wegener's 2000 book. 15. Brendan McKay (1997). "Knight's Tours on an 8 × 8 Chessboard". Technical Report TR-CS-97-03. Department of Computer Science, Australian National University. Archived from the original on 2013-09-28. Retrieved 2013-09-22. 16. Wegener, I. (2000). Branching Programs and Binary Decision Diagrams. Society for Industrial & Applied Mathematics. ISBN 978-0-89871-458-6. 17. Weisstein, Eric W. "Knight Graph". MathWorld. 18. Simon, Dan (2013), Evolutionary Optimization Algorithms, John Wiley & Sons, pp. 449–450, ISBN 9781118659502, The knight's tour problem is a classic combinatorial optimization problem. ... The cardinality Nx of x (the size of the search space) is over 3.3×1013 (Löbbing and Wegener, 1995). We would not want to try to solve this problem using brute force, but by using human insight and ingenuity we can solve the knight's tour without much difficulty. We see that the cardinality of a combinatorial optimization problem is not necessarily indicative of its difficulty. 19. "Enumerating the Knight's Tour". 20. Parberry, Ian (1997). "An Efficient Algorithm for the Knight's Tour Problem" (PDF). Discrete Applied Mathematics. 73 (3): 251–260. doi:10.1016/S0166-218X(96)00010-8. Archived (PDF) from the original on 2022-10-09. 21. Pohl, Ira (July 1967). "A method for finding Hamilton paths and Knight's tours". Communications of the ACM. 10 (7): 446–449. CiteSeerX 10.1.1.412.8410. doi:10.1145/363427.363463. S2CID 14100648. 22. Squirrel, Douglas; Cull, P. (1996). "A Warnsdorff-Rule Algorithm for Knight's Tours on Square Boards" (PDF). GitHub. Retrieved 2011-08-21. 23. Van Horn, Gijs; Olij, Richard; Sleegers, Joeri; Van den Berg, Daan (2018). A Predictive Data Analytic for the Hardness of Hamiltonian Cycle Problem Instances (PDF). DATA ANALYTICS 2018: The Seventh International Conference on Data Analytics. Athens, greece: XPS. pp. 91–96. ISBN 978-1-61208-681-1. Archived (PDF) from the original on 2022-10-09. Retrieved 2018-11-27. 24. Alwan, Karla; Waters, K. (1992). Finding Re-entrant Knight's Tours on N-by-M Boards. ACM Southeast Regional Conference. New York, New York: ACM. pp. 377–382. doi:10.1145/503720.503806. 25. Dally, Simon, ed. (1984). Century/Acorn User Book of Computer Puzzles. ISBN 978-0712605410. 26. Y. Takefuji, K. C. Lee. "Neural network computing for knight's tour problems." Neurocomputing, 4(5):249–254, 1992. External links Wikimedia Commons has media related to Knight's Tours. • OEIS sequence A001230 (Number of undirected closed knight's tours on a 2n X 2n chessboard) • H. C. von Warnsdorf 1823 in Google Books • Introduction to Knight's tours by George Jelliss • Knight's tours complete notes by George Jelliss • Philip, Anish (2013). "A Generalized Pseudo-Knight?s Tour Algorithm for Encryption of an Image". IEEE Potentials. 32 (6): 10–16. doi:10.1109/MPOT.2012.2219651. S2CID 39213422.	Wikipedia
The Knot Atlas The Knot Atlas is a website, an encyclopedia rather than atlas, dedicated to knot theory. It and its predecessor were created by mathematician Dror Bar-Natan, who maintains the current site with Scott Morrison. According to Schiller, the site contains, "beautiful illustrations and detailed information about knots," as does KnotPlot.com.[1] According to the site itself, it is a knot atlas (collection of maps), theory database, knowledge base, and "a home for some computer programs".[2] References 1. Schiller, Christoph (2012). Motion Mountain, The Adventure of Physics – Vol. 5: Motion Inside Matter – Pleasure, Technology and Stars, p. 272. MotionMountain.net. 2. "Knot Atlas:About", The Knot Atlas. External links • Official website • Morrison, Scott (Aug 20, 2007). "Grepping the Knot Atlas". Secret Blogging Seminar. Knot theory (knots and links) Hyperbolic • Figure-eight (41) • Three-twist (52) • Stevedore (61) • 62 • 63 • Endless (74) • Carrick mat (818) • Perko pair (10161) • (−2,3,7) pretzel (12n242) • Whitehead (52 1 ) • Borromean rings (63 2 ) • L10a140 • Conway knot (11n34) Satellite • Composite knots • Granny • Square • Knot sum Torus • Unknot (01) • Trefoil (31) • Cinquefoil (51) • Septafoil (71) • Unlink (02 1 ) • Hopf (22 1 ) • Solomon's (42 1 ) Invariants • Alternating • Arf invariant • Bridge no. • 2-bridge • Brunnian • Chirality • Invertible • Crosscap no. • Crossing no. • Finite type invariant • Hyperbolic volume • Khovanov homology • Genus • Knot group • Link group • Linking no. • Polynomial • Alexander • Bracket • HOMFLY • Jones • Kauffman • Pretzel • Prime • list • Stick no. • Tricolorability • Unknotting no. and problem Notation and operations • Alexander–Briggs notation • Conway notation • Dowker–Thistlethwaite notation • Flype • Mutation • Reidemeister move • Skein relation • Tabulation Other • Alexander's theorem • Berge • Braid theory • Conway sphere • Complement • Double torus • Fibered • Knot • List of knots and links • Ribbon • Slice • Sum • Tait conjectures • Twist • Wild • Writhe • Surgery theory • Category • Commons	Wikipedia
The Large Scale Structure of Space–Time The Large Scale Structure of Space–Time is a 1973 treatise on the theoretical physics of spacetime by the physicist Stephen Hawking and the mathematician George Ellis.[1] It is intended for specialists in general relativity rather than newcomers. The Large Scale Structure of Space–Time Cover of the first edition AuthorsStephen Hawking George Ellis CountryUnited Kingdom LanguageEnglish SubjectGeneral relativity GenreNon-fiction PublisherCambridge University Press Publication date 1973 Media typePrint (Hardcover and Paperback) Pages384 ISBN978-0521200165 Background In the mid-1970s, advances in the technologies of astronomical observations – radio, infrared, and X-ray astronomy – opened up the Universe of exploration. New tools became necessary. In this book, Hawking and Ellis attempt to establish the axiomatic foundation for the geometry of four-dimensional spacetime as described by Albert Einstein's general theory of relativity and to derive its physical consequences for singularities, horizons, and causality. Whereas the tools for studying Euclidean geometry were a straightedge and a compass, those needed to investigate curved spacetime are test particles and light rays.[2] According to the mathematical physicist John Baez from the University of California, Riverside, The Large Scale Structure of Space–Time was "the first book to provide a detailed description of the revolutionary topological methods introduced by Penrose and Hawking in the early seventies."[3] Hawking co-wrote the book with Ellis, while he was postdoctoral fellow at the University of Cambridge. In his 1988 book A Brief History of Time, he describes The Large Scale Structure of Space–Time as "highly technical" and unreadable for the layperson. The book, now considered a classic, has also appeared in paperback format and has been reprinted many times.a A fiftieth anniversary edition was published by Cambridge University Press in February 2023.[4] Table of contents • Preface • 1. The Role of Gravity • 2. Differential Geometry • 3. General Relativity • 4. The Physical Significance of Curvature • 5. Exact Solutions • 6. Causal Structure • 7. The Cauchy Problem in General Relativity • 8. Space–time Singularities • 9. Gravitational Collapse and Black Holes • 10. The Initial Singularity of the Universe • Appendix A: Translation of An Essay by P. S. Laplace • Appendix B: Spherically Symmetric Solutions of Birkhoff's Theorem. • References • Notation • Index Assessment Mathematician Nicholas Michael John Woodhouse at Oxford University considered this book to be an authoritative treatise that could become a classic. He observed that the authors begin with axioms of geometry and physics then derive the consequences in a rigorous fashion. Various well-known exact solutions to Einstein's field equations and their physical meaning are explored. In particular, Hawking and Ellis show that singularities and black holes arise in a large class of plausible solutions. He warned that although this book is self-contained, it is more suitable for specialists rather than new students as it is heavy-going and contains no exercises. He noted that despite the authors' attempt at a rigorous treatment, certain technical terms, such as Lie groups, are used but never explained, and that modern coordinate-free methods are introduced, but not used effectively.[5] Theoretical physicist Rainer Sachs from the University of California, Berkeley, observed that The Large-Scale Structure of Space–Time was published within just a few years as Gravitation and Cosmology by Steven Weinberg and Gravitation by Charles Misner, Kip Thorne, and John Archibald Wheeler. He believed these three books can supplement each other and lead students to the forefront of research. Whereas Hawking and Ellis employ global analysis extensively but say relatively little about perturbative methods, the other two books neglect global analysis and cover in great detail perturbations. He believed Hawking and Ellis did a great job summarizing recent developments in the field (as of 1974) and that the intended audience is a doctoral student (or higher) with a strong mathematical background and prior exposure to general relativity. He argued that the core of the books consists of two chapters, Chapter 4 on the significance of spacetime curvature and Chapter 6 on causal structure, and that the most interesting application is the penultimate chapter on black holes. He noted that mathematical arguments are at times difficult to follow and suggested Techniques of Differential Topology in Relativity by Roger Penrose for reference. He also noticed a small number of errors, though none affect the general conclusions drawn by the authors. He thought that this book is a "model" presentation on the interplay between mathematics and physics that could become highly influential in the future.[6] Theoretical physicist John Archibald Wheeler of Princeton University recommended this book to anyone interested in the implications of general relativity for cosmology, the singularity theorems, and the physics of black holes, presented in an almost Euclidean fashion, though he acknowledged that this is not a textbook due to its lack of examples and exercises. He praised its 62 illustrative diagrams.[2] See also • List of books on general relativity Notes 1. ^ Hawking, S. W.; Ellis, G. F. R. (1973). The Large Scale Structure of Space–Time. Cambridge University Press. ISBN 0-521-09906-4. References 1. Gibbons, G. W.; Shellard, E. P. S.; Rankin, S. J. (23 October 2003). The Future of Theoretical Physics and Cosmology: Celebrating Stephen Hawking's Contributions to Physics. Cambridge University Press. p. 177. ISBN 978-0-521-82081-3. 2. Wheeler, John A. (March–April 1975). "The Large Scale Structure of Space–Time by S. W. Hawking and G. F. R.Ellis". Review. American Scientist. Sigma Xi, The Scientific Research Honor Society. 62 (2): 218. JSTOR 27845370. 3. Baez, John; Hillman, Chris (October 1998). "Guide to Relativity Books". Department of Physics, University of California, Riverside. Retrieved August 25, 2019. 4. Hawking, Stephen (2023). The large scale structure of space-time. George F. R. Ellis, Abhay Ashtekar (50th anniversary ed.). Cambridge, United Kingdom. ISBN 978-1-009-25316-1. OCLC 1347434162.{{cite book}}: CS1 maint: location missing publisher (link) 5. Woodhouse, Nicholas (1974). "The Large Scale Structure of Space–Time". Physical Bulletin. 25 (6): 238. doi:10.1088/0031-9112/25/6/029. 6. Sachs, Rainer (April 1974). "The Large Scale Structure of Space–Time". Physics Today. American Institute of Physics. 27 (4): 91–3. Bibcode:1974PhT....27d..91H. doi:10.1063/1.3128542. S2CID 121949888. Stephen Hawking Physics • Hawking radiation • Black hole thermodynamics • Micro black hole • Chronology protection conjecture • Gibbons–Hawking ansatz • Gibbons–Hawking effect • Gibbons–Hawking space • Gibbons–Hawking–York boundary term • Hartle–Hawking state • Penrose–Hawking singularity theorems • Hawking energy Books Science • The Large Scale Structure of Space–Time (1973) • A Brief History of Time (1988) • Black Holes and Baby Universes and Other Essays (1993) • The Nature of Space and Time (1996) • The Universe in a Nutshell (2001) • On the Shoulders of Giants (2002) • A Briefer History of Time (2005) • God Created the Integers (2005) • The Grand Design (2010) • The Dreams That Stuff Is Made Of (2011) • Brief Answers to the Big Questions (2018) • On the Origin of Time (2023, posthume message) Fiction • George's Secret Key to the Universe (2007) • George's Cosmic Treasure Hunt (2009) • George and the Big Bang (2011) • George and the Unbreakable Code (2014) • George and the Blue Moon (2016) • Unlocking the Universe (2020) Memoirs • My Brief History (2013) Films • A Brief History of Time (1991) • Hawking (2004) • Hawking (2013) • The Theory of Everything (2014) Television • God, the Universe and Everything Else (1988) • Stephen Hawking's Universe (1997 documentary) • Stephen Hawking: Master of the Universe (2008 documentary) • Genius of Britain (2010 series) • Into the Universe with Stephen Hawking (2010 series) • Brave New World with Stephen Hawking (2011 series) • Genius by Stephen Hawking (2016 series) Family • Jane Wilde Hawking (first wife) • Lucy Hawking (daughter) Other • In popular culture • Black hole information paradox • Thorne–Hawking–Preskill bet	Wikipedia
Leibniz operator In abstract algebraic logic, a branch of mathematical logic, the Leibniz operator is a tool used to classify deductive systems, which have a precise technical definition and capture a large number of logics. The Leibniz operator was introduced by Wim Blok and Don Pigozzi, two of the founders of the field, as a means to abstract the well-known Lindenbaum–Tarski process, that leads to the association of Boolean algebras to classical propositional calculus, and make it applicable to as wide a variety of sentential logics as possible. It is an operator that assigns to a given theory of a given sentential logic, perceived as a term algebra with a consequence operation on its universe, the largest congruence on the algebra that is compatible with the theory. Formulation In this article, we introduce the Leibniz operator in the special case of classical propositional calculus, then we abstract it to the general notion applied to an arbitrary sentential logic and, finally, we summarize some of the most important consequences of its use in the theory of abstract algebraic logic. Let ${\mathcal {S}}=\langle {\rm {Fm}},\vdash _{\mathcal {S}}\rangle $ denote the classical propositional calculus. According to the classical Lindenbaum–Tarski process, given a theory $T$ of ${\mathcal {S}}$, if $\equiv _{T}$ denotes the binary relation on the set of formulas of ${\mathcal {S}}$, defined by $\phi \equiv _{T}\psi $ if and only if $T\vdash _{\mathcal {S}}\phi \leftrightarrow \psi ,$ where $\leftrightarrow $ denotes the usual classical propositional equivalence connective, then $\equiv _{T}$ turns out to be a congruence on the formula algebra. Furthermore, the quotient ${\rm {Fm}}/{\equiv _{T}}$ is a Boolean algebra and every Boolean algebra may be formed in this way. Thus, the variety of Boolean algebras, which is, in algebraic logic terminology, the equivalent algebraic semantics (algebraic counterpart) of classical propositional calculus, is the class of all algebras formed by taking appropriate quotients of term algebras by those special kinds of congruences. Notice that the condition $T\vdash _{\mathcal {S}}\phi \leftrightarrow \psi $ that defines $\phi \equiv _{T}\psi $ is equivalent to the condition for every formula $\chi $: $T\vdash _{\mathcal {S}}\phi \leftrightarrow \chi $ if and only if $T\vdash _{\mathcal {S}}\psi \leftrightarrow \chi $. Passing now to an arbitrary sentential logic ${\mathcal {S}}=\langle {\rm {Fm}},\vdash _{\mathcal {S}}\rangle ,$ given a theory $T$, the Leibniz congruence associated with $T$ is denoted by $\Omega (T)$ and is defined, for all $\phi ,\psi \in {\rm {Fm}}$, by $\phi \Omega (T)\psi $ if and only if, for every formula $\alpha (x,{\vec {y}})$ containing a variable $x$ and possibly other variables in the list ${\vec {y}}$, and all formulas ${\vec {\chi }}$ forming a list of the same length as that of ${\vec {y}}$, we have that $T\vdash _{\mathcal {S}}\alpha (\phi ,{\vec {\chi }})$ if and only if $T\vdash _{\mathcal {S}}\alpha (\psi ,{\vec {\chi }})$. It turns out that this binary relation is a congruence relation on the formula algebra and, in fact, may alternatively be characterized as the largest congruence on the formula algebra that is compatible with the theory $T$, in the sense that if $\phi \Omega (T)\psi $ and $T\vdash _{\mathcal {S}}\phi $, then we must have also $T\vdash _{\mathcal {S}}\psi $. It is this congruence that plays the same role as the congruence used in the traditional Lindenbaum–Tarski process described above in the context of an arbitrary sentential logic. It is not, however, the case that for arbitrary sentential logics the quotients of the term algebras by these Leibniz congruences over different theories yield all algebras in the class that forms the natural algebraic counterpart of the sentential logic. This phenomenon occurs only in the case of "nice" logics and one of the main goals of abstract algebraic logic is to make this vague notion of a logic being "nice", in this sense, mathematically precise. The Leibniz operator $\Omega $ is the operator that maps a theory $T$ of a given logic to the Leibniz congruence $\Omega (T),$ associated with the theory. Thus, formally, $\Omega :\mathrm {Th} ({\mathcal {S}})\rightarrow {\rm {Con}}({\rm {Fm}})$ :\mathrm {Th} ({\mathcal {S}})\rightarrow {\rm {Con}}({\rm {Fm}})} is a mapping from the collection ${\rm {Th}}({\mathcal {S}})$ of the theories of a sentential logic ${\mathcal {S}}$ to the collection ${\rm {Con}}({\rm {Fm}})$ of all congruences on the formula algebra ${\rm {Fm}}$ of the sentential logic. Hierarchy The Leibniz operator and the study of various of its properties that may or may not be satisfied for particular sentential logics have given rise to what is now known as the abstract algebraic hierarchy or Leibniz hierarchy of sentential logics. Logics are classified in various levels of this hierarchy depending on how strong a tie exists between the logic and its algebraic counterpart. The properties of the Leibniz operator that help classify the logics are monotonicity, injectivity, continuity and commutativity with inverse substitutions. For instance, protoalgebraic logics, forming the widest class in the hierarchy – i.e., the one that lies in the bottom of the hierarchy and contains all other classes – are characterized by the monotonicity of the Leibniz operator on their theories. Other notable classes are formed by the equivalential logics, the weakly algebraizable logics and the algebraizable logics, among others. There is a generalization of the Leibniz operator, in the context of categorical abstract algebraic logic, that makes it possible to apply a wide variety of techniques that were previously applicable only in the sentential logic framework to logics formalized as $\pi $-institutions. The $\pi $-institution framework is significantly wider in scope than the framework of sentential logics because it allows incorporating multiple signatures and quantifiers in the language and it provides a mechanism for handling logics that are not syntactically based. References • D. Pigozzi (2001). "Abstract algebraic logic". In M. Hazewinkel (ed.). Encyclopaedia of mathematics: Supplement Volume III. Springer. pp. 2–13. ISBN 1-4020-0198-3. • Font, J. M., Jansana, R., Pigozzi, D., (2003), A survey of abstract algebraic logic, Studia Logica 74: 13–79. • Janusz Czelakowski (2001). Protoalgebraic logics. Springer. ISBN 978-0-7923-6940-0. External links • "Algebraic Propositional Logic" entry by Ramon Jansana in the Stanford Encyclopedia of Philosophy	Wikipedia
Lighthouse paradox The lighthouse paradox is a thought experiment in which the speed of light is apparently exceeded. The rotating beam of light from a lighthouse is imagined to be swept from one object to shine on a second object. The farther the two objects are away from the lighthouse, the farther the distance between them crossed by the light beam. If the objects are sufficiently far away from the lighthouse, the places where the beam hits object 2 will traverse the object with an apparent speed faster than light, possibly communicating a signal on object 2 with superluminal velocity, which violates Albert Einstein's theory of special relativity. The solution to this paradox is that superluminal velocities can be observed because no actual particles or information are traveling from object 1 to object 2. The transverse velocity of the beam along the path in the sky between the objects has an apparent speed greater than light, but this represents separate photons of light. No photons are traveling the path from object 1 to object 2; the photons in the light beam are traveling a radial path outward from the lighthouse, at the speed of light. The theory of relativity says information cannot be transmitted faster than light. This experiment does not actually transmit a signal from object 1 to object 2. The time when the light beam strikes object 2 is controlled by the person at the lighthouse, not anyone on object 1, so no one on object 1 can transmit a message to object 2 by this method. Therefore the theory of relativity is not violated. Paradox A lighthouse sends a powerful beam of light which travels significant distances from the point of origin. This light constantly rotates in a circular motion around the lighthouse. This thought experiment proposes that light moving in this situation is actually traveling faster than the speed of light. This presents a paradox because, according to the theory of relativity, the speed of light in vacuum is the same for all observers, regardless of their relative motion or of the motion of the light source, and nothing can travel faster than this speed.[1][2] Moon example A similar example can be explained by the movement of a laser across the face of the Moon.[3] This paradox arises based on a simple principle: if someone stands a distance of "X" away from an object, and shines a laser from one side (A) of the object to the other side (B) of the object, they would have to rotate their hand by an angle "Y". Thus, as X increases, Y would decrease, as the wrist would have to rotate a smaller angle to move the laser from point A to point B. Also, correlated with a smaller angle would be a decreased time it would take to rotate the wrist (it will take a shorter time to rotate the wrist a smaller angle). With respect to distant objects like the Moon, a paradox arises when one is asked to hypothetically move a laser from one side to the other.[3] Turning his wrist half a degree, a person can move a laser from one side of the Moon to the other. It would appear that the laser dot is travelling faster than light, as flicking one's wrist at such a large distance would give the illusion that the object was able to cross the diameter of the Moon (6000 km, due to curvature) in milliseconds.[3] Based on subsequent calculations (distance between points A and B divided by time it takes to move the laser from A to B), it would appear that the dot of light is moving at superluminal velocities, when, in reality, the dots are successive photons being emitted by the source moving across the face of the Moon.[3] Resolution of the paradox in special relativity The paradoxical aspect of each of the described thought experiments arises from Einstein’s theory of special relativity, which proclaims the speed of light (approx. 300,000 km/s) is the upper limit of speed in our universe.[1][4][5] The uniformity of the speed of light is so absolute that regardless of the speed of the observer as well as the speed of the source of light the speed of the light ray should remain constant.[1][4] When considering the movement of an image created by a laser on the Moon, some physical limitations would have to be violated in order to trace out the apparent trajectory at superluminal velocities. To approach the speed of light, and therefore to pass it, an object would have to be accelerated through an infinite potential, an impossibility within the physical universe.[1][4][2] The acceleration process would also cause the object to have infinite mass, which is not only logically impossible, but it would also cause severe gravitational effects in the surrounding spacetime.[1][2] However, these effects that have no empirical evidence leading to the conclusion that there is a simple physical explanation. The fundamental misunderstanding of this paradox is the assumption that the projected image caused by the light ray is a physical object, and therefore must follow physical law. In reality, no physical laws are being broken as there is no physical object travelling faster than light. This paradox uses kinematic processes to explain the motion of this apparent object. However, the projected image on the Moon, or the image created by the lighthouse, is not a real object. The apparent lateral motion across the surface of the Moon is a result of the light source moving through some angular rotation, not superluminal motion across its surface. The angular motion of the source creates a translation of the image projected on the Moon, proportional to the distance between the screen (which in this case is Moon) and the source. Thus, if one was to go close enough to the Moon and rotate the laser through the same angle the image would travel at subluminal speeds even though nothing that should effect its speed has changed. If the image was a physical object, it should be able to travel across the surface of the Moon at the same speed regardless of the distance of the observer. Understanding this, the paradox begins to unravel.[3] It is natural to visualize this phenomenon as an abundance of stationary photons within the same ray of light creating the spot on the Moon. To allow the image to move from one end of the Moon to the other, each photon must move laterally with the movement of the projection. In reality, this is not the case: the ray of light is a collection of moving photons and at each instant a different group of photons, detected by the observers eye, are creating the image seen on the surface of the Moon.[3] The apparent lateral movement is caused by new photons traveling on a different path from the light source to the Moon, caused by the rotation of the source, striking an adjacent position at all instances during the rotation. The movement from point A to point B can be visualized by a collection of photons, each travelling along a different trajectory from Earth to the Moon. The paradox is resolved to be a result of the geometry of the system causing the illusion of superluminal motion, rather than superluminal motion actually occurring.[3] A final issue with this explanation is that there appears to be no delay between the flicking of the wrist and the movement of the image on the Moon, a process that is expected if the photon resolution is correct. This does not invalidate the resolution of the paradox. The apparent simultaneity is a result of the large magnitude of the speed of light, and the observers inability to detect changes that fast. In ideal conditions, the expected delay would be noticeable.[3] References 1. "Maudlin, M. (2011). Quantum non-locality and relativity : metaphysical limitations of modern physics (3rd ed.). Singapore: Blackwell Publishing Ltd 2. Uzan & Leclercg, J.P. & B. (2010). The Natural Laws of the Universe: Understanding Fundamental Constants. Springer Science & Business Media. pp. 43–4. ISBN 978-0-387-73454-5. 3. "How A Laser Appears To Move Faster Than Light (And Why It Really Isn't)". Universe Today. 7 February 2014. Retrieved 2016-04-05. 4. "", Simonetti, J. Virginia Tech Physics: Frequently Asked Questions About Relativity. 5. Jorgensen, Palle E. T. (2008-11-13). "The road to reality: a complete guide to the laws of the universe". The Mathematical Intelligencer. 28 (3): 59–61. doi:10.1007/BF02986885. ISSN 0343-6993. S2CID 117975932.	Wikipedia
MacTutor History of Mathematics Archive The MacTutor History of Mathematics Archive is a website maintained by John J. O'Connor and Edmund F. Robertson and hosted by the University of St Andrews in Scotland. It contains detailed biographies on many historical and contemporary mathematicians, as well as information on famous curves and various topics in the history of mathematics. MacTutor History of Mathematics Archive URLmathshistory.st-andrews.ac.uk The History of Mathematics archive was an outgrowth of Mathematical MacTutor system, a HyperCard database by the same authors,[1] which won them the European Academic Software award in 1994. In the same year, they founded their web site. As of 2015, it has biographies on over 2800 mathematicians and scientists.[2] In 2015, O'Connor and Robertson won the Hirst Prize of the London Mathematical Society for their work.[2][3] The citation for the Hirst Prize calls the archive "the most widely used and influential web-based resource in history of mathematics".[3] See also • Mathematics Genealogy Project • MathWorld • PlanetMath References 1. Review and description of the MacTutor at the CM magazine (Volume III Number 17 April 25, 1997) 2. Mulcahy, Colm (July 19, 2015), "MacTutor History of Mathematics website creators honoured by LMS", The Aperiodical. 3. London Mathematical Society Hirst Prize and Lectureship, University of St Andrews, School of Mathematics and Statistics, July 3, 2015, archived from the original on December 11, 2016, retrieved July 19, 2015. External links • MacTutor History of Mathematics archive • Mathematical MacTutor system	Wikipedia
The Man Who Counted The Man Who Counted (original Portuguese title: O Homem que Calculava) is a book on recreational mathematics and curious word problems by Brazilian writer Júlio César de Mello e Souza, published under the pen name Malba Tahan. Since its first publication in 1938,[1] the book has been immensely popular in Brazil and abroad, not only among mathematics teachers but among the general public as well. The Man Who Counted AuthorJúlio César de Mello e Souza Original titleO Homem que Calculava The book has been published in many other languages, including Catalan, English (in the UK and in the US),[2] German, Italian, and Spanish, and is recommended as a paradidactic source in many countries. It earned its author a prize from the Brazilian Literary Academy. Plot summary First published in Brazil in 1949, O Homem que Calculava is a series of tales in the style of the Arabian Nights, but revolving around mathematical puzzles and curiosities. The book is ostensibly a translation by Brazilian scholar Breno de Alencar Bianco of an original manuscript by Malba Tahan, a thirteenth-century Persian scholar of the Islamic Empire – both equally fictitious. The first two chapters tell how Hanak Tade Maia was traveling from Samarra to Baghdad when he met Beremiz Samir, a young lad from Khoy with amazing mathematical abilities. The traveler then invited Beremiz to come with him to Baghdad, where a man with his abilities will certainly find profitable employment. The rest of the book tells of various incidents that befell the two men along the road and in Baghdad. In all those events, Beremiz Samir uses his abilities with calculation like a magic wand to amaze and entertain people, settle disputes, and find wise and just solutions to seemingly unsolvable problems. In the first incident along their trip (chapter III), Beremiz settles a heated inheritance dispute between three brothers. Their father had left them 35 camels, of which 1/2 (17.5 camels) should go to his eldest son, 1/3 (11.666... camels) to the middle one, and 1/9 (3.888... camels) to the youngest. To solve the brothers dilemma, Beremiz convinces Hanak to donate his only camel to the dead man's estate. Then, with 36 camels, Beremiz gives 18, 12, and 4 animals to the three heirs, making all of them profit with the new share. Of the remaining two camels, one is returned to Hanak, and the other is claimed by Beremiz as his reward. The translator's notes observe that the 17-animal inheritance puzzle, a mathematical puzzle whose first publication is in the works of Muhaqiqi Naraqi, is a variant of this problem, with 17 camels to be divided in the same proportions. It is found in hundreds of recreational mathematics books, such as those of E. Fourrey (1949) and G. Boucheny (1939). However, the 17-camel version leaves only one camel at the end, with no net profit for the estate's executor. At the end of the book, Beremiz uses his abilities to win the hand of his student and secret love Telassim, the daughter of one of the Caliph's advisers. (The caliph mentioned is Al-Musta'sim, the only real character who appears fictitiously; the time period ends with the Abbasid dynasty's collapse.) In the last chapter we learn that Hanak Tade Maia and Beremiz eventually moved to Constantinople following the Siege of Baghdad (Telassim's father died in the fighting), where Beremiz had three sons and Hanak visits him often. Publishing history The "translator's note" signed "B. A. Bianco" is dated from 1965. The preface signed "Malba Tahan" is dated "Baghdad, 19 of the Moon of Ramadan of 1321" (Islamic calendar equivalent of (Gregorian) 8 December 1903). The 1993 English edition published by W.W. Norton & Co. was illustrated by Patricia Reid Baquero. The fifty fourth printing by Editora Record (2001; in Portuguese) contains 164 pages of Malba Tahan's text, plus 60 pages of notes and historical appendices, commented solutions to all the problems, a glossary of Arabic terms, alphabetical index, and other material. The book was translated into Arabic in 2005 by Azza Kubba, an Iraqi from Baghdad (published by Al-Jamel Publishing House, Cologne, Germany). Further reading • Gaston Boucheny, Curiosités et Récréations Mathématiques. Paris, 1939. • E. Fourrey, Récréations Mathématiques. Paris, 1949. References 1. Coppe de Oliveira, Cristiane (2007); A sombra do arco-íris: um estudo histórico/mitocrítico do discurso pedagógico de Malba Tahan. These, Univ. de São Paulo (Br), 2007, 171 pp.; p. 125 2. Tahan, Malba (1993), The Man Who Counted / a collection of mathematical adventures, translated by Leslie Clark; Alastair Reid, W.W. Norton & Co., ISBN 0-393-30934-7 External links • Online copy from The Internet Archive	Wikipedia
Lewis Carroll Charles Lutwidge Dodgson (/ˈlʌtwɪdʒ ˈdɒdʒsən/ LUT-wij DOJ-sən; 27 January 1832 – 14 January 1898), better known by his pen name Lewis Carroll, was an English author, poet and mathematician. His most notable works are Alice's Adventures in Wonderland (1865) and its sequel Through the Looking-Glass (1871). He was noted for his facility with word play, logic, and fantasy. His poems Jabberwocky (1871) and The Hunting of the Snark (1876) are classified in the genre of literary nonsense. Lewis Carroll Carroll in June 1857 BornCharles Lutwidge Dodgson (1832-01-27)27 January 1832 Daresbury, Cheshire, England Died14 January 1898(1898-01-14) (aged 65) Guildford, Surrey, England Resting placeMount Cemetery, Guildford, Surrey, England Occupation • Author • illustrator • poet • mathematician • photographer • teacher • inventor Education • Rugby School • Christ Church, Oxford Genre • Children's literature • fantasy literature • mathematical logic • poetry • literary nonsense • linear algebra • voting theory ParentsCharles Dodgson (father) Relatives • Edwin Dodgson (brother) • Charles Dodgson (great-grandfather) Signature Carroll came from a family of high-church Anglicans, and developed a long relationship with Christ Church, Oxford, where he lived for most of his life as a scholar and teacher. Alice Liddell – a daughter of Henry Liddell, the Dean of Christ Church – is widely identified as the original inspiration for Alice in Wonderland, though Carroll always denied this. An avid puzzler, Carroll created the word ladder puzzle (which he then called "Doublets"), which he published in his weekly column for Vanity Fair magazine between 1879 and 1881. In 1982 a memorial stone to Carroll was unveiled at Poets' Corner in Westminster Abbey. There are societies in many parts of the world dedicated to the enjoyment and promotion of his works.[1][2] Early life Dodgson's family was predominantly northern English, conservative, and high-church Anglican. Most of his male ancestors were army officers or Anglican clergymen. His great-grandfather, Charles Dodgson, had risen through the ranks of the church to become the Bishop of Elphin in rural Ireland.[3] His paternal grandfather, another Charles, had been an army captain, killed in action in Ireland in 1803, when his two sons were hardly more than babies.[4] The older of these sons, yet another Charles Dodgson, was Carroll's father. He went to Westminster School and then to Christ Church, Oxford.[5] He reverted to the other family tradition and took holy orders. He was mathematically gifted and won a double first degree, which could have been the prelude to a brilliant academic career. Instead, he married his first cousin Frances Jane Lutwidge in 1830 and became a country parson.[6][7] Dodgson was born on 27 January 1832 at All Saints' Vicarage in Daresbury, Cheshire,[8] the oldest boy and the third oldest of 11 children. When he was 11, his father was given the living of Croft-on-Tees, Yorkshire, and the whole family moved to the spacious rectory. This remained their home for the next 25 years. Charles' father was an active and highly conservative cleric of the Church of England who later became the Archdeacon of Richmond[9] and involved himself, sometimes influentially, in the intense religious disputes that were dividing the church. He was high-church, inclining toward Anglo-Catholicism, an admirer of John Henry Newman and the Tractarian movement, and did his best to instil such views in his children. However, Charles developed an ambivalent relationship with his father's values and with the Church of England as a whole.[10] During his early youth, Dodgson was educated at home. His "reading lists" preserved in the family archives testify to a precocious intellect: at the age of seven, he was reading books such as The Pilgrim's Progress. He also spoke with a stammer – a condition shared by most of his siblings[11] – that often inhibited his social life throughout his years. At the age of twelve he was sent to Richmond Grammar School (now part of Richmond School) in Richmond, North Yorkshire. In 1846, Dodgson entered Rugby School, where he was evidently unhappy, as he wrote some years after leaving: "I cannot say ... that any earthly considerations would induce me to go through my three years again ... I can honestly say that if I could have been ... secure from annoyance at night, the hardships of the daily life would have been comparative trifles to bear."[12] He did not claim he suffered from bullying, but cited little boys as the main targets of older bullies at Rugby.[13] Stuart Dodgson Collingwood, Dodgson's nephew, wrote that "even though it is hard for those who have only known him as the gentle and retiring don to believe it, it is nevertheless true that long after he left school, his name was remembered as that of a boy who knew well how to use his fists in defence of a righteous cause", which is the protection of the smaller boys.[13] Scholastically, though, he excelled with apparent ease. "I have not had a more promising boy at his age since I came to Rugby", observed mathematics master R. B. Mayor.[14] Francis Walkingame's The Tutor's Assistant; Being a Compendium of Arithmetic – the mathematics textbook that the young Dodgson used – still survives and it contained an inscription in Latin, which translates to: "This book belongs to Charles Lutwidge Dodgson: hands off!"[15] Some pages also included annotations such as the one found on p. 129, where he wrote "Not a fair question in decimals" next to a question.[16] He left Rugby at the end of 1849 and matriculated at the University of Oxford in May 1850 as a member of his father's old college, Christ Church.[17] After waiting for rooms in college to become available, he went into residence in January 1851.[18] He had been at Oxford only two days when he received a summons home. His mother had died of "inflammation of the brain" – perhaps meningitis or a stroke – at the age of 47.[18] His early academic career veered between high promise and irresistible distraction. He did not always work hard, but was exceptionally gifted, and achievement came easily to him. In 1852, he obtained first-class honours in Mathematics Moderations and was soon afterwards nominated to a Studentship by his father's old friend Canon Edward Pusey.[19][20] In 1854, he obtained first-class honours in the Final Honours School of Mathematics, standing first on the list, and thus graduated as Bachelor of Arts.[21][22] He remained at Christ Church studying and teaching, but the next year he failed an important scholarship exam through his self-confessed inability to apply himself to study.[23][24] Even so, his talent as a mathematician won him the Christ Church Mathematical Lectureship in 1855,[25] which he continued to hold for the next 26 years.[26] Despite early unhappiness, Dodgson remained at Christ Church, in various capacities, until his death, including that of Sub-Librarian of the Christ Church library, where his office was close to the Deanery, where Alice Liddell lived.[27] Character and appearance Health problems The young adult Charles Dodgson was about 6 feet (1.83 m) tall and slender, and he had curly brown hair and blue or grey eyes (depending on the account). He was described in later life as somewhat asymmetrical, and as carrying himself rather stiffly and awkwardly, although this might be on account of a knee injury sustained in middle age. As a very young child, he suffered a fever that left him deaf in one ear. At the age of 17, he suffered a severe attack of whooping cough, which was probably responsible for his chronically weak chest in later life. In early childhood, he acquired a stammer, which he referred to as his "hesitation"; it remained throughout his life.[27] The stammer has always been a significant part of the image of Dodgson. While one apocryphal story says that he stammered only in adult company and was free and fluent with children, there is no evidence to support this idea.[28] Many children of his acquaintance remembered the stammer, while many adults failed to notice it. Dodgson himself seems to have been far more acutely aware of it than most people whom he met; it is said that he caricatured himself as the Dodo in Alice's Adventures in Wonderland, referring to his difficulty in pronouncing his last name, but this is one of the many supposed facts often repeated for which no first-hand evidence remains. He did indeed refer to himself as a dodo, but whether or not this reference was to his stammer is simply speculation.[27] Dodgson's stammer did trouble him, but it was never so debilitating that it prevented him from applying his other personal qualities to do well in society. He lived in a time when people commonly devised their own amusements and when singing and recitation were required social skills, and the young Dodgson was well equipped to be an engaging entertainer. He could reportedly sing at a passable level and was not afraid to do so before an audience. He was also adept at mimicry and storytelling, and reputedly quite good at charades.[27] Social connections In the interim between his early published writings and the success of the Alice books, Dodgson began to move in the pre-Raphaelite social circle. He first met John Ruskin in 1857 and became friendly with him. Around 1863, he developed a close relationship with Dante Gabriel Rossetti and his family. He would often take pictures of the family in the garden of the Rossetti's house in Chelsea, London. He also knew William Holman Hunt, John Everett Millais, and Arthur Hughes, among other artists. He knew fairy-tale author George MacDonald well – it was the enthusiastic reception of Alice by the young MacDonald children that persuaded him to submit the work for publication.[27][29] Politics, religion, and philosophy In broad terms, Dodgson has traditionally been regarded as politically, religiously, and personally conservative. Martin Gardner labels Dodgson as a Tory who was "awed by lords and inclined to be snobbish towards inferiors".[30] William Tuckwell, in his Reminiscences of Oxford (1900), regarded him as "austere, shy, precise, absorbed in mathematical reverie, watchfully tenacious of his dignity, stiffly conservative in political, theological, social theory, his life mapped out in squares like Alice's landscape".[31] Dodgson was ordained a deacon in the Church of England on 22 December 1861. In The Life and Letters of Lewis Carroll, the editor states that "his Diary is full of such modest depreciations of himself and his work, interspersed with earnest prayers (too sacred and private to be reproduced here) that God would forgive him the past, and help him to perform His holy will in the future."[32] When a friend asked him about his religious views, Dodgson wrote in response that he was a member of the Church of England, but "doubt[ed] if he was fully a 'High Churchman'". He added: I believe that when you and I come to lie down for the last time, if only we can keep firm hold of the great truths Christ taught us—our own utter worthlessness and His infinite worth; and that He has brought us back to our one Father, and made us His brethren, and so brethren to one another—we shall have all we need to guide us through the shadows. Most assuredly I accept to the full the doctrines you refer to—that Christ died to save us, that we have no other way of salvation open to us but through His death, and that it is by faith in Him, and through no merit of ours, that we are reconciled to God; and most assuredly I can cordially say, "I owe all to Him who loved me, and died on the Cross of Calvary." — Carroll (1897)[33] Dodgson also expressed interest in other fields. He was an early member of the Society for Psychical Research, and one of his letters suggests that he accepted as real what was then called "thought reading".[34] Dodgson wrote some studies of various philosophical arguments. In 1895, he developed a philosophical regressus-argument on deductive reasoning in his article "What the Tortoise Said to Achilles", which appeared in one of the early volumes of Mind.[35] The article was reprinted in the same journal a hundred years later in 1995, with a subsequent article by Simon Blackburn titled "Practical Tortoise Raising".[36] Artistic activities Literature From a young age, Dodgson wrote poetry and short stories, contributing heavily to the family magazine Mischmasch and later sending them to various magazines, enjoying moderate success. Between 1854 and 1856, his work appeared in the national publications The Comic Times and The Train, as well as smaller magazines such as the Whitby Gazette and the Oxford Critic. Most of this output was humorous, sometimes satirical, but his standards and ambitions were exacting. "I do not think I have yet written anything worthy of real publication (in which I do not include the Whitby Gazette or the Oxonian Advertiser), but I do not despair of doing so someday," he wrote in July 1855.[27] Sometime after 1850, he did write puppet plays for his siblings' entertainment, of which one has survived: La Guida di Bragia.[37] In March 1856, he published his first piece of work under the name that would make him famous. A romantic poem called "Solitude" appeared in The Train under the authorship of "Lewis Carroll". This pseudonym was a play on his real name: Lewis was the anglicised form of Ludovicus, which was the Latin for Lutwidge, and Carroll an Irish surname similar to the Latin name Carolus, from which comes the name Charles.[7] The transition went as follows: "Charles Lutwidge" translated into Latin as "Carolus Ludovicus". This was then translated back into English as "Carroll Lewis" and then reversed to make "Lewis Carroll".[38] This pseudonym was chosen by editor Edmund Yates from a list of four submitted by Dodgson, the others being Edgar Cuthwellis, Edgar U. C. Westhill, and Louis Carroll.[39] Alice books In 1856, Dean Henry Liddell arrived at Christ Church, bringing with him his young family, all of whom would figure largely in Dodgson's life over the following years, and would greatly influence his writing career. Dodgson became close friends with Liddell's wife Lorina and their children, particularly the three sisters Lorina, Edith, and Alice Liddell. He was widely assumed for many years to have derived his own "Alice" from Alice Liddell; the acrostic poem at the end of Through the Looking-Glass spells out her name in full, and there are also many superficial references to her hidden in the text of both books. It has been noted that Dodgson himself repeatedly denied in later life that his "little heroine" was based on any real child,[40][41] and he frequently dedicated his works to girls of his acquaintance, adding their names in acrostic poems at the beginning of the text. Gertrude Chataway's name appears in this form at the beginning of The Hunting of the Snark, and it is not suggested that this means that any of the characters in the narrative are based on her.[41] Information is scarce (Dodgson's diaries for the years 1858–1862 are missing), but it seems clear that his friendship with the Liddell family was an important part of his life in the late 1850s, and he grew into the habit of taking the children on rowing trips (first the boy, Harry, and later the three girls) accompanied by an adult friend[42] to nearby Nuneham Courtenay or Godstow.[43] It was on one such expedition on 4 July 1862 that Dodgson invented the outline of the story that eventually became his first and greatest commercial success. He told the story to Alice Liddell and she begged him to write it down, and Dodgson eventually (after much delay) presented her with a handwritten, illustrated manuscript entitled Alice's Adventures Under Ground in November 1864.[43] Before this, the family of friend and mentor George MacDonald read Dodgson's incomplete manuscript, and the enthusiasm of the MacDonald children encouraged Dodgson to seek publication. In 1863, he had taken the unfinished manuscript to Macmillan the publisher, who liked it immediately. After the possible alternative titles were rejected – Alice Among the Fairies and Alice's Golden Hour – the work was finally published as Alice's Adventures in Wonderland in 1865 under the Lewis Carroll pen-name, which Dodgson had first used some nine years earlier.[29] The illustrations this time were by Sir John Tenniel; Dodgson evidently thought that a published book would need the skills of a professional artist. Annotated versions provide insights into many of the ideas and hidden meanings that are prevalent in these books.[44][45] Critical literature has often proposed Freudian interpretations of the book as "a descent into the dark world of the subconscious", as well as seeing it as a satire upon contemporary mathematical advances.[46][47] The overwhelming commercial success of the first Alice book changed Dodgson's life in many ways.[48][49][50] The fame of his alter ego "Lewis Carroll" soon spread around the world. He was inundated with fan mail and with sometimes unwanted attention. Indeed, according to one popular story, Queen Victoria herself enjoyed Alice in Wonderland so much that she commanded that he dedicate his next book to her, and was accordingly presented with his next work, a scholarly mathematical volume entitled An Elementary Treatise on Determinants.[51][52] Dodgson himself vehemently denied this story, commenting "... It is utterly false in every particular: nothing even resembling it has occurred";[52][53] and it is unlikely for other reasons. As T. B. Strong comments in a Times article, "It would have been clean contrary to all his practice to identify [the] author of Alice with the author of his mathematical works".[54][55] He also began earning quite substantial sums of money but continued with his seemingly disliked post at Christ Church.[29] Late in 1871, he published the sequel Through the Looking-Glass, and What Alice Found There. (The title page of the first edition erroneously gives "1872" as the date of publication.[56]) Its somewhat darker mood possibly reflects changes in Dodgson's life. His father's death in 1868 plunged him into a depression that lasted some years.[29] The Hunting of the Snark In 1876, Dodgson produced his next great work, The Hunting of the Snark, a fantastical "nonsense" poem, with illustrations by Henry Holiday, exploring the adventures of a bizarre crew of nine tradesmen and one beaver, who set off to find the snark. It received largely mixed reviews from Carroll's contemporary reviewers,[57] but was enormously popular with the public, having been reprinted seventeen times between 1876 and 1908,[58] and has seen various adaptations into musicals, opera, theatre, plays and music.[59] Painter Dante Gabriel Rossetti reputedly became convinced that the poem was about him.[29] Sylvie and Bruno In 1895, 30 years after the publication of his masterpieces, Carroll attempted a comeback, producing a two-volume tale of the fairy siblings Sylvie and Bruno. Carroll entwines two plots set in two alternative worlds, one set in rural England and the other in the fairytale kingdoms of Elfland, Outland, and others. The fairytale world satirizes English society, and more specifically the world of academia. Sylvie and Bruno came out in two volumes and is considered a lesser work, although it has remained in print for over a century. Photography (1856–1880) In 1856, Dodgson took up the new art form of photography under the influence first of his uncle Skeffington Lutwidge, and later of his Oxford friend Reginald Southey.[60] He soon excelled at the art and became a well-known gentleman-photographer, and he seems even to have toyed with the idea of making a living out of it in his very early years.[29] A study by Roger Taylor and Edward Wakeling exhaustively lists every surviving print, and Taylor calculates that just over half of his surviving work depicts young girls, though about 60% of his original photographic portfolio is now missing.[61] Dodgson also made many studies of men, women, boys, and landscapes; his subjects also include skeletons, dolls, dogs, statues, paintings, and trees.[62] His pictures of children were taken with a parent in attendance and many of the pictures were taken in the Liddell garden because natural sunlight was required for good exposures.[42] He also found photography to be a useful entrée into higher social circles.[63] During the most productive part of his career, he made portraits of notable sitters such as John Everett Millais, Ellen Terry, Dante Gabriel Rossetti, Julia Margaret Cameron, Michael Faraday, Lord Salisbury, and Alfred Tennyson.[29] By the time that Dodgson abruptly ceased photography (1880, after 24 years), he had established his own studio on the roof of Tom Quad, created around 3,000 images, and was an amateur master of the medium, though fewer than 1,000 images have survived time and deliberate destruction. He stopped taking photographs because keeping his studio working was too time-consuming.[64] He used the wet collodion process; commercial photographers who started using the dry-plate process in the 1870s took pictures more quickly.[65] Popular taste changed with the advent of Modernism, affecting the types of photographs that he produced. Inventions To promote letter writing, Dodgson invented "The Wonderland Postage-Stamp Case" in 1889. This was a cloth-backed folder with twelve slots, two marked for inserting the most commonly used penny stamp, and one each for the other current denominations up to one shilling. The folder was then put into a slipcase decorated with a picture of Alice on the front and the Cheshire Cat on the back. It intended to organize stamps wherever one stored their writing implements; Carroll expressly notes in Eight or Nine Wise Words about Letter-Writing it is not intended to be carried in a pocket or purse, as the most common individual stamps could easily be carried on their own. The pack included a copy of a pamphlet version of this lecture.[66][67] Another invention was a writing tablet called the nyctograph that allowed note-taking in the dark, thus eliminating the need to get out of bed and strike a light when one woke with an idea. The device consisted of a gridded card with sixteen squares and a system of symbols representing an alphabet of Dodgson's design, using letter shapes similar to the Graffiti writing system on a Palm device.[68] He also devised a number of games, including an early version of what today is known as Scrabble. Devised some time in 1878, he invented the "doublet" (see word ladder), a form of brain-teaser that is still popular today, changing one word into another by altering one letter at a time, each successive change always resulting in a genuine word.[69] For instance, CAT is transformed into DOG by the following steps: CAT, COT, DOT, DOG.[29] It first appeared in the 29 March 1879 issue of Vanity Fair, with Carroll writing a weekly column for the magazine for two years; the final column dated 9 April 1881.[70] The games and puzzles of Lewis Carroll were the subject of Martin Gardner's March 1960 Mathematical Games column in Scientific American. Other items include a rule for finding the day of the week for any date; a means for justifying right margins on a typewriter; a steering device for a velociman (a type of tricycle); fairer elimination rules for tennis tournaments; a new sort of postal money order; rules for reckoning postage; rules for a win in betting; rules for dividing a number by various divisors; a cardboard scale for the Senior Common Room at Christ Church which, held next to a glass, ensured the right amount of liqueur for the price paid; a double-sided adhesive strip to fasten envelopes or mount things in books; a device for helping a bedridden invalid to read from a book placed sideways; and at least two ciphers for cryptography.[29] He also proposed alternative systems of parliamentary representation. He proposed the so-called Dodgson's method, using the Condorcet method.[71] In 1884, he proposed a proportional representation system based on multi-member districts, each voter casting only a single vote, quotas as minimum requirements to take seats, and votes transferable by candidates through what is now called Liquid democracy.[72] Mathematical work Within the academic discipline of mathematics, Dodgson worked primarily in the fields of geometry, linear and matrix algebra, mathematical logic, and recreational mathematics, producing nearly a dozen books under his real name. Dodgson also developed new ideas in linear algebra (e.g., the first printed proof of the Rouché–Capelli theorem),[73][74] probability, and the study of elections (e.g., Dodgson's method) and committees; some of this work was not published until well after his death. His occupation as Mathematical Lecturer at Christ Church gave him some financial security.[75] Mathematical logic His work in the field of mathematical logic attracted renewed interest in the late 20th century. Martin Gardner's book on logic machines and diagrams and William Warren Bartley's posthumous publication of the second part of Dodgson's symbolic logic book have sparked a reevaluation of Dodgson's contributions to symbolic logic.[76][77][78] It is recognized that in his Symbolic Logic Part II, Dodgson introduced the Method of Trees, the earliest modern use of a truth tree.[79] Algebra Robbins' and Rumsey's investigation[80] of Dodgson condensation, a method of evaluating determinants, led them to the alternating sign matrix conjecture, now a theorem. Recreational mathematics The discovery in the 1990s of additional ciphers that Dodgson had constructed, in addition to his "Memoria Technica", showed that he had employed sophisticated mathematical ideas in their creation.[81] Correspondence Dodgson wrote and received as many as 98,721 letters, according to a special letter register which he devised. He documented his advice about how to write more satisfying letters in a missive entitled "Eight or Nine Wise Words about Letter-Writing".[82] Later years Dodgson's existence remained little changed over the last twenty years of his life, despite his growing wealth and fame. He continued to teach at Christ Church until 1881 and remained in residence there until his death. Public appearances included attending the West End musical Alice in Wonderland (the first major live production of his Alice books) at the Prince of Wales Theatre on 30 December 1886.[83] The two volumes of his last novel, Sylvie and Bruno, were published in 1889 and 1893, but the intricacy of this work was apparently not appreciated by contemporary readers; it achieved nothing like the success of the Alice books, with disappointing reviews and sales of only 13,000 copies.[84][85] The only known occasion on which he travelled abroad was a trip to Russia in 1867 as an ecclesiastic, together with the Reverend Henry Liddon. He recounts the travel in his "Russian Journal", which was first commercially published in 1935.[86] On his way to Russia and back, he also saw different cities in Belgium, Germany, partitioned Poland, and France. Death Dodgson died of pneumonia following influenza on 14 January 1898 at his sisters' home, "The Chestnuts", in Guildford in the county of Surrey, just four days before the death of Henry Liddell. He was two weeks away from turning 66 years old. His funeral was held at the nearby St Mary's Church.[87] His body was buried at the Mount Cemetery in Guildford.[29] He is commemorated at All Saints' Church, Daresbury, in its stained glass windows depicting characters from Alice's Adventures in Wonderland. Controversies and mysteries Sexuality Some late twentieth-century biographers have suggested that Dodgson's interest in children had an erotic element, including Morton N. Cohen in his Lewis Carroll: A Biography (1995),[88] Donald Thomas in his Lewis Carroll: A Portrait with Background (1995), and Michael Bakewell in his Lewis Carroll: A Biography (1996). Cohen, in particular, speculates that Dodgson's "sexual energies sought unconventional outlets", and further writes: We cannot know to what extent sexual urges lay behind Charles's preference for drawing and photographing children in the nude. He contended the preference was entirely aesthetic. But given his emotional attachment to children as well as his aesthetic appreciation of their forms, his assertion that his interest was strictly artistic is naïve. He probably felt more than he dared acknowledge, even to himself.[89] Cohen goes on to note that Dodgson "apparently convinced many of his friends that his attachment to the nude female child form was free of any eroticism", but adds that "later generations look beneath the surface" (p. 229). He argues that Dodgson may have wanted to marry the 11-year-old Alice Liddell and that this was the cause of the unexplained "break" with the family in June 1863,[29] an event for which other explanations are offered. Biographers Derek Hudson and Roger Lancelyn Green stop short of identifying Dodgson as a paedophile (Green also edited Dodgson's diaries and papers), but they concur that he had a passion for small female children and next to no interest in the adult world. Catherine Robson refers to Carroll as "the Victorian era's most famous (or infamous) girl lover".[90] Several other writers and scholars have challenged the evidential basis for Cohen's and others' views about Dodgson's sexual interests. Hugues Lebailly has endeavoured to set Dodgson's child photography within the "Victorian Child Cult", which perceived child nudity as essentially an expression of innocence.[91] Lebailly claims that studies of child nudes were mainstream and fashionable in Dodgson's time and that most photographers made them as a matter of course, including Oscar Gustave Rejlander and Julia Margaret Cameron. Lebailly continues that child nudes even appeared on Victorian Christmas cards, implying a very different social and aesthetic assessment of such material. Lebailly concludes that it has been an error of Dodgson's biographers to view his child-photography with 20th- or 21st-century eyes, and to have presented it as some form of personal idiosyncrasy, when it was a response to a prevalent aesthetic and philosophical movement of the time. Karoline Leach's reappraisal of Dodgson focused in particular on his controversial sexuality. She argues that the allegations of paedophilia rose initially from a misunderstanding of Victorian morals, as well as the mistaken idea – fostered by Dodgson's various biographers – that he had no interest in adult women. She termed the traditional image of Dodgson "the Carroll Myth". She drew attention to the large amounts of evidence in his diaries and letters that he was also keenly interested in adult women, married and single, and enjoyed several relationships with them that would have been considered scandalous by the social standards of his time. She also pointed to the fact that many of those whom he described as "child-friends" were girls in their late teens and even twenties.[92] She argues that suggestions of paedophilia emerged only many years after his death, when his well-meaning family had suppressed all evidence of his relationships with women in an effort to preserve his reputation, thus giving a false impression of a man interested only in little girls. Similarly, Leach points to a 1932 biography by Langford Reed as the source of the dubious claim that many of Carroll's female friendships ended when the girls reached the age of 14.[93] In addition to the biographical works that have discussed Dodgson's sexuality, there are modern artistic interpretations of his life and work that do so as well – in particular, Dennis Potter in his play Alice and his screenplay for the motion picture Dreamchild, and Robert Wilson in his musical Alice. Ordination Dodgson had been groomed for the ordained ministry in the Church of England from a very early age and was expected to be ordained within four years of obtaining his master's degree, as a condition of his residency at Christ Church. He delayed the process for some time but was eventually ordained as a deacon on 22 December 1861. But when the time came a year later to be ordained as a priest, Dodgson appealed to the dean for permission not to proceed. This was against college rules and, initially, Dean Liddell told him that he would have to consult the college ruling body, which would almost certainly have resulted in his being expelled. For unknown reasons, Liddell changed his mind overnight and permitted him to remain at the college in defiance of the rules.[94] Dodgson never became a priest, unique amongst senior students of his time. There is currently no conclusive evidence about why Dodgson rejected the priesthood. Some have suggested that his stammer made him reluctant to take the step because he was afraid of having to preach.[95] Wilson quotes letters by Dodgson describing difficulty in reading lessons and prayers rather than preaching in his own words.[96] But Dodgson did indeed preach in later life, even though not in priest's orders, so it seems unlikely that his impediment was a major factor affecting his choice. Wilson also points out that the Bishop of Oxford, Samuel Wilberforce, who ordained Dodgson, had strong views against clergy going to the theatre, one of Dodgson's great interests. He was interested in minority forms of Christianity (he was an admirer of F. D. Maurice) and "alternative" religions (theosophy).[97] Dodgson became deeply troubled by an unexplained sense of sin and guilt at this time (the early 1860s) and frequently expressed the view in his diaries that he was a "vile and worthless" sinner, unworthy of the priesthood and this sense of sin and unworthiness may well have affected his decision to abandon being ordained to the priesthood.[98] Missing diaries At least four complete volumes and around seven pages of text are missing from Dodgson's 13 diaries.[99] The loss of the volumes remains unexplained; the pages have been removed by an unknown hand. Most scholars assume that the diary material was removed by family members in the interests of preserving the family name, but this has not been proven.[100] Except for one page, material is missing from his diaries for the period between 1853 and 1863 (when Dodgson was 21–31 years old).[101][102] During this period, Dodgson began experiencing great mental and spiritual anguish and confessing to an overwhelming sense of his own sin. This was also the period of time when he composed his extensive love poetry, leading to speculation that the poems may have been autobiographical.[103][104] Many theories have been put forward to explain the missing material. A popular explanation for one missing page (27 June 1863) is that it might have been torn out to conceal a proposal of marriage on that day by Dodgson to the 11-year-old Alice Liddell. However, there has never been any evidence to suggest this, and a paper suggests evidence to the contrary which was discovered by Karoline Leach in the Dodgson family archive in 1996.[105] This paper is known as the "cut pages in diary" document. Carroll's nephew Philip Dodgson Jacques reports that he wrote it well after Carroll's death, based on information from his aunts, who destroyed two diary pages, including the one for 27 June 1863. Jacques did not see the pages himself.[106] The summary for 27 June states that Mrs. Liddell told Dodgson there was gossip circulating about him and the Liddell family's governess, as well as about his relationship with "Ina", presumably Alice's older sister Lorina Liddell. The "break" with the Liddell family that occurred soon after was presumably in response to this gossip.[107][105] Without evidence, Leach suggests an alternative interpretation; Lorina was also the name of Alice Liddell's mother. What is deemed most crucial and surprising is the document seems to imply that Dodgson's break with the family was not connected with Alice at all. Until a primary source is discovered, the events of 27 June 1863 will remain in doubt; however, a 1930 letter from the younger Lorina Liddell to Alice may shed light on the matter. Reporting an interview with an early Dodgson biographer, she wrote: I said his manner became too affectionate to you as you grew older, and that mother spoke to him about it, and that offended him so he ceased coming to visit us again – as one had to find some reason for all intercourse ceasing . . . Mr. D used to take you on his knee . . . I did not say that.[108] Migraine and epilepsy In his diary for 1880, Dodgson recorded experiencing his first episode of migraine with aura, describing very accurately the process of "moving fortifications" that are a manifestation of the aura stage of the syndrome.[109] There is no clear evidence to show whether this was his first experience of migraine per se, or if he may have previously had the far more common form of migraine without aura, although the latter seems most likely, given the fact that migraine most commonly develops in the teens or early adulthood. Another form of migraine aura called Alice in Wonderland syndrome has been named after Dodgson's book of the same name and its titular character because its manifestation can resemble the sudden size-changes in the book. It is also known as micropsia and macropsia, a brain condition affecting the way that objects are perceived by the mind. For example, an affected person may look at a larger object such as a basketball and perceive it as if it were the size of a golf ball. Some authors have suggested that Dodgson may have experienced this type of aura and used it as an inspiration in his work, but there is no evidence that he did.[110][111] Dodgson also had two attacks in which he lost consciousness. He was diagnosed by a Dr. Morshead, Dr. Brooks, and Dr. Stedman, and they believed the attack and a consequent attack to be an "epileptiform" seizure (initially thought to be fainting, but Brooks changed his mind). Some have concluded from this that he had this condition for his entire life, but there is no evidence of this in his diaries beyond the diagnosis of the two attacks already mentioned.[109] Some authors, Sadi Ranson in particular, have suggested that Carroll may have had temporal lobe epilepsy in which consciousness is not always completely lost but altered, and in which the symptoms mimic many of the same experiences as Alice in Wonderland. Carroll had at least one incident in which he suffered full loss of consciousness and awoke with a bloody nose, which he recorded in his diary and noted that the episode left him not feeling himself for "quite sometime afterward". This attack was diagnosed as possibly "epileptiform" and Carroll himself later wrote of his "seizures" in the same diary. Most of the standard diagnostic tests of today were not available in the nineteenth century. Yvonne Hart, consultant neurologist at the John Radcliffe Hospital, Oxford, considered Dodgson's symptoms. Her conclusion, quoted in Jenny Woolf's 2010 The Mystery of Lewis Carroll, is that Dodgson very likely had migraine and may have had epilepsy, but she emphasises that she would have considerable doubt about making a diagnosis of epilepsy without further information.[112] Legacy There are societies in many parts of the world dedicated to the enjoyment and promotion of his works and the investigation of his life.[113] Copenhagen Street in Islington, north London is the location of the Lewis Carroll Children's Library.[114] In 1982, his great-nephew unveiled a memorial stone to him in Poets' Corner, Westminster Abbey.[115] In January 1994, an asteroid, 6984 Lewiscarroll, was discovered and named after Carroll. The Lewis Carroll Centenary Wood near his birthplace in Daresbury opened in 2000.[116] As Carroll was born in All Saints' Vicarage, he is commemorated at All Saints' Church, Daresbury by stained glass windows depicting characters from Alice's Adventures in Wonderland. The Lewis Carroll Centre, attached to the church, was opened in March 2012.[117] Works Literary works • La Guida di Bragia, a Ballad Opera for the Marionette Theatre (around 1850) • "Miss Jones", comic song (1862)[118] • Alice's Adventures in Wonderland (1865) • Phantasmagoria and Other Poems (1869) • Through the Looking-Glass, and What Alice Found There (includes "Jabberwocky" and "The Walrus and the Carpenter") (1871) • The Hunting of the Snark (1876) • Rhyme? And Reason? (1883) – shares some contents with the 1869 collection, including the long poem "Phantasmagoria" • A Tangled Tale (1885) • Sylvie and Bruno (1889) • The Nursery "Alice" (1890) • Sylvie and Bruno Concluded (1893) • Pillow Problems (1893) • What the Tortoise Said to Achilles (1895) • Three Sunsets and Other Poems (1898) • The Manlet (1903)[119] Mathematical works • A Syllabus of Plane Algebraic Geometry (1860) • The Fifth Book of Euclid Treated Algebraically (1858 and 1868) • An Elementary Treatise on Determinants, With Their Application to Simultaneous Linear Equations and Algebraic Equations • Euclid and his Modern Rivals (1879), both literary and mathematical in style • Symbolic Logic Part I • Symbolic Logic Part II (published posthumously) • The Alphabet Cipher (1868) • The Game of Logic (1887) • Curiosa Mathematica I (1888) • Curiosa Mathematica II (1892) • A discussion of the various methods of procedure in conducting elections (1873), Suggestions as to the best method of taking votes, where more than two issues are to be voted on (1874), A method of taking votes on more than two issues (1876), collected as The Theory of Committees and Elections, edited, analysed, and published in 1958 by Duncan Black Other works • Some Popular Fallacies about Vivisection • Eight or Nine Wise Words About Letter-Writing • Notes by an Oxford Chiel • The Principles of Parliamentary Representation (1884) See also • Lewis Carroll identity • Lewis Carroll Shelf Award • RGS Worcester and The Alice Ottley School – Miss Ottley, the first Headmistress of The Alice Ottley School, was a friend of Lewis Carroll. One of the school's houses was named after him. • Carroll diagram • Origins of a Story • The White Knight References 1. "Lewis Carroll Societies". Lewiscarrollsociety.org.uk. Archived from the original on 29 March 2016. Retrieved 7 October 2020. 2. Lewis Carroll Society of North America Inc. Archived 26 March 2022 at the Wayback Machine Charity Navigator. Retrieved 7 October 3. Clark, p. 10 4. Collingwood, pp. 6–7 5. Bakewell, Michael (1996). Lewis Carroll: A Biography. London: Heinemann. p. 2. ISBN 9780434045792. 6. Collingwood, p. 8 7. Cohen, pp. 30–35 8. McCulloch, Fiona (2006). "Lewis Carroll". In Kastan, David Scott (ed.). The Oxford Encyclopedia of British Literature. Vol. 3: Harr—Mirr. Oxford, U.K.: Oxford University Press. p. 386. ISBN 9780195169218. 9. "Charles Lutwidge Dodgson". The MacTutor History of Mathematics archive. Archived from the original on 5 July 2011. Retrieved 8 March 2011. 10. Cohen, pp. 200–202 11. Cohen, p. 4 12. Collingwood, pp. 30–31 13. Woolf, Jenny (2010). The Mystery of Lewis Carroll: Discovering the Whimsical, Thoughtful, and Sometimes Lonely Man Who Created "Alice in Wonderland". New York: St. Martin's Press. pp. 24. ISBN 9780312612986. 14. Collingwood, p. 29 15. Carroll, Lewis (1995). Wakeling, Edward (ed.). Rediscovered Lewis Carroll Puzzles. New York City: Dover Publications. pp. 13. ISBN 0486288617. 16. Lovett, Charlie (2005). Lewis Carroll Among His Books: A Descriptive Catalogue of the Private Library of Charles L. Dodgson. Jefferson, North Carolina: McFarland & Company, Inc., Publishers. p. 329. ISBN 0786421053. 17. Clark, pp. 63–64 18. Clark, pp. 64–65 19. Collingwood, p. 52 20. Clark, p. 74 21. Collingwood, p. 57 22. Wilson, p. 51 23. Cohen, p. 51 24. Clark, p. 79 25. Flood, Raymond; Rice, Adrian; Wilson, Robin (2011). Mathematics in Victorian Britain. Oxfordshire, England: Oxford University Press. p. 41. ISBN 978-0-19-960139-4. OCLC 721931689. 26. Cohen, pp. 414–416 27. Leach, Ch. 2. 28. Leach, p. 91 29. Cohen, pp. 100–4 30. Gardner, Martin (2000). Introduction to The annotated Alice: Alice's adventures in Wonderland & Through the looking glass. W. W. Norton & Company. p. xv. ISBN 0-517-02962-6. 31. Gardner, Martin (2009). Introduction to Alice's Adventures in Wonderland and Through the Looking-Glass. Oxford University Press. p. xvi. ISBN 978-0-517-02962-6. 32. Collingwood 33. Collingwood, Chapter IX 34. Hayness, Renée (1982). The Society for Psychical Research, 1882–1982 A History. London: Macdonald & Co. pp. 13–14. ISBN 0-356-07875-2.{{cite book}}: CS1 maint: multiple names: authors list (link) 35. Carroll, L. (1895). "What the Tortoise Said to Achilles". Mind. IV (14): 278–280. doi:10.1093/mind/IV.14.278. 36. Blackburn, S. (1995). "Practical Tortoise Raising". Mind. 104 (416): 695–711. doi:10.1093/mind/104.416.695. 37. Heath, Peter L. (2007). "Introduction". La Guida Di Bragia, a Ballad Opera for the Marionette Theatre. Lewis Carroll Society of North America. pp. vii–xvi. ISBN 978-0-930326-15-9. 38. Roger Lancelyn Green On-line Encyclopædia Britannica Archived 9 May 2015 at the Wayback Machine 39. Thomas, p. 129 40. Cohen, Morton N. (ed) (1979) The Letters of Lewis Carroll, London: Macmillan. 41. Leach, Ch. 5 "The Unreal Alice" 42. Winchester, Simon (2011). The Alice Behind Wonderland. Oxford University Press. ISBN 978-0-19-539619-5. OCLC 641525313. 43. Leach, Ch. 4 44. Gardner, Martin (2000). "The Annotated Alice. The Definitive Edition". New York: W.W. Norton. 45. Heath, Peter (1974). "The Philosopher's Alice". New York: St. Martin's Press. 46. "Algebra in Wonderland". The New York Times. 7 March 2010. Archived from the original on 12 March 2010. Retrieved 10 February 2017. 47. Bayley, Melanie. "Alice's adventures in algebra: Wonderland solved". New Scientist. Archived from the original on 25 January 2022. Retrieved 14 October 2016. 48. Elster, Charles Harrington (2006). The big book of beastly mispronunciations: the complete opinionated guide for the careful speaker. Houghton Mifflin Harcourt. pp. 158–159. ISBN 061842315X. Archived from the original on 3 January 2017. Retrieved 3 August 2016. 49. Emerson, R. H. (1996). "The Unpronounceables: Difficult Literary Names 1500–1940". English Language Notes. 34 (2): 63–74. ISSN 0013-8282. 50. "Lewis Carroll". Biography in Context. Gale. Archived from the original on 26 March 2022. Retrieved 24 September 2015. 51. Wilson 52. "Lewis Carroll – Logician, Nonsense Writer, Mathematician and Photographer". The Hitchhiker's Guide to the Galaxy. BBC. 26 August 2005. Archived from the original on 3 February 2009. Retrieved 12 February 2009. 53. Dodgson, Charles (1896). Symbolic Logic. 54. Strong, T. B. (27 January 1932). "Mr. Dodgson: Lewis Carroll at Oxford". [The Times]. 55. "Fit for a Queen". Snopes. 26 March 1999. Archived from the original on 26 March 2022. Retrieved 25 March 2011. 56. Cohen, Morton (24 June 2009). Introduction to "Alice in Wonderland and Through the Looking-Glass". Random House. ISBN 978-0-553-21345-4. 57. Cohen, Morton N. (1976). "Hark the Snark". In Guilano, Edward (ed.). Lewis Carroll Observed. New York: Clarkson N. Potter, Inc. pp. 92–110. ISBN 0-517-52497-X. 58. Williams, Sidney Herbert; Madan, Falconer (1979). Handbook of the literature of the Rev. C.L. Dodgson. Folkestone, England: Dawson. p. 68. ISBN 9780712909068. OCLC 5754676. 59. Greenarce, Selwyn (2006) [1876]. "The Listing of the Snark". In Martin Gardner (ed.). The Annotated Hunting of the Snark (Definitive ed.). W. W. Norton. pp. 117–147. ISBN 0-393-06242-2. 60. Clark, p. 93 61. Taylor, Roger; Wakeling, Edward (25 February 2002). Lewis Carroll, Photographer. Princeton University Press. ISBN 978-0-691-07443-6. 62. Cohen, Morton (1999). "Reflections in a Looking Glass." New York: Aperture. 63. Thomas, p. 116 64. Thomas, p. 265 65. Wakeling, Edward (1998). "Lewis Carroll's Photography". An Exhibition From the Jon A. Lindseth Collection of C. L. Dodgson and Lewis Carroll. New York, NY: The Grolier Club. pp. 55–67. ISBN 0-910672-23-7. 66. Flodden W. Heron, "Lewis Carroll, Inventor of Postage Stamp Case" in Stamps, vol. 26, no. 12, 25 March 1939 67. "Carroll Related Stamps". The Lewis Carroll Society. 28 April 2005. Archived from the original on 21 March 2012. Retrieved 10 March 2011. 68. Everson, Michael. (2011) "Alice's Adventures in Wonderland: An edition printed in the Nyctographic Square Alphabet devised by Lewis Carroll". Foreword by Alan Tannenbaum, Éire: Cathair na Mart. ISBN 978-1-904808-78-7 69. Gardner, Martin. "Word Ladders: Lewis Carroll's Doublets". No. Vol. 80, No. 487, Centenary Issue (Mar. 1996). The Mathematical Gazette. JSTOR 3620349. Archived from the original on 20 April 2021. Retrieved 22 March 2022. 70. Deanna Haunsperger, Stephen Kennedy (31 July 2006). The Edge of the Universe: Celebrating Ten Years of Math Horizons. Mathematical Association of America. p. 22. ISBN 0-88385-555-0. 71. Black, Duncan; McLean, Iain; McMillan, Alistair; Monroe, Burt L.; Dodgson, Charles Lutwidge (1996). A Mathematical Approach to Proportional Representation. ISBN 978-0-7923-9620-8. Archived from the original on 4 February 2021. Retrieved 4 October 2020. 72. Charles Dodgson, Principles of Parliamentary Representation (1884) 73. Seneta, Eugene (1984). "Lewis Carroll as a Probabilist and Mathematician" (PDF). The Mathematical Scientist. 9: 79–84. Archived (PDF) from the original on 30 January 2016. Retrieved 1 February 2015. 74. Abeles, Francine F. (1998) Charles L. Dodgson, Mathematician". An Exhibition From the Jon A. Lindseth Collection of C.L. Dodgson and Lewis Carroll". New York: The Grolier Club, pp. 45–54. 75. Wilson, p. 61 76. Gardner, Martin. (1958) "Logic Machines and Diagrams". Brighton, Sussex: Harvester Press 77. Bartley, William Warren III, ed. (1977) "Lewis Carroll's Symbolic Logic". New York: Clarkson N. Potter, 2nd ed 1986. 78. Moktefi, Amirouche. (2008) "Lewis Carroll's Logic", pp. 457–505 in British Logic in the Nineteenth Century, Vol. 4 of Handbook of the History of Logic, Dov M. Gabbay and John Woods (eds.) Amsterdam: Elsevier. 79. "Modern Logic: The Boolean Period: Carroll – Encyclopedia.com". Archived from the original on 3 August 2020. Retrieved 22 July 2020. 80. Robbins, D. P.; Rumsey, H. (1986). "Determinants and alternating sign matrices". Advances in Mathematics. 62 (2): 169. doi:10.1016/0001-8708(86)90099-X. 81. Abeles, F. F. (2005). "Lewis Carroll's ciphers: The literary connections". Advances in Applied Mathematics. 34 (4): 697–708. doi:10.1016/j.aam.2004.06.006. 82. Clark, Dorothy G. (April 2010). "The Place of Lewis Carroll in Children's Literature (review)". The Lion and the Unicorn. 34 (2): 253–258. doi:10.1353/uni.0.0495. S2CID 143924225. Archived from the original on 4 March 2016. Retrieved 21 January 2014. 83. Carroll, Lewis (1979). The Letters of Lewis Carroll, Volumes 1–2. Oxford University Press. p. 657. Dec. 30th.—To London with M—, and took her to "Alice in Wonderland," Mr. Savile Clarke's play at the Prince of Wales's Theatre... as a whole, the play seems a success. 84. Angelica Shirley Carpenter (2002). Lewis Carroll: Through the Looking Glass. Lerner. p. 98.ISBN 978-0822500735. 85. Christensen, Thomas (23 April 1991). "Dodgson's Dodges". rightreading.com. San Francisco, California. Archived from the original on 15 July 2011. 86. "Chronology of Works of Lewis Carroll". Archived from the original on 20 February 2009. Retrieved 20 February 2009. 87. "Lewis Carroll and St Mary's Church – Guildford: This Is Our Town website". 30 October 2013. Archived from the original on 11 November 2016. Retrieved 10 November 2016. 88. Cohen, pp. 166–167, 254–255 89. Cohen, p. 228 90. Robson, Catherine (2001). Men in Wonderland: The Lost Girlhood of the Victorian Gentlemen. Princeton, New Jersey: Princeton University Press. p. 137. ISBN 978-0691004228. 91. "Association for new Lewis Carroll studies". Contrariwise.wild-reality.net. Archived from the original on 7 February 2012. Retrieved 19 October 2019. 92. Leach, pp. 16–17 93. Leach, p. 33 94. Dodgson's MS diaries, volume 8, 22–24 October 1862 95. Cohen, p. 263 96. Wilson, pp. 103–104 97. Leach, p. 134 98. Dodgson's MS diaries, volume 8, see prayers scattered throughout the text 99. Leach, pp. 48, 51 100. Leach, pp. 48–51 101. Leach, p. 52 102. Wakeling, Edward (April 2003). "The Real Lewis Carroll – A Talk given to the Lewis Carroll Society". Archived from the original on 8 July 2006. Retrieved 12 January 2023. 103. Leach p. 54 104. "The Dodgson Family and Their Legacy". Archived from the original on 14 January 2011. Retrieved 5 January 2011. 105. "The cut pages in diary document". Archived from the original on 12 January 2023. Retrieved 12 January 2023. 106. Cohen, Morton N. "When love was young", Times Literary Supplement, 10 September 2004. 107. Leach, pp. 170–2. 108. Cohen, "When love was young" 109. Wakeling, Edward (Ed.) "The Diaries of Lewis Carroll", Vol. 9, p. 52 110. Maudie, F.W. "Migraine and Lewis Carroll". The Migraine Periodical. 17. 111. Podoll, K; Robinson, D (1999). "Lewis Carroll's migraine experiences". The Lancet. 353 (9161): 1366. doi:10.1016/S0140-6736(05)74368-3. PMID 10218566. S2CID 5082284. 112. Woolf, Jenny (4 February 2010). The Mystery of Lewis Carroll. St. Martin's Press. pp. 298–9. ISBN 978-0-312-67371-0. 113. "Lewis Carroll Societies". Lewiscarrollsociety.org.uk. Archived from the original on 29 March 2016. Retrieved 12 September 2013. 114. "'A most curious thing' / Lewis Carroll Library". designbybeam.com. Archived from the original on 3 April 2015. Retrieved 15 March 2013. 115. "LEWIS CARROLL IS HONORED ON 150TH BIRTHDAY". The New York Times. 18 December 1982. Archived from the original on 5 May 2015. Retrieved 30 January 2015. 116. "Lewis Carroll Centenary Wood near Daresbury Runcorn". woodlandtrust.org.uk. Archived from the original on 5 August 2020. Retrieved 27 November 2019. 117. About Us, Lewis Carroll Centre & All Saints Daresbury PCC, archived from the original on 14 April 2012, retrieved 11 April 2012 118. The Carrollian. Lewis Carroll Society. Issue 7–8. p. 7. 2001: "In 1862 when Lewis Carroll sent to Yates the manuscript of the words of a 'melancholy song', entitled 'Miss Jones', he hoped that it would be published and performed by a comedian on a London music-hall stage." Archived 4 August 2020 at the Wayback Machine 119. The Hunting of the Snark and Other Poems and Verses, New York: Harper & Brothers, 1903 Bibliography • Clark, Ann (1979). Lewis Carroll: A Biography. London: J. M. Dent. ISBN 0-460-04302-1. • Cohen, Morton (1996). Lewis Carroll: A Biography. Vintage Books. pp. 30–35. ISBN 0-679-74562-9. • Collingwood, Stuart Dodgson (1898). The Life and Letters of Lewis Carroll. London: T. Fisher Unwin. • Leach, Karoline (1999). In the Shadow of the Dreamchild: A New Understanding of Lewis Carroll. London: Peter Owen. • Pizzati, Giovanni: "An Endless Procession of People in Masquerade". Figure piane in Alice in Wonderland. 1993, Cagliari. • Reed, Langford: The Life of Lewis Carroll (1932. London: W. and G. Foyle) • Taylor, Alexander L., Knight: The White Knight (1952. Edinburgh: Oliver and Boyd) • Taylor, Roger & Wakeling, Edward: Lewis Carroll, Photographer. 2002. Princeton University Press. ISBN 0-691-07443-7. (Catalogues nearly every Carroll photograph known to be still in existence.) • Thomas, Donald (1996). Lewis Carroll: A Biography. Barnes and Noble, Inc. ISBN 978-0-7607-1232-0. • Wilson, Robin (2008). Lewis Carroll in Numberland: His Fantastical Mathematical Logical Life. London: Allen Lane. ISBN 978-0-7139-9757-6. • Woolf, Jenny: The Mystery of Lewis Carroll. 2010. New York: St Martin's Press. ISBN 978-0-312-61298-6 Further reading • Black, Duncan (1958). The Circumstances in which Rev. C. L. Dodgson (Lewis Carroll) wrote his Three Pamphlets and Appendix: Text of Dodgson's Three Pamphlets and of 'The Cyclostyled Sheet' in The Theory of Committees and Elections, Cambridge: Cambridge University Press • Bowman, Isa (1899). The Story of Lewis Carroll: Told for Young People by the Real Alice in Wonderland, Miss Isa Bowman. London: J.M. Dent & Co. • Carroll, Lewis: The Annotated Alice: 150th Anniversary Deluxe Edition. Illustrated by John Tenniel. Edited by Martin Gardner & Mark Burstein. W. W. Norton. 2015. ISBN 978-0-393-24543-1 • Dodgson, Charles L.: Euclid and His Modern Rivals. Macmillan. 1879. • Dodgson, Charles L.: The Pamphlets of Lewis Carroll • Vol. 1: The Oxford Pamphlets. 1993. ISBN 0-8139-1250-4 • Vol. 2: The Mathematical Pamphlets. 1994. ISBN 0-9303-26-09-1 • Vol. 3: The Political Pamphlets. 2001. ISBN 0-930326-14-8 • Vol. 4: The Logic Pamphlets. 2010 ISBN 978-0-930326-25-8. • Douglas-Fairhurst, Robert (2016). The Story of Alice: Lewis Carroll and the Secret History of Wonderland. Harvard University Press. ISBN 9780674970762. • Goodacre, Selwyn (2006). All the Snarks: The Illustrated Editions of the Hunting of the Snark. Oxford: Inky Parrot Press. • Graham-Smith, Darien (2005). Contextualising Carroll, University of Wales, Bangor. PhD thesis. • Edward Guiliano (1982). Lewis Carroll, a Celebration: Essays on the Occasion of the 150th Anniversary of the Birth of Charles Lutwidge Dodgson, C. N. Potter, London. • Huxley, Francis: The Raven and the Writing Desk. 1976. ISBN 0-06-012113-0. • Kelly, Richard: Lewis Carroll. 1990. Boston: Twayne Publishers. • Kelly, Richard (ed.): Alice's Adventures in Wonderland. 2000. Peterborough, Ontario: Broadviewpress. • Lakoff, Robin T.: Lewis Carroll: Subversive Pragmaticist. 2022. Pragmatics : Quarterly Publication of the International Pragmatics Association, pp. 367–85 • Lovett, Charlie: Lewis Carroll Among His Books: A Descriptive Catalogue of the Private Library of Charles L. Dodgson. 2005. ISBN 0-7864-2105-3 • Waggoner, Diane (2020). Lewis Carroll's Photography and Modern Childhood. Princeton: Princeton University Press. ISBN 978-0-691-19318-2. • Wakeling, Edward (2015). The Photographs of Lewis Carroll: A Catalogue Raisonné. Austin: University of Texas Press. ISBN 978-0-292-76743-0. • Wullschläger, Jackie: Inventing Wonderland. ISBN 0-7432-2892-8. – Also looks at Edward Lear (of the "nonsense" verses), J. M. Barrie (Peter Pan), Kenneth Grahame (The Wind in the Willows), and A. A. Milne (Winnie-the-Pooh). • N.N.: Dreaming in Pictures: The Photography of Lewis Carroll. Yale University Press & SFMOMA, 2004. (Places Carroll firmly in the art photography tradition.) • Over the years, many retellings of Lewis Carroll's Alice in Wonderland. This includes examples like: Alice in Zombieland by Gena Showalter. • Schütze, Franziska: Disney in Wonderland: A Comparative Analysis of Disney's Alice in Wonderland Film Adaptations from 1951 and 2010 External links Library resources about Lewis Carroll • Resources in your library • Resources in other libraries Digital collections • Works by Lewis Carroll in eBook form at Standard Ebooks • Works by Lewis Carroll at Project Gutenberg • Works by or about Lewis Carroll at Internet Archive • Works by Lewis Carroll at LibriVox (public domain audiobooks) • Works by Lewis Carroll at Open Library • The Poems of Lewis Carroll • First Editions Physical collections • Guide to Harcourt Amory collection of Lewis Carroll at Houghton Library, Harvard University • Lewis Carroll at the British Library • Lewis Carroll online exhibition at the Harry Ransom Center at the University of Texas at Austin • Lewis Carroll Scrapbook Collection at the Library of Congress • "Archival material relating to Lewis Carroll". UK National Archives. Biographical information and scholarship • Lewis Carroll at victorianweb.org • Contrariwise: the Association for New Lewis Carroll Studies – articles by leading members of the 'new scholarship' • Lewis Carroll's Shifting Reputation • Lewis Carroll: Logic, Internet Encyclopedia of Philosophy Other links • Lewis Carroll at the Internet Book List • Lewis Carroll at the Internet Speculative Fiction Database • Newspaper clippings about Lewis Carroll in the 20th Century Press Archives of the ZBW • The Lewis Carroll Society UK • The Lewis Carroll Society of North America Lewis Carroll Literary works • Alice's Adventures in Wonderland (1865) • Rhyme? And Reason? (1869) • Through the Looking-Glass, and What Alice Found There (1871) • "Jabberwocky" • "The Walrus and the Carpenter" • The Hunting of the Snark (1876) • A Tangled Tale (1885) • Sylvie and Bruno (1889, 1893) • "The Mad Gardener's Song" • The Nursery "Alice" (1890) • "What the Tortoise Said to Achilles" (1895) Mathematical works • Euclid and his Modern Rivals (1879) • "The Alphabet Cipher" (1868) • The Game of Logic (1887) • Symbolic Logic (1896, 1977) Other • Eight or Nine Wise Words About Letter-Writing Related • Charles Dodgson (father) • Charles Dodgson (grandfather) Lewis Carroll's Alice • Alice's Adventures in Wonderland • Through the Looking-Glass Universe Characters Alice's Adventures in Wonderland • Alice • portrayals • Bill the Lizard • Caterpillar • Cheshire Cat • Dodo • Dormouse • Duchess • Gryphon • Hatter • Tarrant Hightopp • King of Hearts • Knave of Hearts • March Hare • Mock Turtle • Mouse • Pat • Puppy • Queen of Hearts • White Rabbit • Minor characters Through the Looking-Glass • Bandersnatch • Humpty Dumpty • Jubjub bird • Red King • Red Queen • The Sheep • The Lion and the Unicorn • Tweedledum and Tweedledee • White King • White Knight • White Queen • Minor characters Locations and events • Wonderland • Looking-glass world • Unbirthday Poems • "All in the golden afternoon..." • "How Doth the Little Crocodile" • "The Mouse's Tale" • "Twinkle, Twinkle, Little Bat" • "You Are Old, Father William" • "'Tis the Voice of the Lobster" • "Jabberwocky" • Vorpal sword • "The Walrus and the Carpenter" • "Haddocks' Eyes" • "The Mock Turtle's Song" • The Hunting of the Snark Related • Alice Liddell • Alice syndrome • Alice's Shop • Illustrators • John Tenniel • Theophilus Carter • The Annotated Alice • Mischmasch • Translations • Alice's Adventures in Wonderland • Through the Looking-Glass Adaptations Stage • Alice in Wonderland (1886 musical) • Alice in Wonderland (1979 opera) • But Never Jam Today (1979 musical) • Through the Looking Glass (2008 opera) • Alice's Adventures in Wonderland (2011 ballet) • Wonderland (2011 musical) • Peter and Alice (2013 play) • Wonder.land (2015 musical) • Alice's Adventures Under Ground (2016 opera) • Alice by Heart (2019 musical) Film • 1903 • 1910 • 1915 • Alice Comedies (1923–1927) • 1931 • 1933 • 1949 • 1951 • Alice of Wonderland in Paris (1966) • 1972 • 1976 • 1976 (Spanish) • Alice or the Last Escapade (1977) • 1981 • 1982 • The Care Bears Adventure in Wonderland (1987) • 1988 (Czechoslovak) • 1988 (Australian) • Malice in Wonderland (2009) • 2010 • Alice in Murderland (2010) • Alice Through the Looking Glass (2016) • Come Away (2020) Television • Alice in Wonderland (1962) • Alice in Wonderland or What's a Nice Kid like You Doing in a Place like This? (1966) • Alice in Wonderland (1966) • Alice Through the Looking Glass (1966) • 1983 (TV film) • Fushigi no Kuni no Alice (1983) • 1985 (TV film) • Alice Through the Looking Glass (1987) • Adventures in Wonderland (1992) • Alice through the Looking Glass (1998) • Alice in Wonderland (1999) • Alice (2009) • Once Upon a Time in Wonderland (2013) • Alice's Wonderland Bakery (2022) Music • "White Rabbit" (1967 song) • "Don't Come Around Here No More" (1985 music video) • Alice in Wonderland (2010) • Almost Alice (2010) • "Follow Me Down" • Alice Through the Looking Glass (2016) Video games • Through the Looking Glass (1984) • Alice in Wonderland (1985) • Märchen Maze (1988) • Wonderland (1990) • Alice: An Interactive Museum (1991) • Alice no Paint Adventure (1995) • Alice in Wonderland (2000) • American McGee's Alice (2000) • Kingdom Hearts (2002) • Alice in the Country of Hearts (2007) • Alice in Wonderland (2010) • Alice: Madness Returns (2011) Sequels • A New Alice in the Old Wonderland (1895) • New Adventures of Alice (1917) • Alice Through the Needle's Eye (1984) • Automated Alice (1996) Retellings • The Nursery "Alice" (1890) • Alice's Adventures in Wonderland Retold in Words of One Syllable (1905) • American McGee's Alice (2000) • Alice in Verse: The Lost Rhymes of Wonderland (2010) • Alice: Madness Returns (2011) Parodies • The Westminster Alice (1902) • Clara in Blunderland (1902) • Lost in Blunderland (1903) • John Bull's Adventures in the Fiscal Wonderland (1904) • Alice in Blunderland: An Iridescent Dream (1904) • The Looking Glass Wars • 2004 • 2007 • 2009 Imitations • Mopsa the Fairy (1869) • Davy and the Goblin (1884) • The Admiral's Caravan (1891) • Gladys in Grammarland (1896) • Rollo in Emblemland (1902) • Alice in Orchestralia (1925) Literary • Alice in Borderland • Alice in the Country of Hearts • Alice in Murderland • Miyuki-chan in Wonderland • Pandora Hearts • Unbirthday: A Twisted Tale Related • Betty in Blunderland (1934 animated short) • Thru the Mirror (1936 animated short) • Jabberwocky (1971 film) • Jabberwocky (1977 film) • Donald in Mathmagic Land (1959 film) • Dungeonland (1983 module) • Dreamchild (1985 film) • The Hunting of the Snark (1991 musical) • How Doth the Little Crocodile (1998 artworks) • Abby in Wonderland (2008 film) • Disney franchise • Category Victorian-era children's literature Authors • Henry Cadwallader Adams • R. M. Ballantyne • Lucy Lyttelton Cameron • Lewis Carroll • Christabel Rose Coleridge • Harry Collingwood • E. E. Cowper • Frank Cowper • Maria Edgeworth • Evelyn Everett-Green • Juliana Horatia Ewing • Frederic W. Farrar • G. E. Farrow • Agnes Giberne • Anna Maria Hall • L. T. Meade • G. A. Henty • Frances Hodgson Burnett • Thomas Hughes • Richard Jefferies • Charles Kingsley • W. H. G. Kingston • Rudyard Kipling • Andrew Lang • Frederick Marryat • George MacDonald • Mary Louisa Molesworth • Kirk Munroe • E. Nesbit • Frances Mary Peard • Beatrix Potter • William Brighty Rands • Talbot Baines Reed • Elizabeth Missing Sewell • Anna Sewell • Mary Martha Sherwood • Flora Annie Steel • Robert Louis Stevenson • Hesba Stretton • Charlotte Elizabeth Tonna • Charlotte Maria Tucker • Charlotte Mary Yonge Illustrators • Eleanor Vere Boyle • Gordon Browne • Randolph Caldecott • Thomas Crane • Walter Crane • George Cruikshank • Thomas Dalziel (engraver) • Richard Doyle • H. H. Emmerson • Edmund Evans (engraver) • Kate Greenaway • Sydney Prior Hall • Edward Lear • Harold Robert Millar • Arthur Rackham • J. G. Sowerby • Millicent Sowerby • John Tenniel Books • List of 19th-century British children's literature titles Types • Toy book Publishers • Blackie & Son • Marcus Ward & Co. • Frederick Warne & Co 19th-century English photographers • William de Wiveleslie Abney • William Makepeace Thackeray • Sarah Angelina Acland • Anna Atkins • William Bambridge • Alexander Bassano • Richard Beard • Robert Jefferson Bingham • Graystone Bird • Samuel Bourne • Sarah Anne Bright • Samuel Buckle • Julia Margaret Cameron • Lewis Carroll • Philip Henry Delamotte • Elliott & Fry • William England • Roger Fenton • Francis Frith • Peter Wickens Fry • William Hayes • Norman Heathcote • John Herschel • Alfred Horsley Hinton • Frederick Hollyer • Alice Hughes • Richard Keene • William Edward Kilburn • Martin Laroche • Richard Cockle Lucas • Farnham Maxwell-Lyte • William Eastman Palmer & Sons • William Pumphrey • James Robertson • Henry Peach Robinson • Alfred Seaman • Alice Seeley Harris • Charles Shepherd • Jane Martha St. John • Francis Meadow Sutcliffe • Constance Fox Talbot • Henry Fox Talbot • Eveleen Myers • Henry Van der Weyde • Carl Vandyk Authority control International • FAST • ISNI • 2 • 3 • 4 • VIAF • 2 • 3 • 4 National • Norway • Chile • 2 • Spain • 2 • France • BnF data • Argentina • 2 • Catalonia • Germany • 2 • Italy • Israel • Finland • United States • 2 • Sweden • Latvia • 2 • Japan • Czech Republic • Australia • Greece • 2 • Korea • Croatia • Netherlands • 2 • Poland • Portugal • Vatican Academics • CiNii • MathSciNet • Mathematics Genealogy Project • zbMATH Artists • MusicBrainz • Museum of Modern Art • Musée d'Orsay • National Gallery of Canada • Photographers' Identities • RKD Artists • Städel • ULAN People • Deutsche Biographie • Trove Other • SNAC • 2 • IdRef	Wikipedia
The Math Book The Math Book (Sterling Publishing, 2009. ISBN 978-1-4027-5796-9) is a book by American author Clifford A. Pickover. Summary The book contains 250 one-page articles on milestones in the history of math. Each article is followed by a related full-page color image. Reception The book has consistently received good reviews.[1][2] The book has been praised by Martin Gardner.[3] The book is the winner of the Von Neumann Prize.[4] The book has been praised by Boing Boing.[5] References 1. johndcook.com 2. johndcook.com 3. wisc.edu 4. warwick.ac.uk 5. boingboing.net	Wikipedia
Mathematical Association The Mathematical Association is a professional society concerned with mathematics education in the UK. The Mathematical Association AbbreviationMA Formation1871 Legal statusNon-profit organisation and registered charity PurposeProfessional organisation for mathematics educators Location • 259 London Road, Leicester, LE2 3BE Region served UK Main organ MA Council President – Professor Nira Chamberlain (2023-2024) Websitehttps://www.m-a.org.uk History It was founded in 1871 as the Association for the Improvement of Geometrical Teaching and renamed to the Mathematical Association in 1894.[1][2] It was the first teachers' subject organisation formed in England. In March 1927, it held a three-day meeting in Grantham to commemorate the bicentenary of the death of Sir Isaac Newton, attended by Sir J. J. Thomson (discoverer of the electron), Sir Frank Watson Dyson – the Astronomer Royal, Sir Horace Lamb, and G. H. Hardy. In 1951, Mary Cartwright became the first female president of the Mathematical Association.[3] In the 1960s, when comprehensive education was being introduced, the Association was in favour of the 11-plus system. For maths teachers training at university, a teaching award that was examined was the Diploma of the Mathematical Association, later known as the Diploma in Mathematical Education of the Mathematical Association. Function It exists to "bring about improvements in the teaching of mathematics and its applications, and to provide a means of communication among students and teachers of mathematics".[4] Since 1894 it has published The Mathematical Gazette. It is one of the participating bodies in the quadrennial British Congress of Mathematics Education, organised by the Joint Mathematical Council, and it holds its annual general meeting as part of the Congress.[5] Structure It is based in the south-east of Leicester on London Road (A6), just south of the Charles Frears campus of De Montfort University. Aside from the Council, it has seven other specialist committees. Regions Its branches are sometimes shared with the Association of Teachers of Mathematics (ATM): • Birmingham • Cambridge • East Midlands • Exeter • Gloucester • Liverpool • London • Greater Manchester • Meridian • Stoke and Staffordshire • Sheffield • Sussex • Yorkshire Past presidents Past presidents of The Association for the Improvement of Geometrical Teaching included: • 1871 Thomas Archer Hirst • 1878 R B Hayward MA, FRS • 1889 Professor G M Minchin MA, FRS • 1891 James Joseph Sylvester • 1892 The Reverend C Taylor DD • 1893 R Wormell MA, DSc • 1895 Joseph Larmor Past presidents of The Mathematical Association have included: • 1897 Alfred Lodge • 1899–1900 Robert Stawell Ball[6] • 1901 John Fletcher Moulton, Baron Moulton • 1903 Andrew Forsyth • 1905 George Ballard Mathews • 1907 George H. Bryan • 1909–1910 Herbert Hall Turner • 1911–1912 E. W. Hobson • 1913–1914 Alfred George Greenhill • 1915–1916 Alfred North Whitehead • 1918–1919 Percy Nunn • 1920 E. T. Whittaker • 1921 James Wilson • 1922–1923 Thomas Little Heath • 1924–1925 G. H. Hardy • 1926–1927 Micaiah John Muller Hill • 1928–1929 William Fleetwood Sheppard • 1930–1931 Arthur Eddington • 1932–1933 G. N. Watson • 1934 Eric Harold Neville[7] • 1935 A W Siddons • 1936 Andrew Forsyth • 1937 Louis Napoleon George Filon • 1938 W Hope-Jones • 1939 W C Fletcher • 1944 C O Tuckey MA • 1945 Sydney Chapman • 1946 Warin Foster Bushell • 1947 George Barker Jeffery • 1948 Harold Spencer Jones • 1949 A Robson MA • 1950 Professor H R Hasse MA, DSc • 1951 Mary Cartwright • 1952 K S Snell MA • 1953 Professor T A A Broadbent MA • 1954 W. V. D. Hodge • 1955 G L Parsons MA • 1956 George Frederick James Temple • 1957 W J Langford JP, MSc • 1958 Max Newman • 1959 Louise Doris Adams • 1960 Edwin A. Maxwell • 1961 J T Combridge MA, MSc • 1962 Professor V C A Ferraro PhD, DIC • 1963 J B Morgan MA • 1964 Ida Busbridge • 1965 Elizabeth Williams • 1966 F W Kellaway BSc • 1967 A.P. Rollett • 1968 Charles Coulson • 1969 Bertha Swirles • 1970 James Lighthill • 1971 B T Bellis MA, FRSE, FIMA • 1972 C T Daltry BSc, FIMA • 1973 William McCrea • 1974 Margaret Hayman • 1975 Reuben Goodstein • 1976 E Kerr BSc, PhD, FIMA, FBCS • 1977 Professor G Matthews MA, PhD, FIMA • 1978 Alan Tammadge • 1979 Clive W. Kilmister • 1980 D A Quadling MA, FIMA, later OBE • 1981 Michael Atiyah • 1982 F J Budden BSc • 1983 Rolph Ludwig Edward Schwarzenberger • 1984 P B Coaker BSc, ARCS, DIC, FIMA, FBCS • 1985 Hilary Shuard • 1986 Anita Straker • 1987 Margaret Rayner • 1988 A.G. Howson • 1989 Mr Peter Reynolds MA • 1990 Margaret Brown • 1991 Alan J. Bishop • 1992 Mr John Hersee MA • 1993 Dr William Wynne-Wilson BA, PhD • 1994 Mary Bradburn • 1995 E. Roy Ashley • 1996 W. P. Richardson MBE • 1997 Tony Gardiner • 1998 Professor J Chris Robson • 1999 John S Berry • 2000 Mr Stephen Abbott BSc, MSc • 2001 Dr Sue Sanders Cert.Ed, BA, MEd, PhD • 2002 Mr Barry Lewis BSc, BA, FIMA • 2003 Christopher Zeeman • 2004 Professor Adam McBride OBE • 2005 Sue Singer • 2006 Mr Doug French • 2007 Rob Eastaway • 2008 Mr Robert Barbour • 2009 Mrs Jane Imrie • 2010 David Acheson • 2011 Dr Paul Andrews • 2012 Professor Marcus Du Sautoy OBE FRS • 2013 Mr Peter Ransom MBE • 2014 Lynne McClure OBE • 2015 Dr Peter M. Neumann OBE • 2016 Dr Jennie Golding • 2017 Mr Tom Roper • 2018 Professor Mike Askew • 2019 Dr Ems Lord • 2020 Professor Hannah Fry • 2021 Dr Chris Pritchard • 2022 Dr Colin Foster • 2023 Professor Nira Chamberlain (President) • 2024 Dr Vicky Neale (President Delegate) Arms Coat of arms of Mathematical Association Adopted Granted 1 June 1965 [8] Crest On a wreath of the colours a dexter hand couped at the wrist holding a crystal cylinder enclosing a like sphere all Proper. Escutcheon Azure a representation of a pentagon with diagonals Or on a chief Argent an open book Proper inscribed with the Greek letters Pi and Epsilon Sable and edged and clasped Or. Motto Tibi Creditum Debes See also • London Mathematical Society • Institute of Mathematics and its Applications References 1. Orton, Anthony (2004). Learning Mathematics: Issues, Theory and Classroom Practice. A&C Black. p. 181. ISBN 0826471137. 2. Flood, Raymond; Rice, Adrian; Wilson, Robin, eds. (2011). Mathematics in Victorian Britain. Oxford University Press. p. 171. ISBN 978-0-19-162794-1. 3. Williams, Mrs. E. M. (October 1966), "Presidential Address: The Changing Role of Mathematics in Education", The Mathematical Gazette, 50 (373): 243–254, doi:10.2307/3614669, JSTOR 3614669 4. The Mathematical Association — supporting mathematics in education 5. BMCE Handbook, accessed 2018-10-09 6. "Court Circular". The Times. No. 36051. London. 29 January 1900. p. 9. 7. MA presidents have served 1 year terms, starting with Neville. 8. "Mathematical Association". Heraldry of the World. Retrieved 16 February 2021. • Siddons, A. W. (1939). "The Mathematical Association—I". Eureka. 1: 13–15. • Siddons, A. W. (1939). "The Mathematical Association—II". Eureka. 2: 18–19. • Michael H Price Mathematics of the Multitude? A History of the Mathematical Association (MA, 1994) External links • The Mathematical Association website • Complete list of Presidents of the Association • The MA's online shop • Annual conference • The Mathematical Gazette No. 1, 30, 31, 37–39, 41, 43 (1901–1904) on the Internet Archive digitised by Google from the Harvard University Library News items • Addressing the downward spiral of UK maths education in February 2004 • Proposal to split Maths GCSE into two in August 2003 Authority control International • ISNI • VIAF National • Israel • United States Mathematics in the United Kingdom Organizations and Projects • International Centre for Mathematical Sciences • Advisory Committee on Mathematics Education • Association of Teachers of Mathematics • British Society for Research into Learning Mathematics • Council for the Mathematical Sciences • Count On • Edinburgh Mathematical Society • HoDoMS • Institute of Mathematics and its Applications • Isaac Newton Institute • United Kingdom Mathematics Trust • Joint Mathematical Council • Kent Mathematics Project • London Mathematical Society • Making Mathematics Count • Mathematical Association • Mathematics and Computing College • Mathematics in Education and Industry • Megamaths • Millennium Mathematics Project • More Maths Grads • National Centre for Excellence in the Teaching of Mathematics • National Numeracy • National Numeracy Strategy • El Nombre • Numbertime • Oxford University Invariant Society • School Mathematics Project • Science, Technology, Engineering and Mathematics Network • Sentinus Maths schools • Exeter Mathematics School • King's College London Mathematics School • Lancaster University School of Mathematics • University of Liverpool Mathematics School Journals • Compositio Mathematica • Eureka • Forum of Mathematics • Glasgow Mathematical Journal • The Mathematical Gazette • Philosophy of Mathematics Education Journal • Plus Magazine Competitions • British Mathematical Olympiad • British Mathematical Olympiad Subtrust • National Cipher Challenge Awards • Chartered Mathematician • Smith's Prize • Adams Prize • Thomas Bond Sprague Prize • Rollo Davidson Prize	Wikipedia
Sunzi Suanjing Sunzi Suanjing (Chinese: 孫子算經; pinyin: Sūnzǐ Suànjīng; Wade–Giles: Sun Tzu Suan Ching; lit. 'The Mathematical Classic of Master Sun/Master Sun's Mathematical Manual') was a mathematical treatise written during 3rd to 5th centuries AD which was listed as one of the Ten Computational Canons during the Tang dynasty. The specific identity of its author Sunzi (lit. "Master Sun") is still unknown but he lived much later than his namesake Sun Tzu, author of The Art of War. From the textual evidence in the book, some scholars concluded that the work was completed during the Southern and Northern Dynasties.[2] Besides describing arithmetic methods and investigating Diophantine equations, the treatise touches upon astronomy and attempts to develop a calendar. Contents The book is divided into three chapters. Chapter 1 Chapter 1 discusses measurement units of length, weight and capacity, and the rules of counting rods. Although counting rods were in use in the Spring and Autumn period and there were many ancient books on mathematics such as Book on Numbers and Computation and The Nine Chapters on the Mathematical Art, no detailed account of the rules was given. For the first time, The Mathematical Classic of Sun Zi provided a detail description of the rules of counting rods: "one must know the position of the counting rods, the units are vertical, the tens horizontal, the hundreds stand, the thousands prostrate",[3] followed by the detailed layout and rules for manipulation of the counting rods in addition, subtraction, multiplication, and division with ample examples. Chapter 2 Chapter 2 deals with operational rules for fractions with rod numerals: the reduction, addition, subtraction, and division of fractions, followed by mechanical algorithm for the extraction of square roots.[4] Chapter 3 Chapter 3 contains the earliest example of Chinese remainder theorem, a key tool to understanding and resolving Diophantine equations. Bibliography Researchers have published a full English translation of the Sūnzĭ Suànjīng: • Fleeting Footsteps; Tracing the Conception of Arithmetic and Algebra in Ancient China, by Lam Lay Yong and Ang Tian Se, Part Two, pp 149–182. World Scientific Publishing Company; June 2004 ISBN 981-238-696-3 The original Chinese text is available on Wikisource. Wikisource has original text related to this article: The Mathematical Classic of Sun Zi External links • Sun Zi at MacTutor References 1. Lam Lay Yong and An Tian Se. "Fleeting Footsteps", p. 4. World Scientific. ISBN 981-02-3696-4. 2. For instance, in problem 33 of volume 3, it is written, "Luoyang is 900 li away from Chang'an". As the name "Chang'an" was first employed during the Han dynasty, this work could not have been written before the 3rd century. Additionally, in problem 3 of volume 3, Sun Tzu writes "We have a board game, 19 rows and 19 columns square. Question: how many stones are there?" Since go made its first appearance in the mid-3rd century, the work was most probably written during the Wei or Jin dynasties.[1] 3. Lam Lay Yong and An Tian Se, Fleeting Footsteps p55, World Scientific, ISBN 981-02-3696-4 4. Lam Lay Yong and An Tian Se, Fleeting Footsteps p65, World Scientific, ISBN 981-02-3696-4	Wikipedia
The Mathematical Coloring Book The Mathematical Coloring Book: Mathematics of Coloring and the Colorful Life of Its Creators is a book on graph coloring, Ramsey theory, and the history of development of these areas, concentrating in particular on the Hadwiger–Nelson problem and on the biography of Bartel Leendert van der Waerden. It was written by Alexander Soifer and published by Springer-Verlag in 2009 (ISBN 978-0-387-74640-1).[1][2] Topics The book "presents mathematics as a human endeavor" and "explores the birth of ideas and moral dilemmas of the times between and during the two World Wars".[1] As such, as well as covering the mathematics of its topics, it includes biographical material and correspondence with many of the people involved in creating it, including in-depth coverage of Issai Schur, Pierre Joseph Henry Baudet, and Bartel Leendert van der Waerden,[2] in particular studying the question of van der Warden's complicity with the Nazis in his war-time service as a professor in Nazi Germany.[3][4] It also includes biographical material on Paul Erdős, Frank P. Ramsey, Emmy Noether, Alfred Brauer, Richard Courant, Kenneth Falconer, Nicolas de Bruijn, Hillel Furstenberg, and Tibor Gallai, among others,[1] as well as many historical photos of these subjects.[2][4] Mathematically, the book considers problems "on the boundary of geometry, combinatorics, and number theory", involving graph coloring problems such as the four color theorem, and generalizations of coloring in Ramsey theory where the use of a too-small number of colors leads to monochromatic structures larger than a single graph edge.[3] Central to the book is the Hadwiger–Nelson problem, the problem of coloring the points of the Euclidean plane in such a way that no two points of the same color are a unit distance apart.[3][4] Other topics covered by the book include Van der Waerden's theorem on monochromatic arithmetic progressions in colorings of the integers[4] and its generalization to Szemerédi's theorem,[1] the Happy ending problem, Rado's theorem,[5] and questions in the foundations of mathematics involving the possibility that different choices of foundational axioms will lead to different answers to some of the coloring questions considered here.[3][4] Reception and audience As a work in graph theory, reviewer Joseph Malkevitch suggests caution over the book's intuitive treatment of graphs that may in many cases be infinite, in comparison with much other work in this area that makes an implicit assumption that every graph is finite.[3] William Gasarch is surprised by the book's omission of some closely related topics, including the proof of the Heawood conjecture on coloring graphs on surfaces by Gerhard Ringel and Ted Youngs.[5] And Günter M. Ziegler complains that many claims are presented without proof.[6] Although Soifer has called the Hadwiger–Nelson problem "the most important problem in all of mathematics",[5] Ziegler disagrees, and suggests that it and the four color theorem are too isolated to be fruitful topics of study.[6] As a work in the history of mathematics, Malkevitch finds the book too credulous of first-person recollections of troubled political times (the lead-up to World War II) and of priority in mathematical discoveries.[3] Ziegler points to several errors of fact in the book's history, takes issue with its insistence that each contribution should be attributed to only one researcher, and doubts Soifer's objectivity with respect to van der Waerden.[6] And reviewer John J. Watkins writes that "Soifer’s book is indeed a treasure trove filled with valuable historical and mathematical information, but a serious reader must also be prepared to sift through a considerable amount of dross" to reach the treasure. And although Watkins is convinced by Soifer's argument that the first conjectural versions of van der Waerden's theorem were due to Schur and Baudet, he finds idiosyncratic Soifer's insistence that this updated credit necessitates a change in the name of the theorem, concluding that "This is a book that needed far better editing."[4] Ziegler agrees, writing "Someone should have also forced him to cut the manuscript, at the long parts and chapters where the investigations into the colorful lives of the creators get out of hand."[6] According to Malkevitch, the book is written for a broad audience, and does not require a graduate-level background in its material, but nevertheless contains much that is of interest to experts as well as beginners.[3] And despite his negative review, Ziegler concurs, writing that it "has interesting parts and a lot of valuable material".[6] Gasarch is much more enthusiastic, writing "This is a Fantastic Book! Go buy it Now!".[5] References 1. Mihók, Peter (2010), "Review of The Mathematical Coloring Book", Mathematical Reviews, MR 2458293 2. Herrera de Figueiredo, Celina Miraglia (January 2009), "Review of The Mathematical Coloring Book", MAA Reviews, Mathematical Association of America 3. Malkevitch, Joseph (August–September 2013), "Review of The Mathematical Coloring Book", American Mathematical Monthly, 120 (7): 670–674, doi:10.4169/amer.math.monthly.120.07.670, JSTOR 10.4169/amer.math.monthly.120.07.670, S2CID 218541540 4. Watkins, John J. (August 2009), "Review of The Mathematical Coloring Book", Historia Mathematica, 36 (3): 275–277, doi:10.1016/j.hm.2009.02.002 5. Gasarch, William (September 2009), "Review of The Mathematical Coloring Book", ACM SIGACT News, 40 (3): 24, doi:10.1145/1620491.1620494, S2CID 20432321 6. Ziegler, Günter M. (September 2014), "Review of The Mathematical Coloring Book", Jahresbericht der Deutschen Mathematiker-Vereinigung, 116 (4): 261–269, doi:10.1365/s13291-014-0101-y, S2CID 256086914	Wikipedia
The Mathematical Diary The Mathematical Diary was an early American mathematical journal and mathematics magazine, published between 1825 and 1833. The Mathematical Diary Cover of the first issue DisciplineMathematics LanguageEnglish Publication details History1825–1833 Standard abbreviations ISO 4 (alt) · Bluebook (alt1 · alt2) NLM · MathSciNet ISO 4Math. Diary The Mathematical Diary was founded by Robert Adrain at Columbia College (now Columbia University) after two unsuccessful attempts, in 1808 and 1814, to start a more purely academic mathematics journal, The Analyst, or, Mathematical Museum. The Mathematical Diary contained, in addition to some serious mathematics, articles of general interest such as mathematical puzzles aimed at the amateur problem-solver, which may have helped it attract more laypeople as subscribers and contributed to its greater longevity. Adrain edited the first four issues; after he left Columbia in 1826 for Rutgers, James Ryan, previously the publisher, took over the editorship. A total of thirteen issues were published. References • Finkel, Benjamin F. (1940). "A History of American Mathematical Journals". National Mathematics Magazine. 14 (8): 461–468. • Parshall, Karen Hunger; David E. Rowe (1994). The Emergence of the American Mathematical Research Community, 1876-1900. American Mathematical Society. p. 44. ISBN 0-8218-9004-2.	Wikipedia
The Mathematical Magpie The Mathematical Magpie is an anthology published in 1962, compiled by Clifton Fadiman as a companion volume to his Fantasia Mathematica (1958).[1] The volume contains stories, cartoons, essays, rhymes, music, anecdotes, aphorisms, and other oddments. Authors include Arthur C. Clarke, Isaac Asimov, Mark Twain, Lewis Carroll, and many other renowned figures. A revised edition was issued in 1981 and again in 1997. Although out of print, it is recommended for undergraduate mathematics libraries by the Mathematical Association of America as part of their Basic Library List.[2] The Mathematical Magpie First edition AuthorClifton Fadiman CountryUnited States LanguageEnglish GenreAnthology PublisherSimon & Schuster Publication date 1962 Media typePrint (Hardcover and Paperback) Pages303 Contents • "Cartoon" by Abner Dean • "Introduction" by Clifton Fadiman A Set of Imaginaries • "Cartoon" by Alan Dunn • "The Feeling of Power" by Isaac Asimov • "The Law" by Robert M. Coates • "The Appendix and the Spectacles" by Miles J. Breuer, MD • "Paul Bunyan Versus the Conveyor Belt" by William Hazlett Upson • "The Pacifist" by Arthur C. Clarke • "The Hermeneutical Doughnut" by H. Nearing Jr. • "Star, Bright" by Mark Clifton • "'FYI"' by James Blish • "The Vanishing Man" by Richard Hughes (writer) • "The Nine Billion Names of God" by Arthur C. Clarke Comic Sections • "Three Mathematical Diversions" by Raymond Queneau • "The Wonderful World of Figures" by Corey Ford • "A B and C – the Human Element in Mathematics" by Stephen Leacock • "Cartoon" by Johnny Hart • "A Note on the Einstein Theory by" Max Beerbohm • "The Achievement of HT Wensel" by H. Allen Smith • "Needed: Feminine Math" by Parke Cummings • "Cartoon" by Alfred Frueh • "Two Extracts" by Mark Twain • "Mathematics for Golfers" by Stephen Leacock • "The Mathematician's Nightmare: The Vision of Professor Squarepunt" by Bertrand Russell • "Milo and the Mathemagician" by Norton Juster Irregular Figures • "Cartoon" by Saul Steinberg • "Sixteen Stones" by Samuel Beckett • "O'Brien's Table" by J.L. Synge • "The Abominable Mr. Gunn" by Robert Graves • "Coconuts" by Ben Ames Williams • "Euclid and the Bright Boy" by J.L. Synge • "The Purse of Fortunatus" (an excerpt from Sylvie and Bruno) by Lewis Carroll • "Cartoon" by Saul Steinberg • "The Symbolic Logic of Murder" by John Reese Simple Harmonic Motions • "Cartoon" James Frankfort • "The Square of the Hypotenuse" Saul Chaplin & Johnny Mercer • "The Ta Ta" Joseph Charles Holbrooke & Sidney H. Sime Dividends and Remainders This section includes a collection of poems, cartoons, anecdotes, and limericks. The final pages describe Mrs. Miniver's problem. References 1. Review of The Mathematical Magpie, Kirkus Reviews, retrieved 2014-04-19. 2. The Mathematical Magpie, Mathematical Association of America, retrieved 2014-04-19.	Wikipedia
The Mathematics of Chip-Firing The Mathematics of Chip-Firing is a textbook in mathematics on chip-firing games and abelian sandpile models. It was written by Caroline Klivans, and published in 2018 by the CRC Press. Topics A chip-firing game, in its most basic form, is a process on an undirected graph, with each vertex of the graph containing some number of chips. At each step, a vertex with more chips than incident edges is selected, and one of its chips is sent to each of its neighbors. If a single vertex is designated as a "black hole", meaning that chips sent to it vanish, then the result of the process is the same no matter what order the other vertices are selected. The stable states of this process are the ones in which no vertex has enough chips to be selected; two stable states can be added by combining their chips and then stabilizing the result. A subset of these states, the so-called critical states, form an abelian group under this addition operation. The abelian sandpile model applies this model to large grid graphs, with the black hole connected to the boundary vertices of the grid; in this formulation, with all eligible vertices selected simultaneously, it can also be interpreted as a cellular automaton. The identity element of the sandpile group often has an unusual fractal structure.[1] The book covers these topics, and is divided into two parts. The first of these parts covers the basic theory outlined above, formulating chip-firing in terms of algebraic graph theory and the Laplacian matrix of the given graph. It describes an equivalence between states of the sandpile group and the spanning trees of the graph, and the group action on spanning trees, as well as similar connections to other combinatorial structures, and applications of these connections in algebraic combinatorics. And it studies chip-firing games on other classes of graphs than grids, including random graphs.[1] The second part of the book has four chapters devoted to more advanced topics in chip-firing. The first of these generalizes chip-firing from Laplacian matrices of graphs to M-matrices, connecting this generalization to root systems and representation theory. The second considers chip-firing on abstract simplicial complexes instead of graphs. The third uses chip-firing to study graph-theoretic analogues of divisor theory and the Riemann–Roch theorem. And the fourth applies methods from commutative algebra to the study of chip-firing.[1][2] The book includes many illustrations, and ends each chapter with a set of exercises making it suitable as a textbook for a course on this topic.[3] Audience and reception Although the book may be readable by some undergraduate mathematics students, reviewer David Perkinson suggests that its main audience should be graduate students in mathematics, for whom it could be used as the basis of a graduate course or seminar. He calls it "a thorough introduction to an exciting and growing subject", with "clear and concise exposition".[1] Reviewer Paul Dreyer calls it a "deep dive" into "incredibly deep mathematics".[3] Another book on the same general topic, published at approximately the same time, is Divisors and Sandpiles: An Introduction to Chip-Firing by Corry and Perkinson (American Mathematical Society, 2018). It is written at a lower level aimed at undergraduate students, covering mainly the material from the first part of The Mathematics of Chip-Firing, and framed more in terms of algebraic geometry than combinatorics.[2] References 1. Perkinson, David (August 2019), "Review of The Mathematics of Chip-Firing", MAA Reviews, Mathematical Association of America 2. Glass, Darren (January 2020), "Review of The Mathematics of Chip-Firing", American Mathematical Monthly, 127 (2): 189–192, doi:10.1080/00029890.2020.1685835 3. Dreyer, Paul A. Jr., "Review of The Mathematics of Chip-Firing", Mathematical Reviews, MR 3889995	Wikipedia
The Mathematics of Games and Gambling The Mathematics of Games and Gambling is a book on probability theory and its application to games of chance. It was written by Edward Packel, and published in 1981 by the Mathematical Association of America as volume 28 of their New Mathematical Library series, with a second edition in 2006. Topics The book has seven chapters. Its first gives a survey of the history of gambling games in western culture, including brief biographies of two famous gamblers, Gerolamo Cardano and Fyodor Dostoevsky,[1] and a review of the games of chance found in Dostoevsky's novel The Gambler.[2] The next four chapters introduce the basic concepts of probability theory, including expectation, binomial distributions and compound distributions, and conditional probability,[1] through games including roulette, keno, craps, chuck-a-luck, backgammon, and blackjack.[3] The sixth chapter of the book moves from probability theory to game theory, including material on tic-tac-toe, matrix representations of zero-sum games, nonzero-sum games such as the prisoner's dilemma, the concept of a Nash equilibrium, game trees, and the minimax method used by computers to play two-player strategy games. A final chapter, "Odds and ends", includes analyses of bluffing in poker, horse racing, and lotteries.[1][4] The second edition adds material on online gambling systems, casino poker machines, and Texas hold 'em poker.[3] It also adds links to online versions of the games, and expands the material on game theory.[5] Audience and reception The book is aimed at students,[1][6] written for a general audience, and does not require any background in mathematics beyond high school algebra.[2][3][5] However, many of its chapters include exercises, making it suitable for teaching high school or undergraduate-level courses using it.[1][3][5] It is also suitable for readers interested in recreational mathematics.[5][7] Although it could also be used to improve readers' ability at games of chance,[7] it is not intended for that, as its overall message is that gambling games are best avoided.[6] Reviewer Sarah Boslaugh notes as a strength of a book the smooth interplay between its mathematical content and the context of the games it describes.[7] Despite noting that the book's description of modern games is based on American practice, and doesn't address the way those games differ in Britain, reviewer Stephen Ainley calls the book "very enjoyable", adding that "it is hard to see how it could be done better or more readably".[4] Reviewer J. Wade Davis calls it "accessible and very entertaining".[5] Recognition The Basic Library List Committee of the Mathematical Association of America has listed this book as essential for inclusion in undergraduate mathematics libraries.[7] It was the 1986 winner of the Beckenbach Book Prize.[8] References 1. Rubel, Laurie (May 2008), "Teaching with games of chance: A review of The Mathematics of Games and Gambling (2nd ed.)", Journal for Research in Mathematics Education, 39 (3): 343–346, doi:10.2307/30034973, JSTOR 30034973 2. Juraschek, William A. (March 1982), "Review of The Mathematics of Games and Gambling (1st ed.)", The Mathematics Teacher, 75 (3): 268–269, JSTOR 27962895 3. Campbell, Paul J. (October 2006), "Review of The Mathematics of Games and Gambling (2nd ed.)", Mathematics Magazine, 79 (4): 318–319, doi:10.2307/27642961, JSTOR 27642961 4. Ainley, Stephen (March 1982), "Review of The Mathematics of Games and Gambling (1st ed.)", The Mathematical Gazette, 66 (435): 82–83, doi:10.2307/3617334, JSTOR 3617334 5. Davis, J. Wade (November 2007), "Review of The Mathematics of Games and Gambling (2nd ed.)", The American Statistician, 61 (4): 372, JSTOR 27643951 6. Wilkins, John (September 2007), "Review of The Mathematics of Games and Gambling (2nd ed.)", The Mathematics Teacher, 101 (2): 159, JSTOR 20876068 7. Boslaugh, Sarah (August 2006), "Review of The Mathematics of Games and Gambling (2nd ed.)", MAA Reviews, Mathematical Association of America 8. Beckenbach Book Prize, Mathematical Association of America, retrieved 2020-04-04	Wikipedia
Measurement of a Circle Measurement of a Circle or Dimension of the Circle (Greek: Κύκλου μέτρησις, Kuklou metrēsis)[1] is a treatise that consists of three propositions by Archimedes, ca. 250 BCE.[2][3] The treatise is only a fraction of what was a longer work.[4][5] Propositions Proposition one Proposition one states: The area of any circle is equal to a right-angled triangle in which one of the sides about the right angle is equal to the radius, and the other to the circumference of the circle. Any circle with a circumference c and a radius r is equal in area with a right triangle with the two legs being c and r. This proposition is proved by the method of exhaustion.[6] Proposition two Proposition two states: The area of a circle is to the square on its diameter as 11 to 14. This proposition could not have been placed by Archimedes, for it relies on the outcome of the third proposition.[6] Proposition three Proposition three states: The ratio of the circumference of any circle to its diameter is greater than $3{\tfrac {10}{71}}$ but less than $3{\tfrac {1}{7}}$. This approximates what we now call the mathematical constant π. He found these bounds on the value of π by inscribing and circumscribing a circle with two similar 96-sided regular polygons.[7] Approximation to square roots This proposition also contains accurate approximations to the square root of 3 (one larger and one smaller) and other larger non-perfect square roots; however, Archimedes gives no explanation as to how he found these numbers.[5] He gives the upper and lower bounds to √3 as 1351/780 > √3 > 265/153.[6] However, these bounds are familiar from the study of Pell's equation and the convergents of an associated continued fraction, leading to much speculation as to how much of this number theory might have been accessible to Archimedes. Discussion of this approach goes back at least to Thomas Fantet de Lagny, FRS (compare Chronology of computation of π) in 1723, but was treated more explicitly by Hieronymus Georg Zeuthen. In the early 1880s, Friedrich Otto Hultsch (1833–1906) and Karl Heinrich Hunrath (b. 1847) noted how the bounds could be found quickly by means of simple binomial bounds on square roots close to a perfect square modelled on Elements II.4, 7; this method is favoured by Thomas Little Heath. Although only one route to the bounds is mentioned, in fact there are two others, making the bounds almost inescapable however the method is worked. But the bounds can also be produced by an iterative geometrical construction suggested by Archimedes' Stomachion in the setting of the regular dodecagon. In this case, the task is to give rational approximations to the tangent of π/12. References 1. Knorr, Wilbur R. (1986-12-01). "Archimedes' dimension of the circle: A view of the genesis of the extant text". Archive for History of Exact Sciences. 35 (4): 281–324. doi:10.1007/BF00357303. ISSN 0003-9519. S2CID 119807724. 2. Lit, L.W.C. (Eric) van (13 November 2012). "Naṣīr al-Dīn al-Ṭūsī's Version of The Measurement of the Circle of Archimedes from his Revision of the Middle Books". Tarikh-e Elm. The measurement of the circle was written by Archimedes (ca. 250 B.C.E.) 3. Knorr, Wilbur R. (1986). The Ancient Tradition of Geometric Problems. Courier Corporation. p. 153. ISBN 9780486675329. Most accounts of Archimedes' works assign this writing to a time relatively late in his career. But this view is the consequence of a plain misunderstanding. 4. Heath, Thomas Little (1921), A History of Greek Mathematics, Boston: Adamant Media Corporation, ISBN 978-0-543-96877-7, retrieved 2008-06-30 5. "Archimedes". Encyclopædia Britannica. 2008. Retrieved 2008-06-30. 6. Heath, Thomas Little (1897), The Works of Archimedes, Cambridge University: Cambridge University Press., pp. lxxvii , 50, retrieved 2008-06-30 7. Heath, Thomas Little (1931), A Manual of Greek Mathematics, Mineola, N.Y.: Dover Publications, p. 146, ISBN 978-0-486-43231-1 Archimedes Written works • Measurement of a Circle • The Sand Reckoner • On the Equilibrium of Planes • Quadrature of the Parabola • On the Sphere and Cylinder • On Spirals • On Conoids and Spheroids • On Floating Bodies • Ostomachion • The Method of Mechanical Theorems • Book of Lemmas (apocryphal) Discoveries and inventions • Archimedean solid • Archimedes's cattle problem • Archimedes' principle • Archimedes's screw • Claw of Archimedes Miscellaneous • Archimedes' heat ray • Archimedes Palimpsest • List of things named after Archimedes • Pseudo-Archimedes Related people • Euclid • Eudoxus of Cnidus • Apollonius of Perga • Hero of Alexandria • Eutocius of Ascalon • Category Ancient Greek mathematics Mathematicians (timeline) • Anaxagoras • Anthemius • Archytas • Aristaeus the Elder • Aristarchus • Aristotle • Apollonius • Archimedes • Autolycus • Bion • Bryson • Callippus • Carpus • Chrysippus • Cleomedes • Conon • Ctesibius • Democritus • Dicaearchus • Diocles • Diophantus • Dinostratus • Dionysodorus • Domninus • Eratosthenes • Eudemus • Euclid • Eudoxus • Eutocius • Geminus • Heliodorus • Heron • Hipparchus • Hippasus • Hippias • Hippocrates • Hypatia • Hypsicles • Isidore of Miletus • Leon • Marinus • Menaechmus • Menelaus • Metrodorus • Nicomachus • Nicomedes • Nicoteles • Oenopides • Pappus • Perseus • Philolaus • Philon • Philonides • Plato • Porphyry • Posidonius • Proclus • Ptolemy • Pythagoras • Serenus • Simplicius • Sosigenes • Sporus • Thales • Theaetetus • Theano • Theodorus • Theodosius • Theon of Alexandria • Theon of Smyrna • Thymaridas • Xenocrates • Zeno of Elea • Zeno of Sidon • Zenodorus Treatises • Almagest • Archimedes Palimpsest • Arithmetica • Conics (Apollonius) • Catoptrics • Data (Euclid) • Elements (Euclid) • Measurement of a Circle • On Conoids and Spheroids • On the Sizes and Distances (Aristarchus) • On Sizes and Distances (Hipparchus) • On the Moving Sphere (Autolycus) • Optics (Euclid) • On Spirals • On the Sphere and Cylinder • Ostomachion • Planisphaerium • Sphaerics • The Quadrature of the Parabola • The Sand Reckoner Problems • Constructible numbers • Angle trisection • Doubling the cube • Squaring the circle • Problem of Apollonius Concepts and definitions • Angle • Central • Inscribed • Axiomatic system • Axiom • Chord • Circles of Apollonius • Apollonian circles • Apollonian gasket • Circumscribed circle • Commensurability • Diophantine equation • Doctrine of proportionality • Euclidean geometry • Golden ratio • Greek numerals • Incircle and excircles of a triangle • Method of exhaustion • Parallel postulate • Platonic solid • Lune of Hippocrates • Quadratrix of Hippias • Regular polygon • Straightedge and compass construction • Triangle center Results In Elements • Angle bisector theorem • Exterior angle theorem • Euclidean algorithm • Euclid's theorem • Geometric mean theorem • Greek geometric algebra • Hinge theorem • Inscribed angle theorem • Intercept theorem • Intersecting chords theorem • Intersecting secants theorem • Law of cosines • Pons asinorum • Pythagorean theorem • Tangent-secant theorem • Thales's theorem • Theorem of the gnomon Apollonius • Apollonius's theorem Other • Aristarchus's inequality • Crossbar theorem • Heron's formula • Irrational numbers • Law of sines • Menelaus's theorem • Pappus's area theorem • Problem II.8 of Arithmetica • Ptolemy's inequality • Ptolemy's table of chords • Ptolemy's theorem • Spiral of Theodorus Centers • Cyrene • Mouseion of Alexandria • Platonic Academy Related • Ancient Greek astronomy • Attic numerals • Greek numerals • Latin translations of the 12th century • Non-Euclidean geometry • Philosophy of mathematics • Neusis construction History of • A History of Greek Mathematics • by Thomas Heath • algebra • timeline • arithmetic • timeline • calculus • timeline • geometry • timeline • logic • timeline • mathematics • timeline • numbers • prehistoric counting • numeral systems • list Other cultures • Arabian/Islamic • Babylonian • Chinese • Egyptian • Incan • Indian • Japanese Ancient Greece portal • Mathematics portal Authority control: National • France • BnF data	Wikipedia
The monkey and the coconuts The monkey and the coconuts is a mathematical puzzle in the field of Diophantine analysis that originated in a magazine fictional short story involving five sailors and a monkey on a desert island who divide up a pile of coconuts; the problem is to find the number of coconuts in the original pile (fractional coconuts not allowed). The problem is notorious for its confounding difficulty to unsophisticated puzzle solvers, though with the proper mathematical approach, the solution is trivial. The problem has become a staple in recreational mathematics collections. General description The problem can be expressed as: There is a pile of coconuts, owned by five men. One man divides the pile into five equal piles, giving the one left over coconut to a passing monkey, and takes away his own share. The second man then repeats the procedure, dividing the remaining pile into five and taking away his share, as do the third, fourth, and fifth, each of them finding one coconut left over when dividing the pile by five, and giving it to a monkey. Finally, the group divide the remaining coconuts into five equal piles: this time no coconuts are left over. How many coconuts were there in the original pile? The monkey and the coconuts is the best known representative of a class of puzzle problems requiring integer solutions structured as recursive division or fractionating of some discretely divisible quantity, with or without remainders, and a final division into some number of equal parts, possibly with a remainder. The problem is so well known that the entire class is often referred to broadly as "monkey and coconut type problems", though most are not closely related to the problem. Another example: "I have a whole number of pounds of cement, I know not how many, but after addition of a ninth and an eleventh, it was partitioned into 3 sacks, each with a whole number of pounds. How many pounds of cement did I have?" Problems ask for either the initial or terminal quantity. Stated or implied is the smallest positive number that could be a solution. There are two unknowns in such problems, the initial number and the terminal number, but only one equation which is an algebraic reduction of an expression for the relation between them. Common to the class is the nature of the resulting equation, which is a linear Diophantine equation in two unknowns. Most members of the class are determinate, but some are not (the monkey and the coconuts is one of the latter). Familiar algebraic methods are unavailing for solving such equations. History The origin of the class of such problems has been attributed to the Indian mathematician Mahāvīra in chapter VI, § 1311⁄2, 1321⁄2 of his Ganita-sara-sangraha (“Compendium of the Essence of Mathematics”), circa 850CE, which dealt with serial division of fruit and flowers with specified remainders.[1] That would make progenitor problems over 1000 years old before their resurgence in the modern era. Problems involving division which invoke the Chinese remainder theorem appeared in Chinese literature as early as the first century CE. Sun Tzu asked: Find a number which leaves the remainders 2, 3 and 2 when divided by 3, 5 and 7, respectively. Diophantus of Alexandria first studied problems requiring integer solutions in the 3rd century CE. The Euclidean algorithm for greatest common divisor which underlies the solution of such problems was discovered by the Greek geometer Euclid and published in his Elements in 300CE. Prof. David Singmaster, a historian of puzzles, traces a series of less plausibly related problems through the middle ages, with a few references as far back as the Babylonian empire circa 1700 BC. They involve the general theme of adding or subtracting fractions of a pile or specific numbers of discrete objects and asking how many there could have been in the beginning. The next reference to a similar problem is in Jacques Ozanam's Récréations mathématiques et physiques, 1725. In the realm of pure mathematics, Lagrange in 1770 expounded his continued fraction theorem and applied it to solution of Diophantine equations. The first description of the problem in close to its modern wording appears in Lewis Carroll's diaries in 1888: it involves a pile of nuts on a table serially divided by four brothers, each time with remainder of one given to a monkey, and the final division coming out even. The problem never appeared in any of Carroll's published works, though from other references it appears the problem was in circulation in 1888. An almost identical problem appeared in W.W. Rouse Ball's Elementary Algebra (1890). The problem was mentioned in works of period mathematicians, with solutions, mostly wrong, indicating that the problem was new and unfamiliar at the time. The problem became notorious when American novelist and short story writer Ben Ames Williams modified an older problem and included it in a story, "Coconuts", in the October 9, 1926, issue of the Saturday Evening Post.[2] Here is how the problem was stated by Williams[3] (condensed and paraphrased): Five men and a monkey were shipwrecked on an island. They spent the first day gathering coconuts for food. During the night, one man woke up, and decided to take his share early. So he divided the coconuts in five piles. He had one coconut left over, and he gave that to the monkey. Then he hid his pile and put the rest back together. By and by each of the five men woke up and did the same thing, one after the other: each one taking a fifth of the coconuts that were in the pile when he woke up, and having one left over for the monkey. In the morning they divided what coconuts were left, and they came out in five equal shares. Of course each one must have known there were coconuts missing; but each one was guilty as the others, so they didn't say anything. How many coconuts were there in the original pile? Williams had not included an answer in the story. The magazine was inundated by more than 2,000 letters pleading for an answer to the problem. The Post editor, Horace Lorimer, famously fired off a telegram to Williams saying: "FOR THE LOVE OF MIKE, HOW MANY COCONUTS? HELL POPPING AROUND HERE". Williams continued to get letters asking for a solution or proposing new ones for the next twenty years.[3] Martin Gardner featured the problem in his April 1958 Mathematical Games column in Scientific American. According to Gardner, Williams had modified an older problem to make it more confounding. In the older version there is a coconut for the monkey on the final division; in Williams's version the final division in the morning comes out even. But the available historical evidence does not indicate which versions Williams had access to.[4] Gardner once told his son Jim that it was his favorite problem.[5] He said that the Monkey and the Coconuts is "probably the most worked on and least often solved" Diophantine puzzle.[2] Since that time the Williams version of the problem has become a staple of recreational mathematics.[6] The original story containing the problem was reprinted in full in Clifton Fadiman's 1962 anthology The Mathematical Magpie,[7] a book that the Mathematical Association of America recommends for acquisition by undergraduate mathematics libraries.[8] Numerous variants which vary the number of sailors, monkeys, or coconuts have appeared in the literature.[9] Solutions A Diophantine problem Diophantine analysis is the study of equations with rational coefficients requiring integer solutions. In Diophantine problems, there are fewer equations than unknowns. The "extra" information required to solve the equations is the condition that the solutions be integers. Any solution must satisfy all equations. Some Diophantine equations have no solution, some have one or a finite number, and others have infinitely many solutions. The monkey and the coconuts reduces to a two-variable linear Diophantine equation of the form ax + by = c, or more generally, (a/d)x + (b/d)y = c/d where d is the greatest common divisor of a and b.[10] By Bézout's identity, the equation is solvable if and only if d divides c. If it does, the equation has infinitely many periodic solutions of the form x = x0 + t · b, y = y0 + t · a where (x0,y0) is a solution and t is a parameter than can be any integer. The problem is not intended to be solved by trial-and-error; there are deterministic methods for solving (x0,y0) in this case (see text). Numerous solutions starting as early as 1928 have been published both for the original problem and Williams modification.[11][12][13][14] Before entering upon a solution to the problem, a couple of things may be noted. If there were no remainders, given there are 6 divisions of 5, 56=15,625 coconuts must be in the pile; on the 6th and last division, each sailor receives 1024 coconuts. No smaller positive number will result in all 6 divisions coming out even. That means that in the problem as stated, any multiple of 15,625 may be added to the pile, and it will satisfy the problem conditions. That also means that the number of coconuts in the original pile is smaller than 15,625, else subtracting 15,625 will yield a smaller solution. But the number in the original pile is not trivially small, like 5 or 10 (that is why this is a hard problem) – it may be in the hundreds or thousands. Unlike trial and error in the case of guessing a polynomial root, trial and error for a Diophantine root will not result in any obvious convergence. There is no simple way of estimating what the solution will be. The original version Martin Gardner's 1958 Mathematical Games column begins its analysis by solving the original problem (with one coconut also remaining in the morning) because it is easier than Williams's version. Let F be the number of coconuts received by each sailor after the final division into 5 equal shares in the morning. Then the number of coconuts left before the morning division is $5F+1$; the number present when the fifth sailor awoke was ${\tfrac {5}{4}}(5F+1)+1={\tfrac {25}{4}}F+{\tfrac {9}{4}}$; the number present when the fourth sailor awoke was ${\tfrac {5}{4}}({\tfrac {25}{4}}F+{\tfrac {9}{4}})+1={\tfrac {125}{16}}F+{\tfrac {241}{16}}$; and so on. We find that the size N of the original pile satisfies the Diophantine equation[3] $1024N=15625F+11529$ Gardner points out that this equation is "much too difficult to solve by trial and error,"[3] but presents a solution he credits to J. H. C. Whitehead (via Paul Dirac):[3] The equation also has solutions in negative integers. Trying out a few small negative numbers it turns out $N=-4,F=-1$ is a solution.[15] We add 15625 to N and 1024 to F to get the smallest positive solution: $N=15621,F=1023$. Williams version Trial and error fails to solve Williams's version, so a more systematic approach is needed. Using a sieve The search space can be reduced by a series of increasingly larger factors by observing the structure of the problem so that a bit of trial and error finds the solution. The search space is much smaller if one starts with the number of coconuts received by each man in the morning division, because that number is much smaller than the number in the original pile. If F is the number of coconuts each sailor receives in the final division in the morning, the pile in the morning is 5F, which must also be divisible by 4, since the last sailor in the night combined 4 piles for the morning division. So the morning pile, call the number n, is a multiple of 20. The pile before the last sailor woke up must have been 5/4(n)+1. If only one sailor woke up in the night, then 5/4(20)+1 = 26 works for the minimum number of coconuts in the original pile. But if two sailors woke up, 26 is not divisible by 4, so the morning pile must be some multiple of 20 that yields a pile divisible by 4 before the last sailor wakes up. It so happens that 320=60 works for two sailors: applying the recursion formula for n twice yields 96 as the smallest number of coconuts in the original pile. 96 is divisible by 4 once more, so for 3 sailors awakening, the pile could have been 121 coconuts. But 121 is not divisible by 4, so for 4 sailors awakening, one needs to make another leap. At this point, the analogy becomes obtuse, because in order to accommodate 4 sailors awakening, the morning pile must be some multiple of 60: if one is persistent, it may be discovered that 1760=1020 does the trick and the minimum number in the original pile would be 2496. A last iteration on 2496 for 5 sailors awakening, i.e. 5/4(2496)+1 brings the original pile to 3121 coconuts. Blue coconuts Another device is to use extra objects to clarify the division process. Suppose that in the evening we add four blue coconuts to the pile. Then the first sailor to wake up will find the pile to be evenly divisible by five, instead of having one coconut left over. The sailor divides the pile into fifths such that each blue coconut is in a different fifth; then he takes the fifth with no blue coconut, gives one of his coconuts to the monkey, and puts the other four fifths (including all four blue coconuts) back together. Each sailor does the same. During the final division in the morning, the blue coconuts are left on the side, belonging to no one. Since the whole pile was evenly divided 5 times in the night, it must have contained 55 coconuts: 4 blue coconuts and 3121 ordinary coconuts. The device of using additional objects to aid in conceptualizing a division appeared as far back as 1912 in a solution due to Norman H. Anning.[3][16] A related device appears in the 17-animal inheritance puzzle: A man wills 17 horses to his three sons, specifying that the eldest son gets half, the next son one-third, and the youngest son, one-ninth of the animals. The sons are confounded, so they consult a wise horse trader. He says, "here, borrow my horse." The sons duly divide the horses, discovering that all the divisions come out even, with one horse left over, which they return to the trader. Base 5 numbering A simple solution appears when the divisions and subtractions are performed in base 5. Consider the subtraction, when the first sailor takes his share (and the monkey's). Let n0,n1,... represent the digits of N, the number of coconuts in the original pile, and s0,s1... represent the digits of the sailor's share S, both base 5. After the monkey's share, the least significant digit of N must now be 0; after the subtraction, the least significant digit of N' left by the first sailor must be 1, hence the following (the actual number of digits in N as well as S is unknown, but they are irrelevant just now): n5n4n3n2n1 0 (N5) s4s3s2s1s0 (S5) 1 (N'5) The digit subtracted from 0 base 5 to yield 1 is 4, so s0=4. But since S is (N-1)/5, and dividing by 55 is just shifting the number right one position, n1=s0=4. So now the subtraction looks like: n5n4n3n2 4 0 s4s3s2s1 4 1 Since the next sailor is going to do the same thing on N', the least significant digit of N' becomes 0 after tossing one to the monkey, and the LSD of S' must be 4 for the same reason; the next digit of N' must also be 4. So now it looks like: n5n4n3n2 4 0 s4s3s2s1 4 4 1 Borrowing 1 from n1 (which is now 4) leaves 3, so s1 must be 4, and therefore n2 as well. So now it looks like: n5n4n3 4 4 0 s4s3s2 4 4 4 1 But the same reasoning again applies to N' as applied to N, so the next digit of N' is 4, so s2 and n3 are also 4, etc. There are 5 divisions; the first four must leave an odd number base 5 in the pile for the next division, but the last division must leave an even number base 5 so the morning division will come out even (in 5s). So there are four 4s in N following a LSD of 1: N=444415=312110 A numerical approach A straightforward numeric analysis goes like this: If N is the initial number, each of 5 sailors transitions the original pile thus: N => 4(N–1)/5 or equivalently, N => 4(N+4)/5 – 4. Repeating this transition 5 times gives the number left in the morning: N => 4(N+4)/5 – 4 => 16(N+4)/25 – 4 => 64(N+4)/125 – 4 => 256(N+4)/625 – 4 => 1024(N+4)/3125 – 4 Since that number must be an integer and 1024 is relatively prime to 3125, N+4 must be a multiple of 3125. The smallest such multiple is 3125 · 1, so N = 3125 – 4 = 3121; the number left in the morning comes to 1020, which is evenly divisible by 5 as required. Modulo congruence A simple succinct solution can be obtained by directly utilizing the recursive structure of the problem: There were five divisions of the coconuts into fifths, each time with one left over (putting aside the final division in the morning). The pile remaining after each division must contain an integral number of coconuts. If there were only one such division, then it is readily apparent that 5 · 1+1=6 is a solution. In fact any multiple of five plus one is a solution, so a possible general formula is 5 · k – 4, since a multiple of 5 plus 1 is also a multiple of 5 minus 4. So 11, 16, etc also work for one division.[17] If two divisions are done, a multiple of 5 · 5=25 rather than 5 must be used, because 25 can be divided by 5 twice. So the number of coconuts that could be in the pile is k · 25 – 4. k=1 yielding 21 is the smallest positive number that can be successively divided by 5 twice with remainder 1. If there are 5 divisions, then multiples of 55=3125 are required; the smallest such number is 3125 – 4 = 3121. After 5 divisions, there are 1020 coconuts left over, a number divisible by 5 as required by the problem. In fact, after n divisions, it can be proven that the remaining pile is divisible by n, a property made convenient use of by the creator of the problem. A formal way of stating the above argument is: The original pile of coconuts will be divided by 5 a total of 5 times with a remainder of 1, not considering the last division in the morning. Let N = number of coconuts in the original pile. Each division must leave the number of nuts in the same congruence class (mod 5). So, $N\equiv 4/5\cdot (N-1)$ (mod 5) (the –1 is the nut tossed to the monkey) $5N\equiv 4N-4$ (mod 5) $N\equiv -4$ (mod 5) (–4 is the congruence class) So if we began in modulo class –4 nuts then we will remain in modulo class –4. Since ultimately we have to divide the pile 5 times or 5^5, the original pile was 5^5 – 4 = 3121 coconuts. The remainder of 1020 coconuts conveniently divides evenly by 5 in the morning. This solution essentially reverses how the problem was (probably) constructed. The Diophantine equation and forms of solution The equivalent Diophantine equation for this version is: $1024N=15625F+8404$ (1) where N is the original number of coconuts, and F is the number received by each sailor on the final division in the morning. This is only trivially different than the equation above for the predecessor problem, and solvability is guaranteed by the same reasoning. Reordering, $1024N-15625F=8404$ (2) This Diophantine equation has a solution which follows directly from the Euclidean algorithm; in fact, it has infinitely many periodic solutions positive and negative. If (x0, y0) is a solution of 1024x–15625y=1, then N0=x0 · 8404, F0=y0 · 8404 is a solution of (2), which means any solution must have the form ${\begin{cases}N=N_{0}+15625\cdot t\\F=F_{0}+1024\cdot t\end{cases}}$(3) where $t$ is an arbitrary parameter that can have any integral value. A reductionist approach One can take both sides of (1) above modulo 1024, so $1024N=15625F+8404\mod 1024$ Another way of thinking about it is that in order for $n$ to be an integer, the RHS of the equation must be an integral multiple of 1024; that property will be unaltered by factoring out as many multiples of 1024 as possible from the RHS. Reducing both sides by multiples of 1024, $0=(15625F-15\cdot 1024F)+(8404-8\cdot 1024)\mod 1024$ subtracting, $0=265F+212\mod 1024$ factoring, $0=53\cdot (5F+4)\mod 1024$ The RHS must still be a multiple of 1024; since 53 is relatively prime to 1024, 5F+4 must be a multiple of 1024. The smallest such multiple is 1 · 1024, so 5F+4=1024 and F=204. Substituting into (1) $1024N=15625\cdot 204+8404\Rightarrow N={\frac {3195904}{1024}}\Rightarrow N=3121$ Euclidean algorithm The Euclidean algorithm is quite tedious but a general methodology for solving rational equations ax+by=c requiring integral answers. From (2) above, it is evident that 1024 (210) and 15625 (56) are relatively prime and therefore their GCD is 1, but we need the reduction equations for back substitution to obtain N and F in terms of these two quantities: First, obtain successive remainders until GCD remains: 15625 = 15·1024 + 265 (a) 1024 = 3·265 + 229 (b) 265 = 1·229 + 36 (c) 229 = 6·36 + 13 (d) 36 = 2·13 + 10 (e) 13 = 1·10 + 3 (f) 10 = 3·3 + 1 (g) (remainder 1 is GCD of 15625 and 1024) 1 = 10 – 3(13–1·10) = 4·10 – 3·13 (reorder (g), substitute for 3 from (f) and combine) 1 = 4·(36 – 2·13) – 3·13 = 4·36 – 11·13 (substitute for 10 from (e) and combine) 1 = 4·36 – 11·(229 – 6·36) = –11·229 + 7036 (substitute for 13 from (d) and combine) 1 = –11·229 + 70·(265 – 1·229) = –81·229 + 70·265 (substitute for 36 from (c) and combine) 1 = –81·(1024 – 3·265) + 70·265 = –81·1024 + 313·265 (substitute for 229 from (b) and combine) 1 = –81·1024 + 313·(15625 – 15·1024) = 313·15625 – 4776·1024 (substitute for 265 from (a) and combine) So the pair (N0,F0) = (-4776·8404, -3138404); the smallest $t$ (see (3) in the previous subsection) that will make both N and F positive is 2569, so: $N=N_{0}+15625\cdot 2569=3121$ $F=F_{0}+1024\cdot 2569=204$ Continued fraction Alternately, one may use a continued fraction, whose construction is based on the Euclidean algorithm. The continued fraction for 1024⁄15625 (0.065536 exactly) is [;15,3,1,6,2,1,3];[18] its convergent terminated after the repetend is 313⁄4776, giving us x0=–4776 and y0=313. The least value of t for which both N and F are non-negative is 2569, so $N=-4776\cdot 8404+15625\cdot 2569=3121$. This is the smallest positive number that satisfies the conditions of the problem. A generalized solution When the number of sailors is a parameter, let it be $m$, rather than a computational value, careful algebraic reduction of the relation between the number of coconuts in the original pile and the number allotted to each sailor in the morning yields an analogous Diophantine relation whose coefficients are expressions in $m$. The first step is to obtain an algebraic expansion of the recurrence relation corresponding to each sailor's transformation of the pile, $n_{i}$ being the number left by the sailor: $n_{i}={\frac {m-1}{m}}(n_{i-1}-1)$ where $n_{i\rightarrow 0}\equiv N$, the number originally gathered, and $n_{i\rightarrow m}$ the number left in the morning. Expanding the recurrence by substituting $n_{i}$ for $n_{i-1}$ $m$ times yields: $n_{m}=\left({\frac {m-1}{m}}\right)^{m}\cdot n_{0}-\left[({\frac {m-1}{m}})^{m}+...+({\frac {m-1}{m}})^{2}+{\frac {m-1}{m}}\right]$ Factoring the latter term, $n_{m}=\left({\frac {m-1}{m}}\right)^{m}\cdot n_{0}-\left({\frac {m-1}{m}}\right)\cdot \left[({\frac {m-1}{m}})^{m-1}+...+{\frac {m-1}{m}}+1\right]$ The power series polynomial in brackets of the form $x^{m-1}+...+x+1$ sums to $(1-x^{m})/(1-x)$ so, $n_{m}=\left({\frac {m-1}{m}}\right)^{m}\cdot n_{0}-\left({\frac {m-1}{m}}\right)\cdot \left[\left(1-({\frac {m-1}{m}})^{m}\right){\bigg /}\left(1-({\frac {m-1}{m}})\right)\right]$ which simplifies to: $n_{m}=\left({\frac {m-1}{m}}\right)^{m}\cdot n_{0}-(m-1)\cdot {\frac {m^{m}-(m-1)^{m}}{m^{m}}}$ But $n_{m}$ is the number left in the morning which is a multiple of $m$ (i.e. $F$, the number allotted to each sailor in the morning): $m\cdot F=\left({\frac {m-1}{m}}\right)^{m}\cdot n_{0}-(m-1)\cdot {\frac {m^{m}-(m-1)^{m}}{m^{m}}}$ Solving for $n_{0}$(=$N$), $N=m^{m}\cdot {\frac {m-1+m\cdot F}{(m-1)^{m}}}-(m-1)$ The equation is a linear Diophantine equation in two variables, $N$ and $F$. $m$ is a parameter that can be any integer. The nature of the equation and the method of its solution do not depend on $m$. Number theoretic considerations now apply. For $N$ to be an integer, it is sufficient that ${\frac {m-1+m\cdot F}{(m-1)^{m}}}$ be an integer, so let it be $r$: $r={\frac {m-1+m\cdot F}{(m-1)^{m}}}$ The equation must be transformed into the form $ax+by=\pm 1$ whose solutions are formulaic. Hence: $(m-1)^{m}\cdot r-m\cdot s=-1$, where $s=1+F$ Because $m$ and $m-1$ are relatively prime, there exist integer solutions $(r,s)$ by Bézout's identity. This equation can be restated as: $(m-1)^{m}\cdot r\equiv -1\mod m$ But (m–1)m is a polynomial Z · m–1 if m is odd and Z · m+1 if m is even, where Z is a polynomial with monomial basis in m. Therefore r0=1 if m is odd and r0=–1 if m is even is a solution. Bézout's identity gives the periodic solution $r=r_{0}+k\cdot m$, so substituting for $r$ in the Diophantine equation and rearranging: $N=r_{0}\cdot m^{m}-(m-1)+k\cdot m^{m+1}$ where $r_{0}=1$ for $m$ odd and $r_{0}=-1$ for $m$ even and $k$ is any integer.[19] For a given $m$, the smallest positive $k$ will be chosen such that $N$ satisfies the constraints of the problem statement. In the William's version of the problem, $m$ is 5 sailors, so $r_{0}$ is 1, and $k$ may be taken to be zero to obtain the lowest positive answer, so N = 1 · 55 – 4 = 3121 for the number of coconuts in the original pile. (It may be noted that the next sequential solution of the equation for k=–1, is –12504, so trial and error around zero will not solve the Williams version of the problem, unlike the original version whose equation, fortuitously, had a small magnitude negative solution). Here is a table of the positive solutions $N$ for the first few $m$ ($k$ is any non-negative integer): $m$ $N$ 2 $11+k\cdot 8$[20] 3 $25+k\cdot 81$ 4 $765+k\cdot 1024$ 5 $3121+k\cdot 15,625$ 6 $233,275+k\cdot 279,936$ 7 $823,537+k\cdot 5,764,801$ 8 $117,440,505+k\cdot 134,217,728$ Other variants and general solutions Other variants, including the putative predecessor problem, have related general solutions for an arbitrary number of sailors. When the morning division also has a remainder of one, the solution is: $N=-(m-1)+k\cdot m^{m+1}$ For $m=5$ and $k=1$ this yields 15,621 as the smallest positive number of coconuts for the pre-William's version of the problem. In some earlier alternate forms of the problem, the divisions came out even, and nuts (or items) were allocated from the remaining pile after division. In these forms, the recursion relation is: $n_{i}={\frac {m-1}{m}}n_{i-1}-1$ The alternate form also had two endings, when the morning division comes out even, and when there is one nut left over for the monkey. When the morning division comes out even, the general solution reduces via a similar derivation to: $N=-m+k\cdot m^{m+1}$ For example, when $m=4$ and $k=1$, the original pile has 1020 coconuts, and after four successive even divisions in the night with a coconut allocated to the monkey after each division, there are 80 coconuts left over in the morning, so the final division comes out even with no coconut left over. When the morning division results in a nut left over, the general solution is: $N=r_{0}\cdot m^{m}-m+k\cdot m^{m+1}$ where $r_{0}=-1$ if $m$ is odd, and $r_{0}=1$ if $m$ is even. For example, when $m=3$, $r_{0}=-1$ and $k=1$, the original pile has 51 coconuts, and after three successive divisions in the night with a coconut allocated to the monkey after each division, there are 13 coconuts left over in the morning, so the final division has a coconut left over for the monkey. Other post-Williams variants which specify different remainders including positive ones (i.e. the monkey adds coconuts to the pile), have been treated in the literature. The solution is: $N=r_{0}\cdot m^{m}-c\cdot (m-1)+k\cdot m^{m+1}$ where $r_{0}=1$ for $m$ odd and $r_{0}=-1$ for $m$ even, $c$ is the remainder after each division (or number of monkeys) and $k$ is any integer ($c$ is negative if the monkeys add coconuts to the pile). Other variants in which the number of men or the remainders vary between divisions, are generally outside the class of problems associated with the monkey and the coconuts, though these similarly reduce to linear Diophantine equations in two variables. Their solutions yield to the same techniques and present no new difficulties. See also • Archimedes's cattle problem, a substantially more difficult Diophantine problem • Fermat's Last Theorem, possibly the most famous Diophantine equation of all • Cannonball problem References 1. Chronology of Recreational Mathematics by David Singmaster 2. Pleacher (2005) 3. Martin Gardner (2001). The Colossal Book of Mathematics. W.W. Norton & Company. pp. 3–9. ISBN 0-393-02023-1. 4. Antonick (2013) 5. Antonick (2013): "I then asked Jim if his father had a favorite puzzle, and he answered almost immediately: 'The monkeys [sic] and the coconuts. He was quite fond of that one.'" 6. Wolfram Mathworld 7. KIRKUS REVIEW of The Mathematical Magpie July 27, 1962 8. The Mathematical Magpie, by Clifton Fadiman, Mathematical Association of America, Springer, 1997 9. Pappas, T. "The Monkey and the Coconuts." The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, pp. 226-227 and 234, 1989. 10. d can be found if necessary via Euclid's algorithm 11. Underwood, R. S., and Robert E. Moritz. "3242." The American Mathematical Monthly 35, no. 1 (1928): 47-48. doi:10.2307/2298601. 12. Kirchner, Roger B. "The Generalized Coconut Problem," The American Mathematical Monthly 67, no. 6 (1960): 516-19. doi:10.2307/2309167. 13. S. Singh and D. Bhattacharya, “On Dividing Coconuts: A Linear Diophantine Problem,” The College Mathematics Journal, May 1997, pp. 203–4 14. G. Salvatore and T. Shima, "Of coconuts and integrity," Crux Mathematicorum, 4 (1978) 182–185 15. Bogomolny (1996) 16. Norman H. Anning (June 1912). "Problem Department (#288)". School Science and Mathematics. 12 (6). 17. A special case is when k=0, so the initial pile contains -4 coconuts. This works because after tossing one positive coconut to the monkey, there are -5 coconuts in the pile. After division, there remain -4 coconuts. No matter how many such divisions are done, the remaining pile will contain -4 coconuts. This is a mathematical anomaly called a "fixed point". Only a few problems have such a point, but when there is one, it makes the problem much easier to solve. All solutions to the problem are multiples of 5 added to or subtracted from the fixed point. 18. See here for an exposition of the method. 19. Gardner gives an equivalent but rather cryptic formulation by inexplicably choosing the non-canonical $r_{0}=-1+1\cdot m$ when $m$ is even, then refactoring the expression in a way that obscures the periodicity: for $m$ odd, $N=(1+m\cdot k)\cdot m^{m}-(m-1)$ for $m$ even, $N=(m-1+m\cdot k)\cdot m^{m}-(m-1)$ where $k$ is a parameter than can be any integer. 20. While N=3 satisfies the equation, 11 is the smallest positive number that gives each sailor a non-zero positive number of coconuts on each division, an implicit condition of the problem. Sources • Antonick, Gary (2013). Martin Gardner’s The Monkey and the Coconuts in Numberplay The New York Times:, October 7, 2013 • Pleacher, David (2005). Problem of the Week: The Monkey and the Coconuts May 16, 2005 • Pappas, Theoni (1993). The Joy of Mathematics: Discovering Mathematics All Around Wide World Publishing, January 23, 1993, ISBN 0933174659 • Wolfram Mathworld: Monkey and Coconut Problem • Kirchner, R. B. "The Generalized Coconut Problem." Amer. Math. Monthly 67, 516-519, 1960. • Fadiman, Clifton (1962). The Mathematical Magpie, Simon & Schuster • Bogomolny, Alexander (1996) Negative Coconuts at cut-the-knot External links • Monkeys and Coconuts – Numberphile video • Coconuts, a copy of the story as it appeared in the Saturday Evening Post	Wikipedia
The Nine Chapters on the Mathematical Art The Nine Chapters on the Mathematical Art is a Chinese mathematics book, composed by several generations of scholars from the 10th–2nd century BCE, its latest stage being from the 2nd century CE. This book is one of the earliest surviving mathematical texts from China, the first being the Suan shu shu (202 BCE – 186 BCE) and Zhoubi Suanjing (compiled throughout the Han until the late 2nd century CE). It lays out an approach to mathematics that centres on finding the most general methods of solving problems, which may be contrasted with the approach common to ancient Greek mathematicians, who tended to deduce propositions from an initial set of axioms. The Nine Chapters on the Mathematical Art A page of The Nine Chapters on the Mathematical Art (1820 edition) Traditional Chinese九章算術 Simplified Chinese九章算术 Literal meaningnine chapters on arithmetic Transcriptions Standard Mandarin Hanyu PinyinJiǔ Zhāng Suànshù Wade–GilesChiu3 Chang1 Suan4-shu4 Middle Chinese Middle Chinese/kɨuX t͡ɕɨɐŋ suɑnX ʑiuɪt̚/ Old Chinese Zhengzhang/kuʔ kjaŋ sloːnʔ ɦljud/ Entries in the book usually take the form of a statement of a problem, followed by the statement of the solution and an explanation of the procedure that led to the solution. These were commented on by Liu Hui in the 3rd century. History Original book The full title of The Nine Chapters on the Mathematical Art appears on two bronze standard measures which are dated to 179 CE, but there is speculation that the same book existed beforehand under different titles.[1] Most scholars believe that Chinese mathematics and the mathematics of the ancient Mediterranean world had developed more or less independently up to the time when The Nine Chapters reached its final form. The method of chapter 7 was not found in Europe until the 13th century, and the method of chapter 8 uses Gaussian elimination before Carl Friedrich Gauss (1777–1855).[2] There is also the mathematical proof given in the treatise for the Pythagorean theorem.[3] The influence of The Nine Chapters greatly assisted the development of ancient mathematics in the regions of Korea and Japan. Its influence on mathematical thought in China persisted until the Qing dynasty era. Liu Hui wrote a very detailed commentary on this book in 263. He analyses the procedures of The Nine Chapters step by step, in a manner which is clearly designed to give the reader confidence that they are reliable, although he is not concerned to provide formal proofs in the Euclidean manner. Liu's commentary is of great mathematical interest in its own right. Liu credits the earlier mathematicians Zhang Cang (fl. 165 BCE – d. 142 BCE) and Geng Shouchang (fl. 75 BCE – 49 BCE) (see armillary sphere) with the initial arrangement and commentary on the book, yet Han dynasty records do not indicate the names of any authors of commentary, as they are not mentioned until the 3rd century.[4] The Nine Chapters is an anonymous work, and its origins are not clear. Until recent years, there was no substantial evidence of related mathematical writing that might have preceded it, with the exception of mathematical work by those such as Jing Fang (78–37 BCE), Liu Xin (d. 23), and Zhang Heng (78–139) and the geometry clauses of the Mozi of the 4th century BCE. This is no longer the case. The Suàn shù shū (算數書) or Writings on Reckonings is an ancient Chinese text on mathematics approximately seven thousand characters in length, written on 190 bamboo strips. It was discovered together with other writings in 1983 when archaeologists opened a tomb in Hubei province. It is among the corpus of texts known as the Zhangjiashan Han bamboo texts. From documentary evidence this tomb is known to have been closed in 186 BCE, early in the Western Han dynasty. While its relationship to The Nine Chapters is still under discussion by scholars, some of its contents are clearly paralleled there. The text of the Suàn shù shū is however much less systematic than The Nine Chapters; and appears to consist of a number of more or less independent short sections of text drawn from a number of sources. The Zhoubi Suanjing, a mathematics and astronomy text, was also compiled during the Han, and was even mentioned as a school of mathematics in and around 180 CE by Cai Yong. Western translations The title of the book has been translated in a wide variety of ways. In 1852, Alexander Wylie referred to it as Arithmetical Rules of the Nine Sections. With only a slight variation, the Japanese historian of mathematics Yoshio Mikami shortened the title to Arithmetic in Nine Sections.[5] David Eugene Smith, in his History of Mathematics (Smith 1923), followed the convention used by Yoshio Mikami. Several years later, George Sarton took note of the book, but only with limited attention and only mentioning the usage of red and black rods for positive and negative numbers. In 1959, Joseph Needham and Wang Ling (historian) translated Jiu Zhang Suan shu as The Nine Chapters on the Mathematical Art for the first time. Later in 1994, Lam Lay Yong used this title in her overview of the book, as did other mathematicians including John N. Crossley and Anthony W.-C Lun in their translation of Li Yan and Du Shiran's Chinese Mathematics: A Concise History (Li and Du 1987).[5] Afterwards, the name The Nine Chapters on the Mathematical Art stuck and became the standard English title for the book. Table of contents Contents of The Nine Chapters are as follows: 1. 方田 Fangtian – Bounding fields. Areas of fields of various shapes, such as rectangles, triangles, trapezoids, and circles; manipulation of vulgar fractions. Liu Hui's commentary includes a method for calculation of π and the approximate value of 3.14159.[6] 2. 粟米 Sumi – Millet and rice. Exchange of commodities at different rates; unit pricing; the Rule of Three for solving proportions, using fractions. 3. 衰分 Cuifen – Proportional distribution. Distribution of commodities and money at proportional rates; deriving arithmetic and geometric sums. 4. 少廣 Shaoguang – Reducing dimensions. Finding the diameter or side of a shape given its volume or area. Division by mixed numbers; extraction of square and cube roots; diameter of sphere, perimeter and diameter of circle. 5. 商功 Shanggong – Figuring for construction. Volumes of solids of various shapes. 6. 均輸 Junshu – Equitable taxation. More advanced word problems on proportion, involving work, distances, and rates. 7. 盈不足 Yingbuzu – Excess and deficit. Linear problems (in two unknowns) solved using the principle known later in the West as the rule of false position. 8. 方程 Fangcheng – The two-sided reference (i.e. Equations). Problems of agricultural yields and the sale of animals that lead to systems of linear equations, solved by a principle indistinguishable from the modern form of Gaussian elimination.[7] 9. 勾股 Gougu – Base and altitude. Problems involving the principle known in the West as the Pythagorean theorem. Major contributions Real number system The Nine Chapters on the Mathematical Art does not discuss natural numbers, that is, positive integers and their operations, but they are widely used and written on the basis of natural numbers. Although it is not a book on fractions, the meaning, nature, and four operations of fractions are fully discussed. For example: combined division (addition), subtraction (subtraction), multiplication (multiplication), warp division (division), division (comparison size), reduction (simplified fraction), and bisector (average).[8] The concept of negative numbers also appears in "Nine Chapters of Arithmetic". In order to cooperate with the algorithm of equations, the rules of addition and subtraction of positive and negative numbers are given. The subtraction is "divide by the same name, benefit by different names. The addition is "divide by different names, benefit from each other by the same name. Among them, "division" is subtraction, "benefit" is addition, and "no entry" means that there is no counter-party, but multiplication and division are not recorded.[8] The Nine Chapters on the Mathematical Art gives a certain discussion on natural numbers, fractions, positive and negative numbers, and some special irrationality. It basically has the prototype of real number system. Gou Gu (Pythagorean) Theorem The geometric figures included in The Nine Chapters on the Mathematical Art are mostly straight and circular figures because of its focus on the applications onto the agricultural fields. In addition, due to the needs of civil architecture, The Nine Chapters on the Mathematical Art also discusses volumetric algorithms of linear and circular 3 dimensional solids. The arrangement of these volumetric algorithms ranges from simple to complex, forming a unique mathematical system.[8] Regarding the direct application of the Gou Gu Theorem, which is precisely the Chinese version of the Pythagorean Theorem, the book divides it into four main categories: Gou Gu mutual seeking, Gou Gu integer, Gou Gu dual capacity, Gou Gu similar. Gou Gu mutual seeking discusses the algorithm of finding the length of a side of the right triangle while knowing the other two. Gou Gu integer is precisely the finding of some significant integer Pythagorean numbers, including famously the triple 3,4,5. Gou Gu dual capacity discusses algorithms for calculating the areas of the inscribed rectangles and other polygons in the circle, which also serves an algorithm to calculate the value of pi. Lastly, Gou Gu similars provide algorithms of calculating heights and lengths of buildings on the mathematical basis of similar right triangles. Completing of squares and solutions of system of equations The methods of completing the squares and cubes as well as solving simultaneous linear equations listed in The Nine Chapters on the Mathematical Art can be regarded one of the major content of ancient Chinese mathematics. The discussion of these algorithms in The Nine Chapters on the Mathematical Art are very detailed. Through these discussions, one can understand the achievements of the development of ancient Chinese mathematics.[8] Completing the squaring and cubes can not only solve systems of two linear equations with two unknowns, but also general quadratic and cubic equations. It is the basis for solving higher-order equations in ancient China, and it also plays an important role in the development of mathematics.[8] The "equations" discussed in the Fang Cheng chapter are equivalent to today's simultaneous linear equations. The solution method called "Fang Cheng Shi" is best known today as Gaussian elimination. Among the eighteen problems listed in the Fang Cheng chapter, some are equivalent to simultaneous linear equations with two unknowns, some are equivalent to simultaneous linear equations with 3 unknowns, and the most complex example analyzes the solution to a system of linear equations with up to 5 unknowns.[8] Significance The word jiu, or "9", means more than just a digit in ancient Chinese. In fact, since it is the largest digit, it often refers to something of a grand scale or a supreme authority. Further, the word zhang, or "chapter", also has more connotations than simply being the "chapter". It may refer to a section, several parts of an article, or an entire treatise.[9] In this light, many scholars of the history of Chinese mathematics compare the significance of The Nine Chapters on the Mathematical Art on the development of Eastern mathematical traditions to that of Euclid's Elements on the Western mathematical traditions.[10][11] However, the influence of The Nine Chapters on the Mathematical Art stops short at the advancement of modern mathematics due to its focus on practical problems and inductive proof methods as opposed to the deductive, axiomatic tradition that Euclid's Elements establishes. However, it is dismissive to say that The Nine Chapters on the Mathematical Art has no impact at all on modern mathematics. The style and structure of The Nine Chapters on the Mathematical Art can be best concluded as "problem, formula, and computation".[12] This process of solving applied mathematical problems is now pretty much the standard approach in the field of applied mathematics. Notable translations • Abridged English translation: Yoshio Mikami: "Arithmetic in Nine Sections", in The Development of Mathematics in China and Japan, 1913. • Highly Abridged English translation: Florian Cajori: "Arithmetic in Nine Sections", in A History of Mathematics, Second Edition, 1919 (possibly copied or paraphrased from Mikami). • Abridged English translation: Lam Lay Yong: Jiu Zhang Suanshu: An Overview, Archive for History of Exact Sciences, Springer Verlag, 1994. • A full translation and study of the Nine Chapters and Liu Hui's commentary is available in Kangshen Shen, The Nine Chapters on the Mathematical Art, Oxford University Press, 1999. ISBN 0-19-853936-3 • A French translation with detailed scholarly addenda and a critical edition of the Chinese text of both the book and its commentary by Karine Chemla and Shuchun Guo is Les neuf chapitres: le classique mathématique de la Chine ancienne et ses commentaires. Paris: Dunod, 2004. ISBN 978-2-10-049589-4. • German translation: Kurt Vogel, Neun Bücher Arithmetischer Technik, Friedrich Vieweg und Sohn Braunsweig, 1968 • Russian translation: E. I Beriozkina, Математика в девяти книгах (Mathematika V Devyati Knigah), Moscow: Nauka, 1980. See also • Haidao Suanjing • History of mathematics • History of geometry Notes 1. Needham, Volume 3, 24–25. 2. Straffin, 164. 3. Needham, Volume 3, 22. 4. Needham, Volume 3, 24. 5. Dauben, Joseph W. (2013). "九章箅术 "Jiu zhang suan shu" (Nine Chapters on the Art of Mathematics) – An Appraisal of the Text, its Editions, and Translations". Sudhoffs Archiv. 97 (2): 199–235. doi:10.25162/sudhoff-2013-0017. ISSN 0039-4564. JSTOR 43694474. S2CID 1159700. 6. O'Connor. 7. http://www.dam.brown.edu/people/mumford/beyond/papers/2010b--Negatives-PrfShts.pdf 8. 中國文明史第三卷秦漢時代中冊. 地球社编辑部. 1992. pp. 515–531. 9. Dauben, Joseph W. (1992), "The "Pythagorean theorem" and Chinese Mathematics Liu Hui's Commentary on the 勾股 (Gou-Gu) Theorem in Chapter Nine of the Jiu Zhang Suan Shu", Amphora, Birkhäuser Basel, pp. 133–155, doi:10.1007/978-3-0348-8599-7_7, ISBN 978-3-0348-9696-2 10. Siu, Man-Keung (December 1993). "Proof and pedagogy in ancient China: Examples from Liu Hui's commentary on JIU ZHANG SUAN SHU". Educational Studies in Mathematics. 24 (4): 345–357. doi:10.1007/bf01273370. ISSN 0013-1954. S2CID 120420378. 11. Dauben, Joseph W. (September 1998). "Ancient Chinese mathematics: the (Jiu Zhang Suan Shu) vs Euclid's Elements. Aspects of proof and the linguistic limits of knowledge". International Journal of Engineering Science. 36 (12–14): 1339–1359. doi:10.1016/s0020-7225(98)00036-6. ISSN 0020-7225. 12. 吴, 文俊 (1982). 九章算术与刘辉. 北京: 北京师范大学出版社. p. 118. References • Needham, Joseph (1986). Science and Civilization in China: Volume 3, Mathematics and the Sciences of the Heavens and the Earth. Taipei: Caves Books, Ltd. • Straffin, Philip D. "Liu Hui and the First Golden Age of Chinese Mathematics", Mathematics Magazine (Volume 71, Number 3, 1998): 163–181. • O'Connor, John J.; Robertson, Edmund F., "Liu Hui", MacTutor History of Mathematics Archive, University of St Andrews External links Chinese Wikisource has original text related to this article: 九章算术 • Full text of the book (Chinese) Han dynasty topics History • Chu–Han Contention (Feast at Swan Goose Gate) • Lü Clan disturbance • Rebellion of the Seven States • Han–Xiongnu War • War of the Heavenly Horses • Han conquest of Gojoseon • Four Commanderies • Southward expansion (Han–Minyue War • Han conquest of Nanyue • Han conquest of Dian • First Chinese domination of Vietnam • Trung sisters' rebellion • Second Chinese domination of Vietnam) • Xin dynasty • Red Eyebrows and Lulin • Chengjia • Disasters of the Partisan Prohibitions • Way of the Five Pecks of Rice • Yellow Turban Rebellion • End of the Han dynasty • Battle of Red Cliffs Society and culture • Ban Gu • Sima Qian • Records of the Grand Historian • Book of Han • Book of the Later Han • Records of the Three Kingdoms • Flying Horse of Gansu • Huainanzi • Eight Immortals of Huainan • Mawangdui Silk Texts • Luxuriant Dew of the Spring and Autumn Annals • Yiwu Zhi • Old Texts • Han poetry • Fu • Eastern Han Chinese Government and military • Ban Chao • Ma Yuan • Emperor • list • Family tree • Three Lords and Nine Ministers • Nine Ministers • Three Ducal Ministers • Kings • Provinces and commanderies • Protectorate of the Western Regions (Chief Official) • Translation of government titles Economy • Coinage • Ancient Chinese coinage • Silk Road • Sino-Roman relations Science and technology • Cai Lun • Ding Huan • Du Shi • Hua Tuo • Wang Chong • Zhang Heng • Zhang Zhongjing Texts • Balanced Discourse • Book of Origins • Book on Numbers and Computation • Fangyan • Essential Prescriptions from the Golden Cabinet • The Nine Chapters on the Mathematical Art • Huangdi Neijing • Shuowen Jiezi • Treatise on Cold Injury and Miscellaneous Disorders • Zhoubi Suanjing Authority control International • FAST • VIAF National • France • BnF data • Germany • Israel • United States Chinese classics and Confucian texts Four Books • Great Learning • Doctrine of the Mean • Analects • Mencius Five Classics • Classic of Poetry • Book of Documents • Book of Rites • I Ching • Spring and Autumn Annals Thirteen Classics • Classic of Poetry • Book of Documents • Rites of Zhou • Etiquette and Ceremonial • Book of Rites • I Ching • Commentary of Zuo • Commentary of Gongyang • Commentary of Guliang • Analects • Erya • Classic of Filial Piety • Mencius San Bai Qian • Three Character Classic • Hundred Family Surnames • Thousand Character Classic Seven Military Classics • The Art of War • The Methods of the Sima • Six Secret Teachings • Wei Liaozi • Wu Zi • Three Strategies of Huang Shigong • Questions and Replies between Tang Taizong and Li Weigong Mathematics • The Nine Chapters on the Mathematical Art Others • Bai Hu Tong • Biographies of Exemplary Women • Classic of Music • Four Books for Women • Lessons for Women • School Sayings of Confucius • The Twenty-four Filial Exemplars • Xunzi	Wikipedia
Numberphile Numberphile is an educational YouTube channel featuring videos that explore topics from a variety of fields of mathematics.[2][3] In the early days of the channel, each video focused on a specific number, but the channel has since expanded its scope, featuring videos on more advanced mathematical concepts such as Fermat's Last Theorem, the Riemann hypothesis[4] and Kruskal's tree theorem.[5] The videos are produced by Brady Haran, a former BBC video journalist and creator of Periodic Videos, Sixty Symbols, and several other YouTube channels.[6] Videos on the channel feature several university professors, maths communicators and famous mathematicians.[7][8] Numberphile The logo of the channel YouTube information Channels • Numberphile • Numberphile2 Created byBrady Haran Presented by • Brady Haran • James Grime • Matt Parker • Tom Scott • See list for more Years active15 September 2011 (2011-09-15) – present GenreEducational entertainment Subscribers • 4.34 million (Numberphile) • 247 thousand (Numberphile2) [1] Total views • 643 million (Numberphile) • 18.7 million (Numberphile2) [1] Associated acts • Periodic Videos • Sixty Symbols • CGP Grey Websitewww.numberphile.com Creator Awards 100,000 subscribers • 2012 (Numberphile) • 2016 (Numberphile2) 1,000,000 subscribers2014 (Numberphile) Last updated: 19 August 2023 In 2018, Haran released a spin-off audio podcast titled The Numberphile Podcast.[9] YouTube channel The Numberphile YouTube channel was started on 15 September 2011. Most videos consist of Haran interviewing an expert on a number, mathematical theorem or other mathematical concept.[10] The expert usually draws out their explanation on a large piece of brown paper and attempts to make the concepts understandable to the average, non-mathematician viewer.[11] It is supported by the Mathematical Sciences Research Institute (MSRI) and Math for America.[12][13] Haran also runs the "Numberphile2" channel, which includes extra footage and further detail than the main channel.[14] Reception Numberphile consistently rates among the top YouTube channels in math and education.[15][16][17][18][19] The channel was nominated for a Shorty Award in Education in 2016.[20] The New York Times said that, "at Numberphile, mathematicians discourse, enthusiastically and winningly, on numbers", and The Independent described the channel as "insanely popular".[21][22] The Sunday Times said, "The mathematical stars of social media, such as James Grime and Matt Parker, entertain legions of fans with glorious videos demonstrating how powerful and playful maths can be."[23] New Scientist listed Numberphile as one of the top ten science channels on YouTube in 2019.[24] Contributors The Numberphile channel has hosted a wide array of mathematicians, computer scientists, scientists and science writers, including:[25] • Federico Ardila[26] • Johnny Ball[27] • Alex Bellos[25] • Elwyn Berlekamp[25] • Andrew Booker[28] • Steven Bradlow[29] • Timothy Browning[30] • Brian Butterworth[25] • John Conway[25] • Ed Copeland[25] • Tom Crawford[31] • Zsuzsanna Dancso[32] • Persi Diaconis[25] • Marcus Du Sautoy[33] • Rob Eastaway[25] • Laurence Eaves[25] • David Eisenbud[25] • Edward Frenkel[25] • Hannah Fry[25] • Lisa Goldberg[25] • James Grime [34] • Ron Graham[35] • Edmund Harriss[36] • Gordon Hamilton[37] • Tim Harford[38] • Don Knuth[25] • Holly Krieger[39] • James Maynard[40] • Barry Mazur[25] • Steve Mould[25] • Colm Mulcahy[41] • Tony Padilla[42] • Simon Pampena[43] • Matt Parker[25] • Roger Penrose[44] • Carl Pomerance[45] • Ken Ribet[25] • Tom Scott[25] • Henry Segerman[46] • Carlo H. Séquin[47] • Jim Simons[48] • Simon Singh[25] • Neil Sloane[49] • Ben Sparks[50] • Katie Steckles[51] • Zvezdelina Stankova[25] • Clifford Stoll[25] • Terence Tao[52] • Tadashi Tokieda[53] • Mariel Vázquez[54] • Cédric Villani[55] • Zandra Vinegar[56] The Numberphile Podcast The Numberphile Podcast Presentation Hosted byBrady Haran GenreInterview LanguageEnglish Length25–75 minutes Production No. of episodes46 (As of 18 January 2023)[57] Publication Original release4 November 2018 (2018-11-04) Related Related shows • Hello Internet • The Unmade Podcast Websitewww.numberphile.com/podcast Haran started a podcast titled The Numberphile Podcast in 2018 as a sister project. The podcast focuses more heavily on the lives and personalities of the subjects of the videos.[58] No.TitleRun TimeOriginal release date 1"The Hope Diamond – with 3Blue1Brown"1:03:204 November 2018 (2018-11-04) 2"Fermat’s Last Theorem – with Ken Ribet"48:2221 November 2018 (2018-11-21) 3"Delicious Problems – with Hannah Fry"52:0016 December 2018 (2018-12-16) 4"The Klein Bottle Guy – with Cliff Stoll"59:088 January 2019 (2019-01-08) 5"The Math Storyteller – with Simon Singh"1:11:3611 February 2019 (2019-02-11) 6"Parker Square – with Matt Parker"52:0424 February 2019 (2019-02-24) 7"A Proof in the Drawer – with David Eisenbud"1:15:207 April 2019 (2019-04-07) 8"The Offensive Lineman – with John Urschel"36:4314 May 2019 (2019-05-14) 9"The Singing Banana – with James Grime"1:13:2120 May 2019 (2019-05-20) 10"The C-Word – talking Calculus with Steven Strogatz"51:1717 June 2019 (2019-06-17) 11"The Number Collector – with Neil Sloane"55:3614 August 2019 (2019-08-14) 12"Fame and Admiration – with Timothy Gowers"54:2522 October 2019 (2019-10-22) 13"The Badly Behaved Prime – with James Maynard"41:3810 November 2019 (2019-11-10) 14"Coffin Problems – with Edward Frenkel"1:21:393 December 2019 (2019-12-03) 15"Champaign Mathematician – with Holly Krieger"40:0013 December 2019 (2019-12-13) 16"Gondor Calls for Aid – with Kit Yates"28:1831 March 2020 (2020-03-31) 17"Crystal Balls and Coronavirus – with Hannah Fry"44:5510 April 2020 (2020-04-10) 18"The Legendary John Conway (1937–2020)"38:0113 April 2020 (2020-04-13) 19"The Accidental Streamer – with 3Blue1Brown"33:0519 April 2020 (2020-04-19) 20"The Parker Quiz – with Matt Parker"55:3421 May 2020 (2020-05-21) 21"The Happy Twin – with Ben Sparks"1:02:2127 May 2020 (2020-05-27) 22"The Numeracy Ambassador – with Simon Pampena"1:00:151 July 2020 (2020-07-01) 23"The Mathematical Showman – Ron Graham (1935–2020)"39:0213 July 2020 (2020-07-13) 24"The Third Cornet – with Katie Steckles"59:4924 July 2020 (2020-07-24) 25"Why Did the Mathematician Cross the Road? – with Roger Penrose"1:05:168 August 2020 (2020-08-08) 26"The Importance of Numbers – with Tim Harford"47:2912 September 2020 (2020-09-12) 27"Nursery Rhymes and Numbers – with Alan Stewart"54:065 October 2020 (2020-10-05) 28"Quiz Shows and Math Anxiety – with Bobby Seagull"1:24:2623 October 2020 (2020-10-23) 29"Club Automatic – with Alex Bellos"54:1725 November 2020 (2020-11-25) 30"Why Study Mathematics – with Vicky Neale"45:118 December 2020 (2020-12-08) 31"Statistics and Saving Lives – with Jennifer Rogers"55:5011 December 2020 (2020-12-11) 32"Rockstar Epidemiologists – with Adam Kucharski"45:102 February 2021 (2021-02-02) 33"The High Jumping Cosmologist – with Katie Mack"54:5325 February 2021 (2021-02-25) 34"Beauty in the Messiness – with Philip Moriarty"39:053 April 2021 (2021-04-03) 35"The Naked Mathematician – with Tom Crawford"58:1231 May 2021 (2021-05-31) 36"A Chance at Immortality – with Marcus du Sautoy"51:2926 July 2021 (2021-07-26) 37"Making Sense of Infinity – with Asaf Karagila"53:2728 August 2021 (2021-08-28) 38"Google's 'DeepMind' does Mathematics"37:022 December 2021 (2021-12-02) 39"The Little Star – with Zvezdelina Stankova"58:2014 January 2022 (2022-01-14) 40"An Infinite Debt – with Christopher Havens (Prisoner #349034)"49:5013 February 2022 (2022-02-13) 41"The First and Last Digits of Pi"42:2614 March 2022 (2022-03-14) 42"A Passion for Big Numbers (and Liverpool FC) – with Tony Padilla"50:4118 April 2022 (2022-04-18) 43"The Orchid Room and Cancer – with Hannah Fry"36:5129 May 2022 (2022-05-29) 44"An Educated Adult – with Tadashi Tokieda"1:13:4511 July 2022 (2022-07-11) 45"Finding a Path – with Tatiana Toro"44:0013 December 2022 (2022-12-13) 46"A Chain of Chance – with Michael Merrifield"1:07:0018 January 2023 (2023-01-18) References 1. "About Numberphile". YouTube. 2. Schultz, Colin. "The Great Debate Over Whether 1+2+3+4..+ ∞ = −1/12". Smithsonian Magazine. 3. Ryan, Jackson. "Earth is getting a black box to record events that lead to the downfall of civilization". CNET. Retrieved 8 December 2021. 4. Lamb, Evelyn. "Does 1+2+3... Really Equal −1/12?". Scientific American Blog Network. Retrieved 8 December 2021. 5. Bennett, Jay (20 October 2017). "The Enormity of the Number TREE(3) Is Beyond Comprehension". Popular Mechanics. Retrieved 8 December 2021. 6. As VidCon gets underway, science presenters rule the Web By Robert Lloyd Television Critic, Los Angeles Times, April 1, 2016 7. Overbye, Dennis (3 February 2014). "In the End, It All Adds Up to – 1/12". The New York Times. Retrieved 8 December 2021. 8. The World of YouTube Math Communication by Scott Hershberger, Notices Of The American Mathematical Society, November 2022 9. "The Numberphile Podcast". Brady Haran. 11 November 2018. Retrieved 25 December 2018. 10. 145 and the melancoil: What do narcissistic numbers and happy numbers share with the wild events that transpire when mathematicians visit the pub? GrrlScientist, The Guardian,Mathematics, 5 Mar 2012 11. ECONOMIC POLICY: The eerie math that could predict terrorist By Ana Swanson, The Washington Post, March 1, 2016 12. Numberphile sponsors Mathematical Sciences Research Institute 13. Numberphile Nominated in Education 8th Annual Shorty Awards 14. Haran, Brady. "Numberphile2". YouTube. Retrieved 17 October 2019. 15. Best YouTube Math Channels By Med Kharbach, PhD, YouTube Channels for Teachers, December 18, 2022 16. Top 10 Best Math YouTube Channels by Jacqueline Michelle, History-Computer.com, December 21, 2022 17. Maths Youtube Channels: The best Maths youtube channels from thousands of youtubers on the web ranked by subscribers, views, and video counts feedspot.com, May 10, 2023 18. 43 Best Math YouTube Channels Ranked by Popularity #2 Numberphile, youneedchannels.com, Nov 20, 2022 19. 17 Best Math Youtube Channels to Study Mathematics Education, Mathematics, January 24, 2023 20. Joy, Alexis (19 January 2016). "The 8th Annual Shorty Award". {{cite journal}}: Cite journal requires \|journal= (help) 21. Hale, Mike (24 April 2012). "Genres Stretch, for Better and Worse, as YouTube Takes On TV". The New York Times. 22. Usborne, Simon (31 October 2014). "Stand-up and Be Counted". Newspapers.com. Retrieved 10 May 2023. 23. Teach children to fall in love with maths and they can count on it for life By Hannah Fry, The Sunday Times, Dec 08, 2019 24. Stokel-Walker, Chris (20 February 2019). "YouTube science videos: The channels you should subscribe to". New Scientist. Retrieved 10 May 2023. 25. Guest speakers on Numberphile MSRI 26. Haran, Brady (6 July 2018). "All in Federico Ardila". Numberphile. 27. "Johnny Ball – Russian Multiplication – Numberphile". YouTube. Retrieved 30 November 2020. 28. "42 is the new 33 – Numberphile". YouTube. 12 March 2019. 29. Qureshi, Zainab (4 May 2020). ""Numberphile" sponsors mathematics professor". The Daily Illini. Retrieved 8 December 2021. 30. "Hasse Principle – Numberphile". YouTube. 1 June 2016. 31. Haran, Brady. "Tom Crawford on Numberphile". YouTube. 32. Haran, Brady. "Zsuzsanna Dancso on Numberphile". YouTube. 33. Guardian Staff (4 December 2021). "From a Sex and the City sequel to Halo Infinite: a complete guide to this week's entertainment". the Guardian. 34. Haran, Brady. "James Grime on Numberphile". YouTube. 35. Haran, Brady. "Ron Graham on Numberphile". YouTube. 36. Haran, Brady (3 June 2019). "All in Edmund Harriss". Numberphile. 37. Haran, Brady. "Gordon Hamilton on Numberphile". YouTube. 38. "Statistics, Storks, and Babies – Numberphile". YouTube. 25 August 2020. 39. Lamb, Evelyn. "Holly Krieger's Favorite Theorem". Scientific American Blog Network. Retrieved 8 December 2021. 40. "Primes without a 7 – Numberphile". YouTube. 20 November 2019. 41. "Little Fibs – Numberphile". YouTube. 2 June 2016. 42. Haran, Brady (19 December 2022). "All in Tony Padilla". Numberphile. 43. Haran, Brady. "Simon Pampena on Numberphile". YouTube. 44. "Why Did The Mathematician Cross The Road? – with Roger Penrose". YouTube. 8 August 2020. 45. "210 is VERY Goldbachy – Numberphile". YouTube. 28 May 2017. 46. Haran, Brady. "Henry Segerman on Numberphile". YouTube. 47. "Mobius Bridges and Buildings – Numberphile". YouTube. 9 April 2014. 48. "Billionaire Mathematician – Numberphile". YouTube. 13 May 2015. 49. Haran, Brady. "Neil Sloane on Numberphile". YouTube. 50. Haran, Brady. "Ben Sparks on Numberphile". YouTube. 51. Haran, Brady. "Katie Steckles on Numberphile". YouTube. 52. "The World's Best Mathematician (*) – Numberphile". YouTube. 14 March 2017. 53. "Stable Rollers – Numberphile". YouTube. 6 March 2017. 54. "The Shape of DNA – Numberphile". YouTube. 26 October 2015. 55. Haran, Brady. "Cedric Villani on Numberphile". YouTube. Retrieved 5 November 2019. 56. "Card Flipping Proof – Numberphile". YouTube. 3 February 2019. 57. "Podcast". Numberphile. Retrieved 29 May 2022. 58. Haran, Brady (11 November 2018). "The Numberphile Podcast". Brady Haran. Retrieved 10 September 2019. External links • Official website	Wikipedia
Oberwolfach problem The Oberwolfach problem is an unsolved problem in mathematics that may be formulated either as a problem of scheduling seating assignments for diners, or more abstractly as a problem in graph theory, on the edge cycle covers of complete graphs. It is named after the Oberwolfach Research Institute for Mathematics, where the problem was posed in 1967 by Gerhard Ringel.[1] It is known to be true for all sufficiently-large complete graphs. Unsolved problem in mathematics: For which 2-regular $n$-vertex graphs $G$ can the complete graph $K_{n}$ be decomposed into edge-disjoint copies of $G$? (more unsolved problems in mathematics) Formulation In conferences held at Oberwolfach, it is the custom for the participants to dine together in a room with circular tables, not all the same size, and with assigned seating that rearranges the participants from meal to meal. The Oberwolfach problem asks how to make a seating chart for a given set of tables so that all tables are full at each meal and all pairs of conference participants are seated next to each other exactly once. An instance of the problem can be denoted as $OP(x,y,z,\dots )$ where $x,y,z,\dots $ are the given table sizes. Alternatively, when some table sizes are repeated, they may be denoted using exponential notation; for instance, $OP(5^{3})$ describes an instance with three tables of size five.[1] Formulated as a problem in graph theory, the pairs of people sitting next to each other at a single meal can be represented as a disjoint union of cycle graphs $C_{x}+C_{y}+C_{z}+\cdots $ of the specified lengths, with one cycle for each of the dining tables. This union of cycles is a 2-regular graph, and every 2-regular graph has this form. If $G$ is this 2-regular graph and has $n$ vertices, the question is whether the complete graph $K_{n}$ of order $n$ can be represented as an edge-disjoint union of copies of $G$.[1] In order for a solution to exist, the total number of conference participants (or equivalently, the total capacity of the tables, or the total number of vertices of the given cycle graphs) must be an odd number. For, at each meal, each participant sits next to two neighbors, so the total number of neighbors of each participant must be even, and this is only possible when the total number of participants is odd. The problem has, however, also been extended to even values of $n$ by asking, for those $n$, whether all of the edges of the complete graph except for a perfect matching can be covered by copies of the given 2-regular graph. Like the ménage problem (a different mathematical problem involving seating arrangements of diners and tables), this variant of the problem can be formulated by supposing that the $n$ diners are arranged into $n/2$ married couples, and that the seating arrangements should place each diner next to each other diner except their own spouse exactly once.[2] Known results The only instances of the Oberwolfach problem that are known not to be solvable are $OP(3^{2})$, $OP(3^{4})$, $OP(4,5)$, and $OP(3,3,5)$.[3] It is widely believed that all other instances have a solution. This conjecture is supported by recent non-constructive and asymptotic solutions for large complete graphs of order greater than a lower bound that is however unquantified.[4][5] Cases for which a constructive solution is known include: • All instances $OP(x^{y})$ except $OP(3^{2})$ and $OP(3^{4})$.[6][7][8][9][2] • All instances in which all of the cycles have even length.[6][10] • All instances (other than the known exceptions) with $n\leq 60$.[11][3] • All instances for certain choices of $n$, belonging to infinite subsets of the natural numbers.[12][13] • All instances $OP(x,y)$ other than the known exceptions $OP(3,3)$ and $OP(4,5)$.[14] Related problems Kirkman's schoolgirl problem, of grouping fifteen schoolgirls into rows of three in seven different ways so that each pair of girls appears once in each triple, is a special case of the Oberwolfach problem, $OP(3^{5})$. The problem of Hamiltonian decomposition of a complete graph $K_{n}$ is another special case, $OP(n)$.[10] Alspach's conjecture, on the decomposition of a complete graph into cycles of given sizes, is related to the Oberwolfach problem, but neither is a special case of the other. If $G$ is a 2-regular graph, with $n$ vertices, formed from a disjoint union of cycles of certain lengths, then a solution to the Oberwolfach problem for $G$ would also provide a decomposition of the complete graph into $(n-1)/2$ copies of each of the cycles of $G$. However, not every decomposition of $K_{n}$ into this many cycles of each size can be grouped into disjoint cycles that form copies of $G$, and on the other hand not every instance of Alspach's conjecture involves sets of cycles that have $(n-1)/2$ copies of each cycle. References 1. Lenz, Hanfried; Ringel, Gerhard (1991), "A brief review on Egmont Köhler's mathematical work", Discrete Mathematics, 97 (1–3): 3–16, doi:10.1016/0012-365X(91)90416-Y, MR 1140782 2. Huang, Charlotte; Kotzig, Anton; Rosa, Alexander (1979), "On a variation of the Oberwolfach problem", Discrete Mathematics, 27 (3): 261–277, doi:10.1016/0012-365X(79)90162-6, MR 0541472 3. Salassa, F.; Dragotto, G.; Traetta, T.; Buratti, M.; Della Croce, F. (2019), Merging Combinatorial Design and Optimization: the Oberwolfach Problem, arXiv:1903.12112, Bibcode:2019arXiv190312112S 4. Glock, Stefan; Joos, Felix; Kim, Jaehoon; Kühn, Daniela; Osthus, Deryk (2021), "Resolution of the Oberwolfach problem", Journal of the European Mathematical Society, 23 (8): 2511–2547, arXiv:1806.04644, doi:10.4171/jems/1060, MR 4269420 5. Keevash, Peter; Staden, Katherine (2022), "The generalised Oberwolfach problem" (PDF), Journal of Combinatorial Theory, Series B, 152: 281–318, doi:10.1016/j.jctb.2021.09.007, MR 4332743 6. Häggkvist, Roland (1985), "A lemma on cycle decompositions", Cycles in graphs (Burnaby, B.C., 1982), North-Holland Math. Stud., vol. 115, Amsterdam: North-Holland, pp. 227–232, doi:10.1016/S0304-0208(08)73015-9, MR 0821524 7. Alspach, Brian; Häggkvist, Roland (1985), "Some observations on the Oberwolfach problem", Journal of Graph Theory, 9 (1): 177–187, doi:10.1002/jgt.3190090114, MR 0785659 8. Alspach, Brian; Schellenberg, P. J.; Stinson, D. R.; Wagner, David (1989), "The Oberwolfach problem and factors of uniform odd length cycles", Journal of Combinatorial Theory, Series A, 52 (1): 20–43, doi:10.1016/0097-3165(89)90059-9, MR 1008157 9. Hoffman, D. G.; Schellenberg, P. J. (1991), "The existence of $C_{k}$-factorizations of $K_{2n}-F$", Discrete Mathematics, 97 (1–3): 243–250, doi:10.1016/0012-365X(91)90440-D, MR 1140806 10. Bryant, Darryn; Danziger, Peter (2011), "On bipartite 2-factorizations of $K_{n}-I$ and the Oberwolfach problem" (PDF), Journal of Graph Theory, 68 (1): 22–37, doi:10.1002/jgt.20538, MR 2833961 11. Deza, A.; Franek, F.; Hua, W.; Meszka, M.; Rosa, A. (2010), "Solutions to the Oberwolfach problem for orders 18 to 40" (PDF), Journal of Combinatorial Mathematics and Combinatorial Computing, 74: 95–102, MR 2675892 12. Bryant, Darryn; Scharaschkin, Victor (2009), "Complete solutions to the Oberwolfach problem for an infinite set of orders", Journal of Combinatorial Theory, Series B, 99 (6): 904–918, doi:10.1016/j.jctb.2009.03.003, MR 2558441 13. Alspach, Brian; Bryant, Darryn; Horsley, Daniel; Maenhaut, Barbara; Scharaschkin, Victor (2016), "On factorisations of complete graphs into circulant graphs and the Oberwolfach problem", Ars Mathematica Contemporanea, 11 (1): 157–173, arXiv:1411.6047, doi:10.26493/1855-3974.770.150, MR 3546656 14. Traetta, Tommaso (2013), "A complete solution to the two-table Oberwolfach problems", Journal of Combinatorial Theory, Series A, 120 (5): 984–997, doi:10.1016/j.jcta.2013.01.003, MR 3033656	Wikipedia
The Petersen Graph The Petersen Graph is a mathematics book about the Petersen graph and its applications in graph theory. It was written by Derek Holton and John Sheehan, and published in 1993 by the Cambridge University Press as volume 7 in their Australian Mathematical Society Lecture Series. This article is about the book. For the graph, see Petersen graph. The Petersen Graph Author • Derek Holton • John Sheehan SeriesAustralian Mathematical Society Lecture Series SubjectThe Petersen graph PublisherCambridge University Press Publication date 1993 Topics The Petersen graph is an undirected graph with ten vertices and fifteen edges, commonly drawn as a pentagram within a pentagon, with corresponding vertices attached to each other. It has many unusual mathematical properties, and has frequently been used as a counterexample to conjectures in graph theory.[1][2] The book uses these properties as an excuse to cover several advanced topics in graph theory where this graph plays an important role.[1][3] It is heavily illustrated, and includes both open problems on the topics it discusses and detailed references to the literature on these problems.[1][4] After an introductory chapter, the second and third chapters concern graph coloring, the history of the four color theorem for planar graphs, its equivalence to 3-edge-coloring of planar cubic graphs, the snarks (cubic graphs that have no such colorings), and the conjecture of W. T. Tutte that every snark has the Petersen graph as a graph minor. Two more chapters concern closely related topics, perfect matchings (the sets of edges that can have a single color in a 3-edge-coloring) and nowhere-zero flows (the dual concept to planar graph coloring). The Petersen graph shows up again in another conjecture of Tutte, that when a bridgeless graph does not have the Petersen graph as a minor, it must have a nowhere-zero 4-flow.[3] Chapter six of the book concerns cages, the smallest regular graphs with no cycles shorter than a given length. The Petersen graph is an example: it is the smallest 3-regular graph with no cycles of length shorter than 5. Chapter seven is on hypohamiltonian graphs, the graphs that do not have a Hamiltonian cycle through all vertices but that do have cycles through every set of all but one vertices; the Petersen graph is the smallest example. The next chapter concerns the symmetries of graphs, and types of graphs defined by their symmetries, including the distance-transitive graphs and strongly regular graphs (of which the Petersen graph is an example)[3] and the Cayley graphs (of which it is not).[1] The book concludes with a final chapter of miscellaneous topics too small for their own chapters.[3] Audience and reception The book assumes that its readers already have some familiarity with graph theory.[3] It can be used as a reference work for researchers in this area,[1][2] or as the basis of an advanced course in graph theory.[2][3] Although Carsten Thomassen describes the book as "elegant",[4] and Robin Wilson evaluates its exposition as "generally good",[2] reviewer Charles H. C. Little takes the opposite view, finding fault with its copyediting, with some of its mathematical notation, and with its failure to discuss the lattice of integer combinations of perfect matchings, in which the number of copies of the Petersen graph in the "bricks" of a certain graph decomposition plays a key role in computing the dimension.[1] Reviewer Ian Anderson notes the superficiality of some of its coverage, but concludes that the book "succeeds in giving an exciting and enthusiastic glimpse" of graph theory.[3] References 1. Little, Charles H. C. (1994), "Review of The Petersen Graph", Mathematical Reviews, MR 1232658 2. Wilson, Robin J. (January 1995), "Review of The Petersen Graph", Bulletin of the London Mathematical Society, 27 (1): 89–89, doi:10.1112/blms/27.1.89 3. Anderson, Ian (March 1995), "Review of The Petersen Graph", The Mathematical Gazette, 79 (484): 239–240, doi:10.2307/3620120, JSTOR 3620120 4. Thomassen, C., "Review of The Petersen Graph", zbMATH, Zbl 0781.05001	Wikipedia
The Principles of Quantum Mechanics The Principles of Quantum Mechanics is an influential monograph on quantum mechanics written by Paul Dirac and first published by Oxford University Press in 1930.[1] Dirac gives an account of quantum mechanics by "demonstrating how to construct a completely new theoretical framework from scratch"; "problems were tackled top-down, by working on the great principles, with the details left to look after themselves".[2] It leaves classical physics behind after the first chapter, presenting the subject with a logical structure. Its 82 sections contain 785 equations with no diagrams.[2] This article is about the book by Paul Dirac. For the book by Ramamurti Shankar, see Principles of Quantum Mechanics. The Principles of Quantum Mechanics Title page of the first edition AuthorPaul Dirac CountryUnited Kingdom LanguageEnglish SubjectQuantum mechanics GenresNon-fiction PublisherOxford University Press Publication date 1930 Media typePrint Pages257 Dirac is credited with developing the subject "particularly in Cambridge and Göttingen between 1925–1927" (Farmelo).[2] History The first and second editions of the book were published in 1930 and 1935.[3] In 1947 the third edition of the book was published, in which the chapter on quantum electrodynamics was rewritten particularly with the inclusion of electron-positron creation.[3] In the fourth edition, 1958, the same chapter was revised, adding new sections on interpretation and applications. Later a revised fourth edition appeared in 1967.[3] Beginning with the third edition (1947), the mathematical descriptions of quantum states and operators were changed to use the Bra–ket notation, introduced in 1939 and largely developed by Dirac himself.[4] Laurie Brown wrote an article describing the book's evolution through its different editions,[5] and Helge Kragh surveyed reviews by physicists (including Heisenberg, Pauli, and others) from the time of Dirac's book's publication.[6] Contents • The principle of superposition • Dynamical variables and observables • Representations • The quantum conditions • The equations of motion • Elementary applications • Perturbation theory • Collision problems • Systems containing several similar particles • Theory of radiation • Relativistic theory of the electron • Quantum electrodynamics See also • The Evolution of Physics (Einstein) • The Feynman Lectures on Physics Vol. III (Feynman) • The Physical Principles of the Quantum Theory (Heisenberg) • Mathematical Foundations of Quantum Mechanics (von Neumann) References 1. "Paul A.M. Dirac – Biography". The Nobel Prize in Physics 1933. Retrieved 26 September 2011. Dirac's publications include ... The Principles of Quantum Mechanics (1930; 3rd ed. 1947). 2. Farmelo, Graham (2 June 1995). "Speaking Volumes: The Principles of Quantum Mechanics" (Book review). Times Higher Education Supplement: 20. Retrieved 26 September 2011. 3. Dalitz, R. H. (1995). The Collected Works of P. A. M. Dirac: Volume 1: 1924–1948. Cambridge University Press. pp. 453–454. ISBN 9780521362313. 4. PAM Dirac (1939). "A new notation for quantum mechanics". Mathematical Proceedings of the Cambridge Philosophical Society. 35 (3): 416–418. Bibcode:1939PCPS...35..416D. doi:10.1017/S0305004100021162. S2CID 121466183. 5. Brown, L.M. (2006), "Paul A.M. Dirac's The Principles of Quantum Mechanics" (PDF), Physics in Perspective, 8 (4): 381–407, Bibcode:2006PhP.....8..381B, doi:10.1007/s00016-006-0276-4, S2CID 59431829, archived from the original (PDF) on 28 February 2020 6. Helge Kragh (2013), Paul Dirac and The Principles of Quantum Mechanics, Research and Pedagogy, Studies 2: A History of Quantum Physics through Its Textbooks, Max-Planck-Gesellschaft zur Förderung der Wissenschaften, ISBN 9783945561249	Wikipedia
Product rule In calculus, the product rule (or Leibniz rule[1] or Leibniz product rule) is a formula used to find the derivatives of products of two or more functions. For two functions, it may be stated in Lagrange's notation as $(u\cdot v)'=u'\cdot v+u\cdot v'$ This article is about the derivative of a product. For the relation between derivatives of 3 dependent variables, see Triple product rule. For a counting principle in combinatorics, see Rule of product. For conditional probabilities, see Chain rule (probability). Part of a series of articles about Calculus • Fundamental theorem • Limits • Continuity • Rolle's theorem • Mean value theorem • Inverse function theorem Differential Definitions • Derivative (generalizations) • Differential • infinitesimal • of a function • total Concepts • Differentiation notation • Second derivative • Implicit differentiation • Logarithmic differentiation • Related rates • Taylor's theorem Rules and identities • Sum • Product • Chain • Power • Quotient • L'Hôpital's rule • Inverse • General Leibniz • Faà di Bruno's formula • Reynolds Integral • Lists of integrals • Integral transform • Leibniz integral rule Definitions • Antiderivative • Integral (improper) • Riemann integral • Lebesgue integration • Contour integration • Integral of inverse functions Integration by • Parts • Discs • Cylindrical shells • Substitution (trigonometric, tangent half-angle, Euler) • Euler's formula • Partial fractions • Changing order • Reduction formulae • Differentiating under the integral sign • Risch algorithm Series • Geometric (arithmetico-geometric) • Harmonic • Alternating • Power • Binomial • Taylor Convergence tests • Summand limit (term test) • Ratio • Root • Integral • Direct comparison • Limit comparison • Alternating series • Cauchy condensation • Dirichlet • Abel Vector • Gradient • Divergence • Curl • Laplacian • Directional derivative • Identities Theorems • Gradient • Green's • Stokes' • Divergence • generalized Stokes Multivariable Formalisms • Matrix • Tensor • Exterior • Geometric Definitions • Partial derivative • Multiple integral • Line integral • Surface integral • Volume integral • Jacobian • Hessian Advanced • Calculus on Euclidean space • Generalized functions • Limit of distributions Specialized • Fractional • Malliavin • Stochastic • Variations Miscellaneous • Precalculus • History • Glossary • List of topics • Integration Bee • Mathematical analysis • Nonstandard analysis or in Leibniz's notation as ${\frac {d}{dx}}(u\cdot v)={\frac {du}{dx}}\cdot v+u\cdot {\frac {dv}{dx}}.$ The rule may be extended or generalized to products of three or more functions, to a rule for higher-order derivatives of a product, and to other contexts. Discovery Discovery of this rule is credited to Gottfried Leibniz, who demonstrated it using differentials.[2] (However, J. M. Child, a translator of Leibniz's papers,[3] argues that it is due to Isaac Barrow.) Here is Leibniz's argument: Let u(x) and v(x) be two differentiable functions of x. Then the differential of uv is ${\begin{aligned}d(u\cdot v)&{}=(u+du)\cdot (v+dv)-u\cdot v\\&{}=u\cdot dv+v\cdot du+du\cdot dv.\end{aligned}}$ Since the term du·dv is "negligible" (compared to du and dv), Leibniz concluded that $d(u\cdot v)=v\cdot du+u\cdot dv$ and this is indeed the differential form of the product rule. If we divide through by the differential dx, we obtain ${\frac {d}{dx}}(u\cdot v)=v\cdot {\frac {du}{dx}}+u\cdot {\frac {dv}{dx}}$ which can also be written in Lagrange's notation as $(u\cdot v)'=v\cdot u'+u\cdot v'.$ Examples • Suppose we want to differentiate f(x) = x2 sin(x). By using the product rule, one gets the derivative f′(x) = 2x sin(x) + x2 cos(x) (since the derivative of x2 is 2x and the derivative of the sine function is the cosine function). • One special case of the product rule is the constant multiple rule, which states: if c is a number and f(x) is a differentiable function, then cf(x) is also differentiable, and its derivative is (cf)′(x) = c f′(x). This follows from the product rule since the derivative of any constant is zero. This, combined with the sum rule for derivatives, shows that differentiation is linear. • The rule for integration by parts is derived from the product rule, as is (a weak version of) the quotient rule. (It is a "weak" version in that it does not prove that the quotient is differentiable, but only says what its derivative is if it is differentiable.) Proofs Limit definition of derivative Let h(x) = f(x)g(x) and suppose that f and g are each differentiable at x. We want to prove that h is differentiable at x and that its derivative, h′(x), is given by f′(x)g(x) + f(x)g′(x). To do this, $f(x)g(x+\Delta x)-f(x)g(x+\Delta x)$ (which is zero, and thus does not change the value) is added to the numerator to permit its factoring, and then properties of limits are used. ${\begin{aligned}h'(x)&=\lim _{\Delta x\to 0}{\frac {h(x+\Delta x)-h(x)}{\Delta x}}\\[5pt]&=\lim _{\Delta x\to 0}{\frac {f(x+\Delta x)g(x+\Delta x)-f(x)g(x)}{\Delta x}}\\[5pt]&=\lim _{\Delta x\to 0}{\frac {f(x+\Delta x)g(x+\Delta x)-f(x)g(x+\Delta x)+f(x)g(x+\Delta x)-f(x)g(x)}{\Delta x}}\\[5pt]&=\lim _{\Delta x\to 0}{\frac {{\big [}f(x+\Delta x)-f(x){\big ]}\cdot g(x+\Delta x)+f(x)\cdot {\big [}g(x+\Delta x)-g(x){\big ]}}{\Delta x}}\\[5pt]&=\lim _{\Delta x\to 0}{\frac {f(x+\Delta x)-f(x)}{\Delta x}}\cdot \underbrace {\lim _{\Delta x\to 0}g(x+\Delta x)} _{\text{See the note below.}}+\lim _{\Delta x\to 0}f(x)\cdot \lim _{\Delta x\to 0}{\frac {g(x+\Delta x)-g(x)}{\Delta x}}\\[5pt]&=f'(x)g(x)+f(x)g'(x).\end{aligned}}$ The fact that $\lim _{\Delta x\to 0}g(x+\Delta x)=g(x)$ follows from the fact that differentiable functions are continuous. Linear approximations By definition, if $f,g:\mathbb {R} \to \mathbb {R} $ are differentiable at $x$, then we can write linear approximations: $f(x+h)=f(x)+f'(x)h+\varepsilon _{1}(h)$ and $g(x+h)=g(x)+g'(x)h+\varepsilon _{2}(h),$ where the error terms are small with respect to h: that is, $ \lim _{h\to 0}{\frac {\varepsilon _{1}(h)}{h}}=\lim _{h\to 0}{\frac {\varepsilon _{2}(h)}{h}}=0,$ also written $\varepsilon _{1},\varepsilon _{2}\sim o(h)$. Then: ${\begin{aligned}f(x+h)g(x+h)-f(x)g(x)&=(f(x)+f'(x)h+\varepsilon _{1}(h))(g(x)+g'(x)h+\varepsilon _{2}(h))-f(x)g(x)\\[.5em]&=f(x)g(x)+f'(x)g(x)h+f(x)g'(x)h-f(x)g(x)+{\text{error terms}}\\[.5em]&=f'(x)g(x)h+f(x)g'(x)h+o(h).\end{aligned}}$ The "error terms" consist of items such as $f(x)\varepsilon _{2}(h),f'(x)g'(x)h^{2}$ and $hf'(x)\varepsilon _{1}(h)$ which are easily seen to have magnitude $o(h).$ Dividing by $h$ and taking the limit $h\to 0$ gives the result. Quarter squares This proof uses the chain rule and the quarter square function $q(x)={\tfrac {1}{4}}x^{2}$ with derivative $q'(x)={\tfrac {1}{2}}x$. We have: $uv=q(u+v)-q(u-v),$ and differentiating both sides gives: ${\begin{aligned}f'&=q'(u+v)(u'+v')-q'(u-v)(u'-v')\\[4pt]&=\left({\tfrac {1}{2}}(u+v)(u'+v')\right)-\left({\tfrac {1}{2}}(u-v)(u'-v')\right)\\[4pt]&={\tfrac {1}{2}}(uu'+vu'+uv'+vv')-{\tfrac {1}{2}}(uu'-vu'-uv'+vv')\\[4pt]&=vu'+uv'.\end{aligned}}$ Multivariable chain rule The product rule can be considered a special case of the chain rule for several variables, applied to the multiplication function $m(u,v)=uv$: ${d(uv) \over dx}={\frac {\partial (uv)}{\partial u}}{\frac {du}{dx}}+{\frac {\partial (uv)}{\partial v}}{\frac {dv}{dx}}=v{\frac {du}{dx}}+u{\frac {dv}{dx}}.$ Non-standard analysis Let u and v be continuous functions in x, and let dx, du and dv be infinitesimals within the framework of non-standard analysis, specifically the hyperreal numbers. Using st to denote the standard part function that associates to a finite hyperreal number the real infinitely close to it, this gives ${\begin{aligned}{\frac {d(uv)}{dx}}&=\operatorname {st} \left({\frac {(u+du)(v+dv)-uv}{dx}}\right)\\&=\operatorname {st} \left({\frac {uv+u\cdot dv+v\cdot du+du\cdot dv-uv}{dx}}\right)\\&=\operatorname {st} \left({\frac {u\cdot dv+v\cdot du+du\cdot dv}{dx}}\right)\\&=\operatorname {st} \left(u{\frac {dv}{dx}}+(v+dv){\frac {du}{dx}}\right)\\&=u{\frac {dv}{dx}}+v{\frac {du}{dx}}.\end{aligned}}$ This was essentially Leibniz's proof exploiting the transcendental law of homogeneity (in place of the standard part above). Smooth infinitesimal analysis In the context of Lawvere's approach to infinitesimals, let $dx$ be a nilsquare infinitesimal. Then $du=u'\ dx$ and $dv=v'\ dx$, so that ${\begin{aligned}d(uv)&=(u+du)(v+dv)-uv\\&=uv+u\cdot dv+v\cdot du+du\cdot dv-uv\\&=u\cdot dv+v\cdot du+du\cdot dv\\&=u\cdot dv+v\cdot du\,\!\end{aligned}}$ since $du\,dv=u'v'(dx)^{2}=0.$ Dividing by $dx$ then gives ${\frac {d(uv)}{dx}}=u{\frac {dv}{dx}}+v{\frac {du}{dx}}$ or $(uv)'=u\cdot v'+v\cdot u'$. Logarithmic differentiation Let $h(x)=f(x)g(x)$. Taking the absolute value of each function and the natural log of both sides of the equation, $\ln \|h(x)\|=\ln \|f(x)g(x)\|$ Applying properties of the absolute value and logarithms, $\ln \|h(x)\|=\ln \|f(x)\|+\ln \|g(x)\|$ Taking the logarithmic derivative of both sides and then solving for $h'(x)$: ${\frac {h'(x)}{h(x)}}={\frac {f'(x)}{f(x)}}+{\frac {g'(x)}{g(x)}}$ Solving for $h'(x)$ and substituting back $f(x)g(x)$ for $h(x)$ gives: ${\begin{aligned}h'(x)&=h(x)\left({\frac {f'(x)}{f(x)}}+{\frac {g'(x)}{g(x)}}\right)\\&=f(x)g(x)\left({\frac {f'(x)}{f(x)}}+{\frac {g'(x)}{g(x)}}\right)\\&=f'(x)g(x)+f(x)g'(x).\end{aligned}}$ Note: Taking the absolute value of the functions is necessary for the logarithmic differentiation of functions that may have negative values, as logarithms are only real-valued for positive arguments. This works because ${\tfrac {d}{dx}}(\ln \|u\|)={\tfrac {u'}{u}}$, which justifies taking the absolute value of the functions for logarithmic differentiation. Generalizations Product of more than two factors The product rule can be generalized to products of more than two factors. For example, for three factors we have ${\frac {d(uvw)}{dx}}={\frac {du}{dx}}vw+u{\frac {dv}{dx}}w+uv{\frac {dw}{dx}}.$ For a collection of functions $f_{1},\dots ,f_{k}$, we have ${\frac {d}{dx}}\left[\prod _{i=1}^{k}f_{i}(x)\right]=\sum _{i=1}^{k}\left(\left({\frac {d}{dx}}f_{i}(x)\right)\prod _{j=1,j\neq i}^{k}f_{j}(x)\right)=\left(\prod _{i=1}^{k}f_{i}(x)\right)\left(\sum _{i=1}^{k}{\frac {f'_{i}(x)}{f_{i}(x)}}\right).$ The logarithmic derivative provides a simpler expression of the last form, as well as a direct proof that does not involve any recursion. The logarithmic derivative of a function f, denoted here Logder(f), is the derivative of the logarithm of the function. It follows that $\operatorname {Logder} (f)={\frac {f'}{f}}.$ Using that the logarithm of a product is the sum of the logarithms of the factors, the sum rule for derivatives gives immediately $\operatorname {Logder} (f_{1}\cdots f_{k})=\sum _{i=1}^{k}\operatorname {Logder} (f_{i}).$ The last above expression of the derivative of a product is obtained by multiplying both members of this equation by the product of the $f_{i}.$ Higher derivatives Main article: General Leibniz rule It can also be generalized to the general Leibniz rule for the nth derivative of a product of two factors, by symbolically expanding according to the binomial theorem: $d^{n}(uv)=\sum _{k=0}^{n}{n \choose k}\cdot d^{(n-k)}(u)\cdot d^{(k)}(v).$ Applied at a specific point x, the above formula gives: $(uv)^{(n)}(x)=\sum _{k=0}^{n}{n \choose k}\cdot u^{(n-k)}(x)\cdot v^{(k)}(x).$ Furthermore, for the nth derivative of an arbitrary number of factors, one has a similar formula with multinomial coefficients: $\left(\prod _{i=1}^{k}f_{i}\right)^{\!\!(n)}=\sum _{j_{1}+j_{2}+\cdots +j_{k}=n}{n \choose j_{1},j_{2},\ldots ,j_{k}}\prod _{i=1}^{k}f_{i}^{(j_{i})}.$ Higher partial derivatives For partial derivatives, we have[4] ${\partial ^{n} \over \partial x_{1}\,\cdots \,\partial x_{n}}(uv)=\sum _{S}{\partial ^{\|S\|}u \over \prod _{i\in S}\partial x_{i}}\cdot {\partial ^{n-\|S\|}v \over \prod _{i\not \in S}\partial x_{i}}$ where the index S runs through all 2n subsets of {1, ..., n}, and \|S\| is the cardinality of S. For example, when n = 3, ${\begin{aligned}&{\partial ^{3} \over \partial x_{1}\,\partial x_{2}\,\partial x_{3}}(uv)\\[6pt]={}&u\cdot {\partial ^{3}v \over \partial x_{1}\,\partial x_{2}\,\partial x_{3}}+{\partial u \over \partial x_{1}}\cdot {\partial ^{2}v \over \partial x_{2}\,\partial x_{3}}+{\partial u \over \partial x_{2}}\cdot {\partial ^{2}v \over \partial x_{1}\,\partial x_{3}}+{\partial u \over \partial x_{3}}\cdot {\partial ^{2}v \over \partial x_{1}\,\partial x_{2}}\\[6pt]&+{\partial ^{2}u \over \partial x_{1}\,\partial x_{2}}\cdot {\partial v \over \partial x_{3}}+{\partial ^{2}u \over \partial x_{1}\,\partial x_{3}}\cdot {\partial v \over \partial x_{2}}+{\partial ^{2}u \over \partial x_{2}\,\partial x_{3}}\cdot {\partial v \over \partial x_{1}}+{\partial ^{3}u \over \partial x_{1}\,\partial x_{2}\,\partial x_{3}}\cdot v.\end{aligned}}$ Banach space Suppose X, Y, and Z are Banach spaces (which includes Euclidean space) and B : X × Y → Z is a continuous bilinear operator. Then B is differentiable, and its derivative at the point (x,y) in X × Y is the linear map D(x,y)B : X × Y → Z given by $(D_{\left(x,y\right)}\,B)\left(u,v\right)=B\left(u,y\right)+B\left(x,v\right)\qquad \forall (u,v)\in X\times Y.$ This result can be extended[5] to more general topological vector spaces. In vector calculus Main article: Vector calculus identities § First derivative identities The product rule extends to various product operations of vector functions on $\mathbb {R} ^{n}$:[6] • For scalar multiplication: $(f\cdot \mathbf {g} )'=f'\cdot \mathbf {g} +f\cdot \mathbf {g} '$ • For dot product: $(\mathbf {f} \cdot \mathbf {g} )'=\mathbf {f} '\cdot \mathbf {g} +\mathbf {f} \cdot \mathbf {g} '$ • For cross product of vector functions on $\mathbb {R} ^{3}$: $(\mathbf {f} \times \mathbf {g} )'=\mathbf {f} '\times \mathbf {g} +\mathbf {f} \times \mathbf {g} '$ There are also analogues for other analogs of the derivative: if f and g are scalar fields then there is a product rule with the gradient: $\nabla (f\cdot g)=\nabla f\cdot g+f\cdot \nabla g$ Such a rule will hold for any continuous bilinear product operation. Let B : X × Y → Z be a continuous bilinear map between vector spaces, and let f and g be differentiable functions into X and Y, respectively. The only properties of multiplication used in the proof using the limit definition of derivative is that multiplication is continuous and bilinear. So for any continuous bilinear operation, $H(f,g)'=H(f',g)+H(f,g').$ This is also a special case of the product rule for bilinear maps in Banach space. Derivations in abstract algebra and differential geometry In abstract algebra, the product rule is the defining property of a derivation. In this terminology, the product rule states that the derivative operator is a derivation on functions. In differential geometry, a tangent vector to a manifold M at a point p may be defined abstractly as an operator on real-valued functions which behaves like a directional derivative at p: that is, a linear functional v which is a derivation, $v(fg)=v(f)\,g(p)+f(p)\,v(g).$ Generalizing (and dualizing) the formulas of vector calculus to an n-dimensional manifold M, one may take differential forms of degrees k and l, denoted $\alpha \in \Omega ^{k}(M),\beta \in \Omega ^{\ell }(M)$, with the wedge or exterior product operation $\alpha \wedge \beta \in \Omega ^{k+\ell }(M)$, as well as the exterior derivative $d:\Omega ^{m}(M)\to \Omega ^{m+1}(M)$. Then one has the graded Leibniz rule: $d(\alpha \wedge \beta )=d\alpha \wedge \beta +(-1)^{k}\alpha \wedge d\beta .$ Applications Among the applications of the product rule is a proof that ${d \over dx}x^{n}=nx^{n-1}$ when n is a positive integer (this rule is true even if n is not positive or is not an integer, but the proof of that must rely on other methods). The proof is by mathematical induction on the exponent n. If n = 0 then xn is constant and nxn − 1 = 0. The rule holds in that case because the derivative of a constant function is 0. If the rule holds for any particular exponent n, then for the next value, n + 1, we have ${\begin{aligned}{d \over dx}x^{n+1}&{}={d \over dx}\left(x^{n}\cdot x\right)\\[12pt]&{}=x{d \over dx}x^{n}+x^{n}{d \over dx}x\qquad {\mbox{(the product rule is used here)}}\\[12pt]&{}=x\left(nx^{n-1}\right)+x^{n}\cdot 1\qquad {\mbox{(the induction hypothesis is used here)}}\\[12pt]&{}=(n+1)x^{n}.\end{aligned}}$ Therefore, if the proposition is true for n, it is true also for n + 1, and therefore for all natural n. See also • Differentiation of integrals – Problem in mathematics • Differentiation of trigonometric functions – Mathematical process of finding the derivative of a trigonometric function • Differentiation rules – Rules for computing derivatives of functions • Distribution (mathematics) – Mathematical analysis term similar to generalized function • General Leibniz rule – Generalization of the product rule in calculus • Integration by parts – Mathematical method in calculus • Inverse functions and differentiation – Calculus identityPages displaying short descriptions of redirect targets • Linearity of differentiation – Calculus property • Power rule – Method of differentiating single term polynomials • Quotient rule – Formula for the derivative of a ratio of functions • Table of derivatives – Rules for computing derivatives of functionsPages displaying short descriptions of redirect targets • Vector calculus identities – Mathematical identities References 1. "Leibniz rule – Encyclopedia of Mathematics". 2. Michelle Cirillo (August 2007). "Humanizing Calculus". The Mathematics Teacher. 101 (1): 23–27. doi:10.5951/MT.101.1.0023. 3. Leibniz, G. W. (2005) [1920], The Early Mathematical Manuscripts of Leibniz (PDF), translated by J.M. Child, Dover, p. 28, footnote 58, ISBN 978-0-486-44596-0 4. Micheal Hardy (January 2006). "Combinatorics of Partial Derivatives" (PDF). The Electronic Journal of Combinatorics. 13. arXiv:math/0601149. Bibcode:2006math......1149H. 5. Kreigl, Andreas; Michor, Peter (1997). The Convenient Setting of Global Analysis (PDF). American Mathematical Society. p. 59. ISBN 0-8218-0780-3. 6. Stewart, James (2016), Calculus (8 ed.), Cengage, Section 13.2. 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Prüfer group In mathematics, specifically in group theory, the Prüfer p-group or the p-quasicyclic group or p∞-group, Z(p∞), for a prime number p is the unique p-group in which every element has p different p-th roots. Algebraic structure → Ring theory Ring theory Basic concepts Rings • Subrings • Ideal • Quotient ring • Fractional ideal • Total ring of fractions • Product of rings • Free product of associative algebras • Tensor product of algebras Ring homomorphisms • Kernel • Inner automorphism • Frobenius endomorphism Algebraic structures • Module • Associative algebra • Graded ring • Involutive ring • Category of rings • Initial ring $\mathbb {Z} $ • Terminal ring $0=\mathbb {Z} _{1}$ Related structures • Field • Finite field • Non-associative ring • Lie ring • Jordan ring • Semiring • Semifield Commutative algebra Commutative rings • Integral domain • Integrally closed domain • GCD domain • Unique factorization domain • Principal ideal domain • Euclidean domain • Field • Finite field • Composition ring • Polynomial ring • Formal power series ring Algebraic number theory • Algebraic number field • Ring of integers • Algebraic independence • Transcendental number theory • Transcendence degree p-adic number theory and decimals • Direct limit/Inverse limit • Zero ring $\mathbb {Z} _{1}$ • Integers modulo pn $\mathbb {Z} /p^{n}\mathbb {Z} $ • Prüfer p-ring $\mathbb {Z} (p^{\infty })$ • Base-p circle ring $\mathbb {T} $ • Base-p integers $\mathbb {Z} $ • p-adic rationals $\mathbb {Z} [1/p]$ • Base-p real numbers $\mathbb {R} $ • p-adic integers $\mathbb {Z} _{p}$ • p-adic numbers $\mathbb {Q} _{p}$ • p-adic solenoid $\mathbb {T} _{p}$ Algebraic geometry • Affine variety Noncommutative algebra Noncommutative rings • Division ring • Semiprimitive ring • Simple ring • Commutator Noncommutative algebraic geometry Free algebra Clifford algebra • Geometric algebra Operator algebra The Prüfer p-groups are countable abelian groups that are important in the classification of infinite abelian groups: they (along with the group of rational numbers) form the smallest building blocks of all divisible groups. The groups are named after Heinz Prüfer, a German mathematician of the early 20th century. Constructions of Z(p∞) The Prüfer p-group may be identified with the subgroup of the circle group, U(1), consisting of all pn-th roots of unity as n ranges over all non-negative integers: $\mathbf {Z} (p^{\infty })=\{\exp(2\pi im/p^{n})\mid 0\leq m<p^{n},\,n\in \mathbf {Z} ^{+}\}=\{z\in \mathbf {C} \mid z^{(p^{n})}=1{\text{ for some }}n\in \mathbf {Z} ^{+}\}.\;$ The group operation here is the multiplication of complex numbers. There is a presentation $\mathbf {Z} (p^{\infty })=\langle \,g_{1},g_{2},g_{3},\ldots \mid g_{1}^{p}=1,g_{2}^{p}=g_{1},g_{3}^{p}=g_{2},\dots \,\rangle .$ Here, the group operation in Z(p∞) is written as multiplication. Alternatively and equivalently, the Prüfer p-group may be defined as the Sylow p-subgroup of the quotient group Q/Z, consisting of those elements whose order is a power of p: $\mathbf {Z} (p^{\infty })=\mathbf {Z} [1/p]/\mathbf {Z} $ (where Z[1/p] denotes the group of all rational numbers whose denominator is a power of p, using addition of rational numbers as group operation). For each natural number n, consider the quotient group Z/pnZ and the embedding Z/pnZ → Z/pn+1Z induced by multiplication by p. The direct limit of this system is Z(p∞): $\mathbf {Z} (p^{\infty })=\varinjlim \mathbf {Z} /p^{n}\mathbf {Z} .$ If we perform the direct limit in the category of topological groups, then we need to impose a topology on each of the $\mathbf {Z} /p^{n}\mathbf {Z} $, and take the final topology on $\mathbf {Z} (p^{\infty })$. If we wish for $\mathbf {Z} (p^{\infty })$ to be Hausdorff, we must impose the discrete topology on each of the $\mathbf {Z} /p^{n}\mathbf {Z} $, resulting in $\mathbf {Z} (p^{\infty })$ to have the discrete topology. We can also write $\mathbf {Z} (p^{\infty })=\mathbf {Q} _{p}/\mathbf {Z} _{p}$ where Qp denotes the additive group of p-adic numbers and Zp is the subgroup of p-adic integers. Properties The complete list of subgroups of the Prüfer p-group Z(p∞) = Z[1/p]/Z is: $0\subsetneq \left({1 \over p}\mathbf {Z} \right)/\mathbf {Z} \subsetneq \left({1 \over p^{2}}\mathbf {Z} \right)/\mathbf {Z} \subsetneq \left({1 \over p^{3}}\mathbf {Z} \right)/\mathbf {Z} \subsetneq \cdots \subsetneq \mathbf {Z} (p^{\infty })$ (Here $\left({1 \over p^{n}}\mathbf {Z} \right)/\mathbf {Z} $ is a cyclic subgroup of Z(p∞) with pn elements; it contains precisely those elements of Z(p∞) whose order divides pn and corresponds to the set of pn-th roots of unity.) The Prüfer p-groups are the only infinite groups whose subgroups are totally ordered by inclusion. This sequence of inclusions expresses the Prüfer p-group as the direct limit of its finite subgroups. As there is no maximal subgroup of a Prüfer p-group, it is its own Frattini subgroup. Given this list of subgroups, it is clear that the Prüfer p-groups are indecomposable (cannot be written as a direct sum of proper subgroups). More is true: the Prüfer p-groups are subdirectly irreducible. An abelian group is subdirectly irreducible if and only if it is isomorphic to a finite cyclic p-group or to a Prüfer group. The Prüfer p-group is the unique infinite p-group that is locally cyclic (every finite set of elements generates a cyclic group). As seen above, all proper subgroups of Z(p∞) are finite. The Prüfer p-groups are the only infinite abelian groups with this property.[1] The Prüfer p-groups are divisible. They play an important role in the classification of divisible groups; along with the rational numbers they are the simplest divisible groups. More precisely: an abelian group is divisible if and only if it is the direct sum of a (possibly infinite) number of copies of Q and (possibly infinite) numbers of copies of Z(p∞) for every prime p. The (cardinal) numbers of copies of Q and Z(p∞) that are used in this direct sum determine the divisible group up to isomorphism.[2] As an abelian group (that is, as a Z-module), Z(p∞) is Artinian but not Noetherian.[3] It can thus be used as a counterexample against the idea that every Artinian module is Noetherian (whereas every Artinian ring is Noetherian). The endomorphism ring of Z(p∞) is isomorphic to the ring of p-adic integers Zp.[4] In the theory of locally compact topological groups the Prüfer p-group (endowed with the discrete topology) is the Pontryagin dual of the compact group of p-adic integers, and the group of p-adic integers is the Pontryagin dual of the Prüfer p-group.[5] See also • p-adic integers, which can be defined as the inverse limit of the finite subgroups of the Prüfer p-group. • Dyadic rational, rational numbers of the form a/2b. The Prüfer 2-group can be viewed as the dyadic rationals modulo 1. • Cyclic group (finite analogue) • Circle group (uncountably infinite analogue) Notes 1. See Vil'yams (2001) 2. See Kaplansky (1965) 3. See also Jacobson (2009), p. 102, ex. 2. 4. See Vil'yams (2001) 5. D. L. Armacost and W. L. Armacost,"On p-thetic groups", Pacific J. Math., 41, no. 2 (1972), 295–301 References • Jacobson, Nathan (2009). Basic algebra. Vol. 2 (2nd ed.). Dover. ISBN 978-0-486-47187-7. • Pierre Antoine Grillet (2007). Abstract algebra. Springer. ISBN 978-0-387-71567-4. • Kaplansky, Irving (1965). Infinite Abelian Groups. University of Michigan Press. • N.N. Vil'yams (2001) [1994], "Quasi-cyclic group", Encyclopedia of Mathematics, EMS Press	Wikipedia
The Pursuit of Perfect Packing The Pursuit of Perfect Packing is a book on packing problems in geometry. It was written by physicists Tomaso Aste and Denis Weaire, and published in 2000 by Institute of Physics Publishing (doi:10.1887/0750306483, ISBN 0-7503-0648-3) with a second edition published in 2008 by Taylor & Francis (ISBN 978-1-4200-6817-7). Topics The mathematical topics described in the book include sphere packing (including the Tammes problem, the Kepler conjecture, and higher-dimensional sphere packing), the Honeycomb conjecture and the Weaire–Phelan structure, Voronoi diagrams and Delaunay triangulations, Apollonian gaskets, random sequential adsorption,[1][2][3] and the physical realizations of some of these structures by sand, soap bubbles, the seeds of plants, and columnar basalt.[4][2] A broader theme involves the contrast between locally ordered and locally disordered structures, and the interplay between local and global considerations in optimal packings.[1] As well, the book includes biographical sketches of some of the contributors to this field, and histories of their work in this area, including Johannes Kepler, Stephen Hales, Joseph Plateau, Lord Kelvin, Osborne Reynolds, and J. D. Bernal.[4][3] Audience and reception The book is aimed at a general audience rather than to professional mathematicians.[4][5] Therefore, it avoids mathematical proofs and is otherwise not very technical. However, it contains pointers to the mathematical literature where readers more expert in these topics can find more detail.[4][2] Avoiding proof may have been a necessary decision as some proofs in this area defy summarization: the proof by Thomas Hales of the Kepler conjecture on optimal sphere packing in three dimensions, announced shortly before the publication of the book and one of its central topics, is hundreds of pages long.[3] Reviewer Johann Linhart complains that (in the first edition) some figures are inaccurately drawn.[4] And although finding the book "entertaining and easy to read", William Satzer finds it "frustrating" in the lack of detail in its stories.[1] Nevertheless, Linhart and reviewer Stephen Blundell highly recommend the book,[4][3] and reviewer Charles Radin calls it "a treasure trove of intriguing examples" and "a real gem".[5] And despite complaining about a format that mixes footnote markers into mathematical formulas, and the illegibility of some figures, Michael Fox recommends it to "any mathematics or science library".[2] References 1. "Review of The Pursuit of Perfect Packing", MAA Reviews, Mathematical Association of America, August 2008 2. Fox, Michael (Jul 2001), "Review of The Pursuit of Perfect Packing", The Mathematical Gazette, 85 (503): 370–372, doi:10.2307/3622070, JSTOR 3622070 3. Blundell, S. J. (January 2010), "Review of The Pursuit of Perfect Packing", Contemporary Physics, 51 (1): 94–95, doi:10.1080/00107510903021467 4. Linhart, Johann (2001), "Review of The Pursuit of Perfect Packing", Mathematical Reviews, MR 1786410 5. Radin, Charles (January 2006), "Review of The Pursuit of Perfect Packing", American Mathematical Monthly, 113 (1): 87–90, doi:10.2307/27641857, JSTOR 27641857	Wikipedia
Runge–Kutta methods In numerical analysis, the Runge–Kutta methods (English: /ˈrʊŋəˈkʊtɑː/ (listen) RUUNG-ə-KUUT-tah[1]) are a family of implicit and explicit iterative methods, which include the Euler method, used in temporal discretization for the approximate solutions of simultaneous nonlinear equations.[2] These methods were developed around 1900 by the German mathematicians Carl Runge and Wilhelm Kutta. Differential equations Scope Fields • Natural sciences • Engineering • Astronomy • Physics • Chemistry • Biology • Geology Applied mathematics • Continuum mechanics • Chaos theory • Dynamical systems Social sciences • Economics • Population dynamics List of named differential equations Classification Types • Ordinary • Partial • Differential-algebraic • Integro-differential • Fractional • Linear • Non-linear By variable type • Dependent and independent variables • Autonomous • Coupled / Decoupled • Exact • Homogeneous / Nonhomogeneous Features • Order • Operator • Notation Relation to processes • Difference (discrete analogue) • Stochastic • Stochastic partial • Delay Solution Existence and uniqueness • Picard–Lindelöf theorem • Peano existence theorem • Carathéodory's existence theorem • Cauchy–Kowalevski theorem General topics • Initial conditions • Boundary values • Dirichlet • Neumann • Robin • Cauchy problem • Wronskian • Phase portrait • Lyapunov / Asymptotic / Exponential stability • Rate of convergence • Series / Integral solutions • Numerical integration • Dirac delta function Solution methods • Inspection • Method of characteristics • Euler • Exponential response formula • Finite difference (Crank–Nicolson) • Finite element • Infinite element • Finite volume • Galerkin • Petrov–Galerkin • Green's function • Integrating factor • Integral transforms • Perturbation theory • Runge–Kutta • Separation of variables • Undetermined coefficients • Variation of parameters People List • Isaac Newton • Gottfried Leibniz • Jacob Bernoulli • Leonhard Euler • Józef Maria Hoene-Wroński • Joseph Fourier • Augustin-Louis Cauchy • George Green • Carl David Tolmé Runge • Martin Kutta • Rudolf Lipschitz • Ernst Lindelöf • Émile Picard • Phyllis Nicolson • John Crank The Runge–Kutta method The most widely known member of the Runge–Kutta family is generally referred to as "RK4", the "classic Runge–Kutta method" or simply as "the Runge–Kutta method". Let an initial value problem be specified as follows: ${\frac {dy}{dt}}=f(t,y),\quad y(t_{0})=y_{0}.$ Here $y$ is an unknown function (scalar or vector) of time $t$, which we would like to approximate; we are told that ${\frac {dy}{dt}}$, the rate at which $y$ changes, is a function of $t$ and of $y$ itself. At the initial time $t_{0}$ the corresponding $y$ value is $y_{0}$. The function $f$ and the initial conditions $t_{0}$, $y_{0}$ are given. Now we pick a step-size h > 0 and define: ${\begin{aligned}y_{n+1}&=y_{n}+{\frac {h}{6}}\left(k_{1}+2k_{2}+2k_{3}+k_{4}\right),\\t_{n+1}&=t_{n}+h\\\end{aligned}}$ for n = 0, 1, 2, 3, ..., using[3] ${\begin{aligned}k_{1}&=\ f(t_{n},y_{n}),\\k_{2}&=\ f\!\left(t_{n}+{\frac {h}{2}},y_{n}+h{\frac {k_{1}}{2}}\right),\\k_{3}&=\ f\!\left(t_{n}+{\frac {h}{2}},y_{n}+h{\frac {k_{2}}{2}}\right),\\k_{4}&=\ f\!\left(t_{n}+h,y_{n}+hk_{3}\right).\end{aligned}}$ (Note: the above equations have different but equivalent definitions in different texts).[4] Here $y_{n+1}$ is the RK4 approximation of $y(t_{n+1})$, and the next value ($y_{n+1}$) is determined by the present value ($y_{n}$) plus the weighted average of four increments, where each increment is the product of the size of the interval, h, and an estimated slope specified by function f on the right-hand side of the differential equation. • $k_{1}$ is the slope at the beginning of the interval, using $y$ (Euler's method); • $k_{2}$ is the slope at the midpoint of the interval, using $y$ and $k_{1}$; • $k_{3}$ is again the slope at the midpoint, but now using $y$ and $k_{2}$; • $k_{4}$ is the slope at the end of the interval, using $y$ and $k_{3}$. In averaging the four slopes, greater weight is given to the slopes at the midpoint. If $f$ is independent of $y$, so that the differential equation is equivalent to a simple integral, then RK4 is Simpson's rule.[5] The RK4 method is a fourth-order method, meaning that the local truncation error is on the order of $O(h^{5})$, while the total accumulated error is on the order of $O(h^{4})$. In many practical applications the function $f$ is independent of $t$ (so called autonomous system, or time-invariant system, especially in physics), and their increments are not computed at all and not passed to function $f$, with only the final formula for $t_{n+1}$ used. Explicit Runge–Kutta methods The family of explicit Runge–Kutta methods is a generalization of the RK4 method mentioned above. It is given by $y_{n+1}=y_{n}+h\sum _{i=1}^{s}b_{i}k_{i},$ where[6] ${\begin{aligned}k_{1}&=f(t_{n},y_{n}),\\k_{2}&=f(t_{n}+c_{2}h,y_{n}+(a_{21}k_{1})h),\\k_{3}&=f(t_{n}+c_{3}h,y_{n}+(a_{31}k_{1}+a_{32}k_{2})h),\\&\ \ \vdots \\k_{s}&=f(t_{n}+c_{s}h,y_{n}+(a_{s1}k_{1}+a_{s2}k_{2}+\cdots +a_{s,s-1}k_{s-1})h).\end{aligned}}$ (Note: the above equations may have different but equivalent definitions in some texts).[4] To specify a particular method, one needs to provide the integer s (the number of stages), and the coefficients aij (for 1 ≤ j < i ≤ s), bi (for i = 1, 2, ..., s) and ci (for i = 2, 3, ..., s). The matrix [aij] is called the Runge–Kutta matrix, while the bi and ci are known as the weights and the nodes.[7] These data are usually arranged in a mnemonic device, known as a Butcher tableau (after John C. Butcher): $0$ $c_{2}$$a_{21}$ $c_{3}$$a_{31}$$a_{32}$ $\vdots $$\vdots $$\ddots $ $c_{s}$ $a_{s1}$ $a_{s2}$ $\cdots $ $a_{s,s-1}$ $b_{1}$$b_{2}$$\cdots $$b_{s-1}$$b_{s}$ A Taylor series expansion shows that the Runge–Kutta method is consistent if and only if $\sum _{i=1}^{s}b_{i}=1.$ There are also accompanying requirements if one requires the method to have a certain order p, meaning that the local truncation error is O(hp+1). These can be derived from the definition of the truncation error itself. For example, a two-stage method has order 2 if b1 + b2 = 1, b2c2 = 1/2, and b2a21 = 1/2.[8] Note that a popular condition for determining coefficients is [9] $\sum _{j=1}^{i-1}a_{ij}=c_{i}{\text{ for }}i=2,\ldots ,s.$ This condition alone, however, is neither sufficient, nor necessary for consistency. [10] In general, if an explicit $s$-stage Runge–Kutta method has order $p$, then it can be proven that the number of stages must satisfy $s\geq p$, and if $p\geq 5$, then $s\geq p+1$.[11] However, it is not known whether these bounds are sharp in all cases; for example, all known methods of order 8 have at least 11 stages, though it is possible that there are methods with fewer stages. (The bound above suggests that there could be a method with 9 stages; but it could also be that the bound is simply not sharp.) Indeed, it is an open problem what the precise minimum number of stages $s$ is for an explicit Runge–Kutta method to have order $p$ in those cases where no methods have yet been discovered that satisfy the bounds above with equality. Some values which are known are:[12] ${\begin{array}{c\|cccccccc}p&1&2&3&4&5&6&7&8\\\hline \min s&1&2&3&4&6&7&9&11\end{array}}$ The provable bounds above then imply that we can not find methods of orders $p=1,2,\ldots ,6$ that require fewer stages than the methods we already know for these orders. However, it is conceivable that we might find a method of order $p=7$ that has only 8 stages, whereas the only ones known today have at least 9 stages as shown in the table. Examples The RK4 method falls in this framework. Its tableau is[13] 0 1/21/2 1/201/2 10 01 1/61/31/31/6 A slight variation of "the" Runge–Kutta method is also due to Kutta in 1901 and is called the 3/8-rule.[14] The primary advantage this method has is that almost all of the error coefficients are smaller than in the popular method, but it requires slightly more FLOPs (floating-point operations) per time step. Its Butcher tableau is 0 1/31/3 2/3-1/31 11 −11 1/83/83/81/8 However, the simplest Runge–Kutta method is the (forward) Euler method, given by the formula $y_{n+1}=y_{n}+hf(t_{n},y_{n})$. This is the only consistent explicit Runge–Kutta method with one stage. The corresponding tableau is 0 1 Second-order methods with two stages An example of a second-order method with two stages is provided by the explicit midpoint method: $y_{n+1}=y_{n}+hf\left(t_{n}+{\frac {1}{2}}h,y_{n}+{\frac {1}{2}}hf(t_{n},\ y_{n})\right).$ The corresponding tableau is 0 1/21/2 01 The midpoint method is not the only second-order Runge–Kutta method with two stages; there is a family of such methods, parameterized by α and given by the formula[15] $y_{n+1}=y_{n}+h{\bigl (}(1-{\tfrac {1}{2\alpha }})f(t_{n},y_{n})+{\tfrac {1}{2\alpha }}f(t_{n}+\alpha h,y_{n}+\alpha hf(t_{n},y_{n})){\bigr )}.$ Its Butcher tableau is 0 $\alpha $$\alpha $ $(1-{\tfrac {1}{2\alpha }})$${\tfrac {1}{2\alpha }}$ In this family, $\alpha ={\tfrac {1}{2}}$ gives the midpoint method, $\alpha =1$ is Heun's method,[5] and $\alpha ={\tfrac {2}{3}}$ is Ralston's method. Use As an example, consider the two-stage second-order Runge–Kutta method with α = 2/3, also known as Ralston method. It is given by the tableau 0 2/32/3 1/43/4 with the corresponding equations ${\begin{aligned}k_{1}&=f(t_{n},\ y_{n}),\\k_{2}&=f(t_{n}+{\tfrac {2}{3}}h,\ y_{n}+{\tfrac {2}{3}}hk_{1}),\\y_{n+1}&=y_{n}+h\left({\tfrac {1}{4}}k_{1}+{\tfrac {3}{4}}k_{2}\right).\end{aligned}}$ This method is used to solve the initial-value problem ${\frac {dy}{dt}}=\tan(y)+1,\quad y_{0}=1,\ t\in [1,1.1]$ with step size h = 0.025, so the method needs to take four steps. The method proceeds as follows: $t_{0}=1\colon $ $y_{0}=1$ $t_{1}=1.025\colon $ $y_{0}=1$$k_{1}=2.557407725$$k_{2}=f(t_{0}+{\tfrac {2}{3}}h,\ y_{0}+{\tfrac {2}{3}}hk_{1})=2.7138981400$ $y_{1}=y_{0}+h({\tfrac {1}{4}}k_{1}+{\tfrac {3}{4}}k_{2})={\underline {1.066869388}}$ $t_{2}=1.05\colon $ $y_{1}=1.066869388$$k_{1}=2.813524695$$k_{2}=f(t_{1}+{\tfrac {2}{3}}h,\ y_{1}+{\tfrac {2}{3}}hk_{1})$ $y_{2}=y_{1}+h({\tfrac {1}{4}}k_{1}+{\tfrac {3}{4}}k_{2})={\underline {1.141332181}}$ $t_{3}=1.075\colon $ $y_{2}=1.141332181$$k_{1}=3.183536647$$k_{2}=f(t_{2}+{\tfrac {2}{3}}h,\ y_{2}+{\tfrac {2}{3}}hk_{1})$ $y_{3}=y_{2}+h({\tfrac {1}{4}}k_{1}+{\tfrac {3}{4}}k_{2})={\underline {1.227417567}}$ $t_{4}=1.1\colon $ $y_{3}=1.227417567$$k_{1}=3.796866512$$k_{2}=f(t_{3}+{\tfrac {2}{3}}h,\ y_{3}+{\tfrac {2}{3}}hk_{1})$ $y_{4}=y_{3}+h({\tfrac {1}{4}}k_{1}+{\tfrac {3}{4}}k_{2})={\underline {1.335079087}}.$ The numerical solutions correspond to the underlined values. Adaptive Runge–Kutta methods Adaptive methods are designed to produce an estimate of the local truncation error of a single Runge–Kutta step. This is done by having two methods, one with order $p$ and one with order $p-1$. These methods are interwoven, i.e., they have common intermediate steps. Thanks to this, estimating the error has little or negligible computational cost compared to a step with the higher-order method. During the integration, the step size is adapted such that the estimated error stays below a user-defined threshold: If the error is too high, a step is repeated with a lower step size; if the error is much smaller, the step size is increased to save time. This results in an (almost) optimal step size, which saves computation time. Moreover, the user does not have to spend time on finding an appropriate step size. The lower-order step is given by $y_{n+1}^{}=y_{n}+h\sum _{i=1}^{s}b_{i}^{}k_{i},$ where $k_{i}$ are the same as for the higher-order method. Then the error is $e_{n+1}=y_{n+1}-y_{n+1}^{}=h\sum _{i=1}^{s}(b_{i}-b_{i}^{})k_{i},$ which is $O(h^{p})$. The Butcher tableau for this kind of method is extended to give the values of $b_{i}^{}$: 0 $c_{2}$$a_{21}$ $c_{3}$$a_{31}$$a_{32}$ $\vdots $$\vdots $$\ddots $ $c_{s}$ $a_{s1}$ $a_{s2}$ $\cdots $ $a_{s,s-1}$ $b_{1}$$b_{2}$$\cdots $$b_{s-1}$$b_{s}$ $b_{1}^{}$$b_{2}^{}$$\cdots $$b_{s-1}^{}$$b_{s}^{}$ The Runge–Kutta–Fehlberg method has two methods of orders 5 and 4. Its extended Butcher tableau is: 0 1/41/4 3/83/329/32 12/131932/2197−7200/21977296/2197 1439/216−83680/513-845/4104 1/2−8/272−3544/25651859/4104−11/40 16/13506656/1282528561/56430−9/502/55 25/21601408/25652197/4104−1/50 However, the simplest adaptive Runge–Kutta method involves combining Heun's method, which is order 2, with the Euler method, which is order 1. Its extended Butcher tableau is: 0 11 1/21/2 10 Other adaptive Runge–Kutta methods are the Bogacki–Shampine method (orders 3 and 2), the Cash–Karp method and the Dormand–Prince method (both with orders 5 and 4). Nonconfluent Runge–Kutta methods A Runge–Kutta method is said to be nonconfluent [16] if all the $c_{i},\,i=1,2,\ldots ,s$ are distinct. Runge–Kutta–Nyström methods Runge–Kutta–Nyström methods are specialized Runge-Kutta methods that are optimized for second-order differential equations of the following form:[17][18] ${\frac {d^{2}y}{dt^{2}}}=f(y,{\dot {y}},t).$ Implicit Runge–Kutta methods All Runge–Kutta methods mentioned up to now are explicit methods. Explicit Runge–Kutta methods are generally unsuitable for the solution of stiff equations because their region of absolute stability is small; in particular, it is bounded.[19] This issue is especially important in the solution of partial differential equations. The instability of explicit Runge–Kutta methods motivates the development of implicit methods. An implicit Runge–Kutta method has the form $y_{n+1}=y_{n}+h\sum _{i=1}^{s}b_{i}k_{i},$ where $k_{i}=f\left(t_{n}+c_{i}h,\ y_{n}+h\sum _{j=1}^{s}a_{ij}k_{j}\right),\quad i=1,\ldots ,s.$ [20] The difference with an explicit method is that in an explicit method, the sum over j only goes up to i − 1. This also shows up in the Butcher tableau: the coefficient matrix $a_{ij}$ of an explicit method is lower triangular. In an implicit method, the sum over j goes up to s and the coefficient matrix is not triangular, yielding a Butcher tableau of the form[13] ${\begin{array}{c\|cccc}c_{1}&a_{11}&a_{12}&\dots &a_{1s}\\c_{2}&a_{21}&a_{22}&\dots &a_{2s}\\\vdots &\vdots &\vdots &\ddots &\vdots \\c_{s}&a_{s1}&a_{s2}&\dots &a_{ss}\\\hline &b_{1}&b_{2}&\dots &b_{s}\\&b_{1}^{}&b_{2}^{}&\dots &b_{s}^{}\\\end{array}}={\begin{array}{c\|c}\mathbf {c} &A\\\hline &\mathbf {b^{T}} \\\end{array}}$ See Adaptive Runge-Kutta methods above for the explanation of the $b^{*}$ row. The consequence of this difference is that at every step, a system of algebraic equations has to be solved. This increases the computational cost considerably. If a method with s stages is used to solve a differential equation with m components, then the system of algebraic equations has ms components. This can be contrasted with implicit linear multistep methods (the other big family of methods for ODEs): an implicit s-step linear multistep method needs to solve a system of algebraic equations with only m components, so the size of the system does not increase as the number of steps increases.[21] Examples The simplest example of an implicit Runge–Kutta method is the backward Euler method: $y_{n+1}=y_{n}+hf(t_{n}+h,\ y_{n+1}).\,$ The Butcher tableau for this is simply: ${\begin{array}{c\|c}1&1\\\hline &1\\\end{array}}$ This Butcher tableau corresponds to the formulae $k_{1}=f(t_{n}+h,\ y_{n}+hk_{1})\quad {\text{and}}\quad y_{n+1}=y_{n}+hk_{1},$ which can be re-arranged to get the formula for the backward Euler method listed above. Another example for an implicit Runge–Kutta method is the trapezoidal rule. Its Butcher tableau is: ${\begin{array}{c\|cc}0&0&0\\1&{\frac {1}{2}}&{\frac {1}{2}}\\\hline &{\frac {1}{2}}&{\frac {1}{2}}\\&1&0\\\end{array}}$ The trapezoidal rule is a collocation method (as discussed in that article). All collocation methods are implicit Runge–Kutta methods, but not all implicit Runge–Kutta methods are collocation methods.[22] The Gauss–Legendre methods form a family of collocation methods based on Gauss quadrature. A Gauss–Legendre method with s stages has order 2s (thus, methods with arbitrarily high order can be constructed).[23] The method with two stages (and thus order four) has Butcher tableau: ${\begin{array}{c\|cc}{\frac {1}{2}}-{\frac {1}{6}}{\sqrt {3}}&{\frac {1}{4}}&{\frac {1}{4}}-{\frac {1}{6}}{\sqrt {3}}\\{\frac {1}{2}}+{\frac {1}{6}}{\sqrt {3}}&{\frac {1}{4}}+{\frac {1}{6}}{\sqrt {3}}&{\frac {1}{4}}\\\hline &{\frac {1}{2}}&{\frac {1}{2}}\\&{\frac {1}{2}}+{\frac {1}{2}}{\sqrt {3}}&{\frac {1}{2}}-{\frac {1}{2}}{\sqrt {3}}\end{array}}$ [21] Stability The advantage of implicit Runge–Kutta methods over explicit ones is their greater stability, especially when applied to stiff equations. Consider the linear test equation $y'=\lambda y$. A Runge–Kutta method applied to this equation reduces to the iteration $y_{n+1}=r(h\lambda )\,y_{n}$, with r given by $r(z)=1+zb^{T}(I-zA)^{-1}e={\frac {\det(I-zA+zeb^{T})}{\det(I-zA)}},$ [24] where e stands for the vector of ones. The function r is called the stability function.[25] It follows from the formula that r is the quotient of two polynomials of degree s if the method has s stages. Explicit methods have a strictly lower triangular matrix A, which implies that det(I − zA) = 1 and that the stability function is a polynomial.[26] The numerical solution to the linear test equation decays to zero if \| r(z) \| < 1 with z = hλ. The set of such z is called the domain of absolute stability. In particular, the method is said to be absolute stable if all z with Re(z) < 0 are in the domain of absolute stability. The stability function of an explicit Runge–Kutta method is a polynomial, so explicit Runge–Kutta methods can never be A-stable.[26] If the method has order p, then the stability function satisfies $r(z)={\textrm {e}}^{z}+O(z^{p+1})$ as $z\to 0$. Thus, it is of interest to study quotients of polynomials of given degrees that approximate the exponential function the best. These are known as Padé approximants. A Padé approximant with numerator of degree m and denominator of degree n is A-stable if and only if m ≤ n ≤ m + 2.[27] The Gauss–Legendre method with s stages has order 2s, so its stability function is the Padé approximant with m = n = s. It follows that the method is A-stable.[28] This shows that A-stable Runge–Kutta can have arbitrarily high order. In contrast, the order of A-stable linear multistep methods cannot exceed two.[29] B-stability The A-stability concept for the solution of differential equations is related to the linear autonomous equation $y'=\lambda y$. Dahlquist proposed the investigation of stability of numerical schemes when applied to nonlinear systems that satisfy a monotonicity condition. The corresponding concepts were defined as G-stability for multistep methods (and the related one-leg methods) and B-stability (Butcher, 1975) for Runge–Kutta methods. A Runge–Kutta method applied to the non-linear system $y'=f(y)$, which verifies $\langle f(y)-f(z),\ y-z\rangle \leq 0$, is called B-stable, if this condition implies $\\|y_{n+1}-z_{n+1}\\|\leq \\|y_{n}-z_{n}\\|$ for two numerical solutions. Let $B$, $M$ and $Q$ be three $s\times s$ matrices defined by $B=\operatorname {diag} (b_{1},b_{2},\ldots ,b_{s}),\,M=BA+A^{T}B-bb^{T},\,Q=BA^{-1}+A^{-T}B-A^{-T}bb^{T}A^{-1}.$ A Runge–Kutta method is said to be algebraically stable[30] if the matrices $B$ and $M$ are both non-negative definite. A sufficient condition for B-stability[31] is: $B$ and $Q$ are non-negative definite. Derivation of the Runge–Kutta fourth-order method In general a Runge–Kutta method of order $s$ can be written as: $y_{t+h}=y_{t}+h\cdot \sum _{i=1}^{s}a_{i}k_{i}+{\mathcal {O}}(h^{s+1}),$ where: $k_{i}=y_{t}+h\cdot \sum _{j=1}^{s}\beta _{ij}f\left(k_{j},\ t_{n}+\alpha _{i}h\right)$ are increments obtained evaluating the derivatives of $y_{t}$ at the $i$-th order. We develop the derivation[32] for the Runge–Kutta fourth-order method using the general formula with $s=4$ evaluated, as explained above, at the starting point, the midpoint and the end point of any interval $(t,\ t+h)$; thus, we choose: ${\begin{aligned}&\alpha _{i}&&\beta _{ij}\\\alpha _{1}&=0&\beta _{21}&={\frac {1}{2}}\\\alpha _{2}&={\frac {1}{2}}&\beta _{32}&={\frac {1}{2}}\\\alpha _{3}&={\frac {1}{2}}&\beta _{43}&=1\\\alpha _{4}&=1&&\\\end{aligned}}$ and $\beta _{ij}=0$ otherwise. We begin by defining the following quantities: ${\begin{aligned}y_{t+h}^{1}&=y_{t}+hf\left(y_{t},\ t\right)\\y_{t+h}^{2}&=y_{t}+hf\left(y_{t+h/2}^{1},\ t+{\frac {h}{2}}\right)\\y_{t+h}^{3}&=y_{t}+hf\left(y_{t+h/2}^{2},\ t+{\frac {h}{2}}\right)\end{aligned}}$ where $y_{t+h/2}^{1}={\dfrac {y_{t}+y_{t+h}^{1}}{2}}$ and $y_{t+h/2}^{2}={\dfrac {y_{t}+y_{t+h}^{2}}{2}}$. If we define: ${\begin{aligned}k_{1}&=f(y_{t},\ t)\\k_{2}&=f\left(y_{t+h/2}^{1},\ t+{\frac {h}{2}}\right)=f\left(y_{t}+{\frac {h}{2}}k_{1},\ t+{\frac {h}{2}}\right)\\k_{3}&=f\left(y_{t+h/2}^{2},\ t+{\frac {h}{2}}\right)=f\left(y_{t}+{\frac {h}{2}}k_{2},\ t+{\frac {h}{2}}\right)\\k_{4}&=f\left(y_{t+h}^{3},\ t+h\right)=f\left(y_{t}+hk_{3},\ t+h\right)\end{aligned}}$ and for the previous relations we can show that the following equalities hold up to ${\mathcal {O}}(h^{2})$: ${\begin{aligned}k_{2}&=f\left(y_{t+h/2}^{1},\ t+{\frac {h}{2}}\right)=f\left(y_{t}+{\frac {h}{2}}k_{1},\ t+{\frac {h}{2}}\right)\\&=f\left(y_{t},\ t\right)+{\frac {h}{2}}{\frac {d}{dt}}f\left(y_{t},\ t\right)\\k_{3}&=f\left(y_{t+h/2}^{2},\ t+{\frac {h}{2}}\right)=f\left(y_{t}+{\frac {h}{2}}f\left(y_{t}+{\frac {h}{2}}k_{1},\ t+{\frac {h}{2}}\right),\ t+{\frac {h}{2}}\right)\\&=f\left(y_{t},\ t\right)+{\frac {h}{2}}{\frac {d}{dt}}\left[f\left(y_{t},\ t\right)+{\frac {h}{2}}{\frac {d}{dt}}f\left(y_{t},\ t\right)\right]\\k_{4}&=f\left(y_{t+h}^{3},\ t+h\right)=f\left(y_{t}+hf\left(y_{t}+{\frac {h}{2}}k_{2},\ t+{\frac {h}{2}}\right),\ t+h\right)\\&=f\left(y_{t}+hf\left(y_{t}+{\frac {h}{2}}f\left(y_{t}+{\frac {h}{2}}f\left(y_{t},\ t\right),\ t+{\frac {h}{2}}\right),\ t+{\frac {h}{2}}\right),\ t+h\right)\\&=f\left(y_{t},\ t\right)+h{\frac {d}{dt}}\left[f\left(y_{t},\ t\right)+{\frac {h}{2}}{\frac {d}{dt}}\left[f\left(y_{t},\ t\right)+{\frac {h}{2}}{\frac {d}{dt}}f\left(y_{t},\ t\right)\right]\right]\end{aligned}}$ where: ${\frac {d}{dt}}f(y_{t},\ t)={\frac {\partial }{\partial y}}f(y_{t},\ t){\dot {y}}_{t}+{\frac {\partial }{\partial t}}f(y_{t},\ t)=f_{y}(y_{t},\ t){\dot {y}}+f_{t}(y_{t},\ t):={\ddot {y}}_{t}$ is the total derivative of $f$ with respect to time. If we now express the general formula using what we just derived we obtain: ${\begin{aligned}y_{t+h}={}&y_{t}+h\left\lbrace a\cdot f(y_{t},\ t)+b\cdot \left[f(y_{t},\ t)+{\frac {h}{2}}{\frac {d}{dt}}f(y_{t},\ t)\right]\right.+\\&{}+c\cdot \left[f(y_{t},\ t)+{\frac {h}{2}}{\frac {d}{dt}}\left[f\left(y_{t},\ t\right)+{\frac {h}{2}}{\frac {d}{dt}}f(y_{t},\ t)\right]\right]+\\&{}+d\cdot \left[f(y_{t},\ t)+h{\frac {d}{dt}}\left[f(y_{t},\ t)+{\frac {h}{2}}{\frac {d}{dt}}\left[f(y_{t},\ t)+\left.{\frac {h}{2}}{\frac {d}{dt}}f(y_{t},\ t)\right]\right]\right]\right\rbrace +{\mathcal {O}}(h^{5})\\={}&y_{t}+a\cdot hf_{t}+b\cdot hf_{t}+b\cdot {\frac {h^{2}}{2}}{\frac {df_{t}}{dt}}+c\cdot hf_{t}+c\cdot {\frac {h^{2}}{2}}{\frac {df_{t}}{dt}}+\\&{}+c\cdot {\frac {h^{3}}{4}}{\frac {d^{2}f_{t}}{dt^{2}}}+d\cdot hf_{t}+d\cdot h^{2}{\frac {df_{t}}{dt}}+d\cdot {\frac {h^{3}}{2}}{\frac {d^{2}f_{t}}{dt^{2}}}+d\cdot {\frac {h^{4}}{4}}{\frac {d^{3}f_{t}}{dt^{3}}}+{\mathcal {O}}(h^{5})\end{aligned}}$ and comparing this with the Taylor series of $y_{t+h}$ around $t$: ${\begin{aligned}y_{t+h}&=y_{t}+h{\dot {y}}_{t}+{\frac {h^{2}}{2}}{\ddot {y}}_{t}+{\frac {h^{3}}{6}}y_{t}^{(3)}+{\frac {h^{4}}{24}}y_{t}^{(4)}+{\mathcal {O}}(h^{5})=\\&=y_{t}+hf(y_{t},\ t)+{\frac {h^{2}}{2}}{\frac {d}{dt}}f(y_{t},\ t)+{\frac {h^{3}}{6}}{\frac {d^{2}}{dt^{2}}}f(y_{t},\ t)+{\frac {h^{4}}{24}}{\frac {d^{3}}{dt^{3}}}f(y_{t},\ t)\end{aligned}}$ we obtain a system of constraints on the coefficients: ${\begin{cases}&a+b+c+d=1\\[6pt]&{\frac {1}{2}}b+{\frac {1}{2}}c+d={\frac {1}{2}}\\[6pt]&{\frac {1}{4}}c+{\frac {1}{2}}d={\frac {1}{6}}\\[6pt]&{\frac {1}{4}}d={\frac {1}{24}}\end{cases}}$ which when solved gives $a={\frac {1}{6}},b={\frac {1}{3}},c={\frac {1}{3}},d={\frac {1}{6}}$ as stated above. See also • Euler's method • List of Runge–Kutta methods • Numerical methods for ordinary differential equations • Runge–Kutta method (SDE) • General linear methods • Lie group integrator Notes 1. "Runge-Kutta method". Dictionary.com. Retrieved 4 April 2021. 2. DEVRIES, Paul L. ; HASBUN, Javier E. A first course in computational physics. Second edition. Jones and Bartlett Publishers: 2011. p. 215. 3. Press et al. 2007, p. 908; Süli & Mayers 2003, p. 328 4. Atkinson (1989, p. 423), Hairer, Nørsett & Wanner (1993, p. 134), Kaw & Kalu (2008, §8.4) and Stoer & Bulirsch (2002, p. 476) leave out the factor h in the definition of the stages. Ascher & Petzold (1998, p. 81), Butcher (2008, p. 93) and Iserles (1996, p. 38) use the y values as stages. 5. Süli & Mayers 2003, p. 328 6. Press et al. 2007, p. 907 7. Iserles 1996, p. 38 8. Iserles 1996, p. 39 9. Iserles 1996, p. 39 10. As a counterexample, consider any explicit 2-stage Runge-Kutta scheme with $b_{1}=b_{2}=1/2$ and $c_{1}$ and $a_{21}$ randomly chosen. This method is consistent and (in general) first-order convergent. On the other hand, the 1-stage method with $b_{1}=1/2$ is inconsistent and fails to converge, even though it trivially holds that $\sum _{j=1}^{i-1}a_{ij}=c_{i}{\text{ for }}i=2,\ldots ,s.$. 11. Butcher 2008, p. 187 12. Butcher 2008, pp. 187–196 13. Süli & Mayers 2003, p. 352 14. Hairer, Nørsett & Wanner (1993, p. 138) refer to Kutta (1901). 15. Süli & Mayers 2003, p. 327 16. Lambert 1991, p. 278 17. Dormand, J. R.; Prince, P. J. (October 1978). "New Runge–Kutta Algorithms for Numerical Simulation in Dynamical Astronomy". Celestial Mechanics. 18 (3): 223–232. Bibcode:1978CeMec..18..223D. doi:10.1007/BF01230162. S2CID 120974351. 18. Fehlberg, E. (October 1974). Classical seventh-, sixth-, and fifth-order Runge–Kutta–Nyström formulas with stepsize control for general second-order differential equations (Report) (NASA TR R-432 ed.). Marshall Space Flight Center, AL: National Aeronautics and Space Administration. 19. Süli & Mayers 2003, pp. 349–351 20. Iserles 1996, p. 41; Süli & Mayers 2003, pp. 351–352 21. Süli & Mayers 2003, p. 353 22. Iserles 1996, pp. 43–44 23. Iserles 1996, p. 47 24. Hairer & Wanner 1996, pp. 40–41 25. Hairer & Wanner 1996, p. 40 26. Iserles 1996, p. 60 27. Iserles 1996, pp. 62–63 28. Iserles 1996, p. 63 29. This result is due to Dahlquist (1963). 30. Lambert 1991, p. 275 31. Lambert 1991, p. 274 32. Lyu, Ling-Hsiao (August 2016). "Appendix C. Derivation of the Numerical Integration Formulae" (PDF). Numerical Simulation of Space Plasmas (I) Lecture Notes. Institute of Space Science, National Central University. Retrieved 17 April 2022. References • Runge, Carl David Tolmé (1895), "Über die numerische Auflösung von Differentialgleichungen", Mathematische Annalen, Springer, 46 (2): 167–178, doi:10.1007/BF01446807, S2CID 119924854. • Kutta, Wilhelm (1901), "Beitrag zur näherungsweisen Integration totaler Differentialgleichungen", Zeitschrift für Mathematik und Physik, 46: 435–453. • Ascher, Uri M.; Petzold, Linda R. (1998), Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations, Philadelphia: Society for Industrial and Applied Mathematics, ISBN 978-0-89871-412-8. • Atkinson, Kendall A. (1989), An Introduction to Numerical Analysis (2nd ed.), New York: John Wiley & Sons, ISBN 978-0-471-50023-0. • Butcher, John C. (May 1963), "Coefficients for the study of Runge-Kutta integration processes", Journal of the Australian Mathematical Society, 3 (2): 185–201, doi:10.1017/S1446788700027932. • Butcher, John C. (May 1964), "On Runge-Kutta processes of high order", Journal of the Australian Mathematical Society, 4 (2): 179–194, doi:10.1017/S1446788700023387 • Butcher, John C. (1975), "A stability property of implicit Runge-Kutta methods", BIT, 15 (4): 358–361, doi:10.1007/bf01931672, S2CID 120854166. • Butcher, John C. (2000), "Numerical methods for ordinary differential equations in the 20th century", J. Comput. Appl. Math., 125 (1–2): 1–29, doi:10.1016/S0377-0427(00)00455-6. • Butcher, John C. (2008), Numerical Methods for Ordinary Differential Equations, New York: John Wiley & Sons, ISBN 978-0-470-72335-7. • Cellier, F.; Kofman, E. (2006), Continuous System Simulation, Springer Verlag, ISBN 0-387-26102-8. • Dahlquist, Germund (1963), "A special stability problem for linear multistep methods", BIT, 3: 27–43, doi:10.1007/BF01963532, hdl:10338.dmlcz/103497, ISSN 0006-3835, S2CID 120241743. • Forsythe, George E.; Malcolm, Michael A.; Moler, Cleve B. (1977), Computer Methods for Mathematical Computations, Prentice-Hall (see Chapter 6). • Hairer, Ernst; Nørsett, Syvert Paul; Wanner, Gerhard (1993), Solving ordinary differential equations I: Nonstiff problems, Berlin, New York: Springer-Verlag, ISBN 978-3-540-56670-0. • Hairer, Ernst; Wanner, Gerhard (1996), Solving ordinary differential equations II: Stiff and differential-algebraic problems (2nd ed.), Berlin, New York: Springer-Verlag, ISBN 978-3-540-60452-5. • Iserles, Arieh (1996), A First Course in the Numerical Analysis of Differential Equations, Cambridge University Press, ISBN 978-0-521-55655-2. • Lambert, J.D (1991), Numerical Methods for Ordinary Differential Systems. The Initial Value Problem, John Wiley & Sons, ISBN 0-471-92990-5 • Kaw, Autar; Kalu, Egwu (2008), Numerical Methods with Applications (1st ed.), autarkaw.com. • Press, William H.; Teukolsky, Saul A.; Vetterling, William T.; Flannery, Brian P. (2007), "Section 17.1 Runge-Kutta Method", Numerical Recipes: The Art of Scientific Computing (3rd ed.), Cambridge University Press, ISBN 978-0-521-88068-8. Also, Section 17.2. Adaptive Stepsize Control for Runge-Kutta. • Stoer, Josef; Bulirsch, Roland (2002), Introduction to Numerical Analysis (3rd ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-387-95452-3. • Süli, Endre; Mayers, David (2003), An Introduction to Numerical Analysis, Cambridge University Press, ISBN 0-521-00794-1. • Tan, Delin; Chen, Zheng (2012), "On A General Formula of Fourth Order Runge-Kutta Method" (PDF), Journal of Mathematical Science & Mathematics Education, 7 (2): 1–10. • advance discrete maths ignou reference book (code- mcs033) • John C. Butcher: "B-Series : Algebraic Analysis of Numerical Methods", Springer（SSCM, volume 55), ISBN 978-3030709556 (April, 2021). External links • "Runge-Kutta method", Encyclopedia of Mathematics, EMS Press, 2001 [1994] • Runge–Kutta 4th-Order Method • Tracker Component Library Implementation in Matlab — Implements 32 embedded Runge Kutta algorithms in RungeKStep, 24 embedded Runge-Kutta Nyström algorithms in RungeKNystroemSStep and 4 general Runge-Kutta Nyström algorithms in RungeKNystroemGStep. Numerical methods for integration First-order methods • Euler method • Backward Euler • Semi-implicit Euler • Exponential Euler Second-order methods • Verlet integration • Velocity Verlet • Trapezoidal rule • Beeman's algorithm • Midpoint method • Heun's method • Newmark-beta method • Leapfrog integration Higher-order methods • Exponential integrator • Runge–Kutta methods • List of Runge–Kutta methods • Linear multistep method • General linear methods • Backward differentiation formula • Yoshida • Gauss–Legendre method Theory • Symplectic integrator	Wikipedia
The Schoolmaster's Assistant, Being a Compendium of Arithmetic Both Practical and Theoretical The Schoolmaster's Assistant, Being a Compendium of Arithmetic both Practical and Theoretical was an early and popular English arithmetic textbook, written by Thomas Dilworth and first published in England in 1743. An American edition was published in 1769; by 1786 it had reached 23 editions, and through 1800 it was the most popular mathematics text in America.[1] The Schoolmaster's Assistant, Being a Compendium of Arithmetic Both Practical and Theoretical AuthorThomas Dilworth CountryEngland SubjectArithmetic Published1743 Although different editions of the book varied in content according to the whims of their publishers, most editions of the book reached from the introductory topics to the advanced in five sections:[2] • Section I, Whole Numbers included the basis of the four operations and proceeded to topics on interest, rebates, partnership, weights and measures, the double rule of three, alligation, mediation and permutations. • Section II dealt with common fractions. • Section III dealt with decimal fraction operations and included roots up to the fourth power, and work on annuities and pensions. • Section IV was a collection of 104 word problems to be solved. As was common in many older texts, the questions were sometimes stated in rhyme. Lessons for students were for memorization and recitation. • Section V was on duodecimals, working with fractions in which the only denominators were twelfths. These types of problems continue in textbooks and appear in the 1870 edition of White's Complete Arithmetic in the appendix. The definition states: A Duodecimal is a denominate number in which twelve units of any denomination make a unit of the next higher denomination. Duodecimals are used by artificers in measuring surfaces and solids. References 1. Monroe, Walter S. (September 1912), "A Chapter in the Development of Arithmetic Teaching in the United States. III", The Elementary School Teacher, 13 (1): 17–24, JSTOR 993627. 2. Nietz, John A. (1967), "Evolution of old secondary-school arithmetic textbooks", The Mathematics Teacher, 60 (4): 387–393, JSTOR 27957584.	Wikipedia
Charles Hutton Charles Hutton FRS FRSE LLD (14 August 1737 – 27 January 1823) was an English mathematician and surveyor. He was professor of mathematics at the Royal Military Academy, Woolwich from 1773 to 1807. He is remembered for his calculation of the density of the earth from Nevil Maskelyne's measurements collected during the Schiehallion experiment. Charles Hutton Born14 August 1737 Newcastle upon Tyne, England Died27 January 1823(1823-01-27) (aged 85) London, England, UK NationalityBritish AwardsCopley Medal 1778 Scientific career Fieldsmathematics InstitutionsRoyal Military Academy InfluencedJohn Scott Life Hutton was born on Percy Street in Newcastle upon Tyne[1] in the north of England, the son of a superintendent of mines, who died when he was still very young.[2] He was educated at a school at Jesmond, kept by Mr Ivison, an Anglican clergyman. There is reason to believe, on the evidence of two pay-bills, that for a short time in 1755 and 1756 Hutton worked in the colliery at Old Long Benton. Following Ivison's promotion to a church living, Hutton took over the Jesmond school, which, in consequence of his increasing number of pupils, he relocated to nearby Stotes Hall, since demolished. While he taught during the day at Stotes Hall, which overlooked Jesmond Dene, he studied mathematics in the evening at a school in Newcastle. In 1760 he married, and began teaching on a larger scale in Newcastle, where his pupils included John Scott, later Lord Eldon, who became Lord High Chancellor of Great Britain.[3] In 1764 Hutton published his first work, The Schoolmasters Guide, or a Complete System of Practical Arithmetic, which was followed by his Treatise on Mensuration both in Theory and Practice in 1770.[3] At around this time he was employed by the mayor and corporation of Newcastle to make a survey of the town and its environs. He drew up a map for the corporation; a smaller one, of the town only, was engraved and published.[4] In 1772 he brought out a tract on The Principles of Bridges, a subject suggested by the destruction of the sole Newcastle bridge by the Great Flood of 1771.[3] Hutton left Newcastle in 1773, following his appointment as professor of mathematics at the Royal Military Academy, Woolwich.[3] He was elected a Fellow of the Royal Society in July, 1774[5] He was asked by the society to perform the calculations necessary to work out the mass and density of the earth from the results of the Schiehallion experiment – a set of observations of the gravitational pull of a mountain in Perthshire made by the Astronomer Royal, Nevil Maskelyne,[6] in 1774–76.[3] Hutton's results appeared in the society's Philosophical Transactions for 1778, and were later reprinted in the second volume of Hutton's Tracts on Mathematical and Philosophical Subjects. His work on the question procured for him the degree of LL.D. from the University of Edinburgh. He became the foreign secretary of the Royal Society in 1779. His resignation from the society in 1783 was brought about by tensions between its president Sir Joseph Banks and the mathematicians amongst its members.[3] He was elected a Foreign Honorary Member of the American Academy of Arts and Sciences in 1788.[7] While working on the Schiehallion experiment, Hutton recorded 23 Gaelic place-names on or near his measurement contour. Less than half are to be found on the modern Ordnance Survey map.[8] After his Tables of the Products and Powers of Numbers, 1781, and his Mathematical Tables of 1785 (second edition 1794), Hutton issued, for the use of the Royal Military Academy, in 1787 Elements of Conic Sections, and in 1798 his Course of Mathematics. His Mathematical and Philosophical Dictionary, a valuable contribution to scientific biography, was published in 1795 and the four volumes of Recreations in Mathematics and Natural Philosophy, mostly translated from the French, in 1803. One of his most laborious works was the abridgment, in conjunction with G. Shaw and R. Pearson, of the Royal Society's Philosophical Transactions. This undertaking, the mathematical and scientific parts of which fell to Hutton, was completed in 1809, and filled 18 quarto volumes.[3] From 1764 he contributed to The Ladies' Diary (a poetical and mathematical almanac established in 1704), and became its editor in 1773–4, retaining the post until 1817.[9] He had previously begun a small periodical called Miscellane Mathematica, of which only 13 numbers appeared; he subsequently published five volumes of The Diarian Miscellany which contained substantial extracts from the Diary.[3] Due to ill health, Hutton resigned his professorship in 1807,[3] although he served as the principal examiner of the Royal Military Academy, and also to the Addiscombe Military Seminary for some years after his retirement. The Board of Ordnance had granted him a pension of £500 a year.[2] During his last years, he worked on new editions of his earlier works.[10] He died on 27 January 1823, and was buried in the family vault at Charlton, in Kent.[2] During the last year of his life a group of his friends set up a fund to pay to have a marble bust made of him. It was executed by the sculptor Sebastian Gahagan. The subscription exceeded the amount necessary, and a medal was also produced, engraved by Benjamin Wyon, showing Hutton's head on one side and emblems representing his discoveries about the force of gunpowder, and the density of the earth on the other.[2] References 1. Biographical Index of Former Fellows of the Royal Society of Edinburgh 1783–2002 (PDF). The Royal Society of Edinburgh. July 2006. ISBN 0-902-198-84-X. Archived from the original (PDF) on 24 January 2013. Retrieved 21 November 2016. 2. "Charles Hutton, LL.D. F.R.S." The European Magazine, and London Review. 83: 482–7. June 1823. 3. Chisholm 1911. 4. Bruce 1823, p.13 5. "Library and Archive Catalogue". Royal Society. Retrieved 24 November 2010. 6. "Background to Boys' experiment to determine G". University of Oxford Department of Physics. Archived from the original on 24 September 2015. Retrieved 15 March 2013. 7. "Book of Members, 1780–2010: Chapter H" (PDF). American Academy of Arts and Sciences. Retrieved 28 July 2014. 8. Murray, John (2019), Reading the Gaelic Landscape: Leughadh Aghaidh na Tire, Whittles Publishing, pp. 23 & 24. 9. Niccolò Guicciardini, 'Hutton, Charles (1737–1823)’, Oxford Dictionary of National Biography, Oxford University Press, 2004 accessed 9 April 2015 10. Bruce 1823, p.27 Sources • Bruce, John (1823). A Memoir of Charles Hutton LLD FRS. Newcastle. • Wardhaugh, Benjamin (2019). Gunpowder & Geometry. The Life of Charles Hutton: Pit Boy, Mathematician and Scientific Rebel. London: William Collins. • This article incorporates text from a publication now in the public domain: Chisholm, Hugh, ed. (1911). "Hutton, Charles". Encyclopædia Britannica. Vol. 14 (11th ed.). Cambridge University Press. p. 15. Works • A mathematical and philosophical dictionary Vol. I London : Printed by J. Davis for J. Johnson and G. G. and J. Robinson 1795 at Internet Archive • A mathematical and philosophical dictionary Vol. II London : Printed by J. Davis for J. Johnson and G. G. and J. Robinson 1795 at Internet Archive • A mathematical and philosophical dictionary Vols. I and II London : Printed by J. Davis for J. Johnson and G. G. and J. Robinson 1795 at the Archimedes Project • A mathematical and philosophical dictionary Vol. I London, Printed for the author [etc.] 1815 at Internet Archive • A mathematical and philosophical dictionary Vol. II London, Printed for the author [etc.] 1815 at Google Books • Charles Hutton Tracts on Mathematical and Philosophical Subjects (F. & C. Rivington, London, 1812) • Charles Hutton A Course of Mathematics For the Use of Academies... (volume 1) (Campbell & sons, New York, 1825) • Charles Hutton A Course of Mathematics For the Use of Academies... (volume 2) (Dean, New York, 1831) • Charles Hutton A Treatise on Mensuration both in Theory and in practice (Newcastle upon Tyne, 1770) • Charles Hutton Mathematical tables (F. & C. Rivington, London, 1811) External links Wikimedia Commons has media related to Charles Hutton. • Works by or about Charles Hutton at Wikisource • O'Connor, John J.; Robertson, Edmund F., "Charles Hutton", MacTutor History of Mathematics Archive, University of St Andrews • The Correspondence of Charles Hutton in EMLO Copley Medallists (1751–1800) • John Canton (1751) • John Pringle (1752) • Benjamin Franklin (1753) • William Lewis (1754) • John Huxham (1755) • Charles Cavendish (1757) • John Dollond (1758) • John Smeaton (1759) • Benjamin Wilson (1760) • John Canton (1764) • William Brownrigg / Edward Delaval / Henry Cavendish (1766) • John Ellis (1767) • Peter Woulfe (1768) • William Hewson (1769) • William Hamilton (1770) • Matthew Raper (1771) • Joseph Priestley (1772) • John Walsh (1773) • Nevil Maskelyne (1775) • James Cook (1776) • John Mudge (1777) • Charles Hutton (1778) • Samuel Vince (1780) • William Herschel (1781) • Richard Kirwan (1782) • John Goodricke / Thomas Hutchins (1783) • Edward Waring (1784) • William Roy (1785) • John Hunter (1787) • Charles Blagden (1788) • William Morgan (1789) • James Rennell / Jean-André Deluc (1791) • Benjamin Thompson (1792) • Alessandro Volta (1794) • Jesse Ramsden (1795) • George Atwood (1796) • George Shuckburgh-Evelyn / Charles Hatchett (1798) • John Hellins (1799) • Edward Charles Howard (1800) Authority control International • FAST • ISNI • VIAF • 2 National • Norway • Spain • France • BnF data • Germany • Israel • Belgium • United States • Japan • Australia • Netherlands • Poland • Portugal • Vatican Academics • MathSciNet • zbMATH People • Deutsche Biographie • Trove Other • SNAC • IdRef	Wikipedia
The Secrets of Triangles The Secrets of Triangles: A Mathematical Journey is a popular mathematics book on the geometry of triangles. It was written by Alfred S. Posamentier and Ingmar Lehmann, and published in 2012 by Prometheus Books. Topics The book consists of ten chapters,[1] with the first six concentrating on triangle centers while the final four cover more diverse topics including the area of triangles, inequalities involving triangles, straightedge and compass constructions, and fractals.[2] Beyond the classical triangle centers (the circumcenter, incenter, orthocenter, and centroid) the book covers other centers including the Brocard points, Fermat point, Gergonne point, and other geometric objects associated with triangle centers such as the Euler line, Simson line, and nine-point circle.[1] The chapter on areas includes both trigonometric formulas and Heron's formula for computing the area of a triangle from its side lengths, and the chapter on inequalities includes the Erdős–Mordell inequality on sums of distances from the sides of a triangle and Weitzenböck's inequality relating the area of a triangle to that of squares on its sides.[2] Under constructions, the book considers 95 different triples of elements from which a triangle's shape may be determined (taken from its side lengths, angles, medians, heights, or angle bisectors) and describes how to find a triangle with each combination for which this is possible.[3] Triangle-related fractals in the final chapter include the Sierpiński triangle and Koch snowflake.[2] Audience and reception Reviewer Alasdair McAndrew criticizes the book as being too "breathless" in its praise of the geometry it discusses and too superficial to be of interest to professional mathematicians,[2] and Patricia Baggett writes that it little of its content would be of use in high school mathematics education. However, Baggett suggests that it may be usable as a reference work,[4] and similarly Robert Dawson suggests using its chapter on inequalities in this way.[5] The book is written at a level suitable for high school students and interested amateurs,[1][3] and McAndrew recommends the book to them.[2] Both Baggett and Gerry Leversha find the chapter on fractals (written by Robert A. Chaffer)[6] to be the weakest part of the book,[1][4] and Joop van der Vaart calls this chapter interesting but not a good fit for the rest of the book.[3] Leversha calls the chapter on area "a bit of a mish-mash". Otherwise, Baggett evaluates the book as "well written and well illustrated", although lacking a glossary.[4] Robert Dawson calls the book "very readable", and recommends it to any mathematics library.[5] See also • Encyclopedia of Triangle Centers • 99 Points of Intersection References 1. Leversha, Gerry (July 2014), "Review of The Secrets of Triangles", The Mathematical Gazette, 98 (542): 371–373, doi:10.1017/s0025557200001571, JSTOR 24496691 2. McAndrew, Alasdair (2014), "Review of The Secrets of Triangles", Australian Mathematical Society Gazette, 41 (4): 244–247 3. van der Vaart, Joop (2018), "Review of The Secrets of Triangles" (PDF), Nieuw Archief voor Wiskunde, 5th series (in Dutch), 19 (1): 60–61 4. Baggett, Patricia (2014), "Review of The Secrets of Triangles", The Mathematics Teacher, 107 (8): 636, doi:10.5951/mathteacher.107.8.0636, JSTOR 10.5951/mathteacher.107.8.0636 5. Dawson, Robert, "Review of The Secrets of Triangles", zbMATH, Zbl 1266.00008 6. Bumcrot, R. J., "Review of The Secrets of Triangles", MathSciNet, MR 2963520	Wikipedia
Shortest path problem In graph theory, the shortest path problem is the problem of finding a path between two vertices (or nodes) in a graph such that the sum of the weights of its constituent edges is minimized. The problem of finding the shortest path between two intersections on a road map may be modeled as a special case of the shortest path problem in graphs, where the vertices correspond to intersections and the edges correspond to road segments, each weighted by the length of the segment. Definition The shortest path problem can be defined for graphs whether undirected, directed, or mixed. It is defined here for undirected graphs; for directed graphs the definition of path requires that consecutive vertices be connected by an appropriate directed edge. Two vertices are adjacent when they are both incident to a common edge. A path in an undirected graph is a sequence of vertices $P=(v_{1},v_{2},\ldots ,v_{n})\in V\times V\times \cdots \times V$ such that $v_{i}$ is adjacent to $v_{i+1}$ for $1\leq i<n$. Such a path $P$ is called a path of length $n-1$ from $v_{1}$ to $v_{n}$. (The $v_{i}$ are variables; their numbering here relates to their position in the sequence and needs not to relate to any canonical labeling of the vertices.) Let $e_{i,j}$ be the edge incident to both $v_{i}$ and $v_{j}$. Given a real-valued weight function $f:E\rightarrow \mathbb {R} $, and an undirected (simple) graph $G$, the shortest path from $v$ to $v'$ is the path $P=(v_{1},v_{2},\ldots ,v_{n})$ (where $v_{1}=v$ and $v_{n}=v'$) that over all possible $n$ minimizes the sum $\sum _{i=1}^{n-1}f(e_{i,i+1}).$ When each edge in the graph has unit weight or $f:E\rightarrow \{1\}$, this is equivalent to finding the path with fewest edges. The problem is also sometimes called the single-pair shortest path problem, to distinguish it from the following variations: • The single-source shortest path problem, in which we have to find shortest paths from a source vertex v to all other vertices in the graph. • The single-destination shortest path problem, in which we have to find shortest paths from all vertices in the directed graph to a single destination vertex v. This can be reduced to the single-source shortest path problem by reversing the arcs in the directed graph. • The all-pairs shortest path problem, in which we have to find shortest paths between every pair of vertices v, v' in the graph. These generalizations have significantly more efficient algorithms than the simplistic approach of running a single-pair shortest path algorithm on all relevant pairs of vertices. Algorithms The most important algorithms for solving this problem are: • Dijkstra's algorithm solves the single-source shortest path problem with non-negative edge weight. • Bellman–Ford algorithm solves the single-source problem if edge weights may be negative. • A* search algorithm solves for single-pair shortest path using heuristics to try to speed up the search. • Floyd–Warshall algorithm solves all pairs shortest paths. • Johnson's algorithm solves all pairs shortest paths, and may be faster than Floyd–Warshall on sparse graphs. • Viterbi algorithm solves the shortest stochastic path problem with an additional probabilistic weight on each node. Additional algorithms and associated evaluations may be found in Cherkassky, Goldberg & Radzik (1996). Single-source shortest paths Undirected graphs WeightsTime complexityAuthor $\mathbb {R} $+O(V2)Dijkstra 1959 $\mathbb {R} $+O((E + V) log V)Johnson 1977 (binary heap) $\mathbb {R} $+O(E + V log V)Fredman & Tarjan 1984 (Fibonacci heap) $\mathbb {N} $O(E)Thorup 1999 (requires constant-time multiplication) Unweighted graphs AlgorithmTime complexityAuthor Breadth-first searchO(E + V) Directed acyclic graphs (DAGs) An algorithm using topological sorting can solve the single-source shortest path problem in time Θ(E + V) in arbitrarily-weighted DAGs.[1] Directed graphs with nonnegative weights The following table is taken from Schrijver (2004), with some corrections and additions. A green background indicates an asymptotically best bound in the table; L is the maximum length (or weight) among all edges, assuming integer edge weights. WeightsAlgorithmTime complexityAuthor $\mathbb {R} $$O(V^{2}EL)$Ford 1956 $\mathbb {R} $Bellman–Ford algorithm$O(VE)$Shimbel 1955, Bellman 1958, Moore 1959 $\mathbb {R} $$O(V^{2}\log {V})$Dantzig 1960 $\mathbb {R} $Dijkstra's algorithm with list$O(V^{2})$Leyzorek et al. 1957, Dijkstra 1959, Minty (see Pollack & Wiebenson 1960), Whiting & Hillier 1960 $\mathbb {R} $Dijkstra's algorithm with binary heap$O((E+V)\log {V})$Johnson 1977 $\mathbb {R} $Dijkstra's algorithm with Fibonacci heap$O(E+V\log {V})$Fredman & Tarjan 1984, Fredman & Tarjan 1987 $\mathbb {N} $Dial's algorithm[2] (Dijkstra's algorithm using a bucket queue with L buckets)$O(E+LV)$Dial 1969 $O(E\log {\log {L}})$Johnson 1981, Karlsson & Poblete 1983 Gabow's algorithm$O(E\log _{E/V}L)$Gabow 1983, Gabow 1985 $O(E+V{\sqrt {\log {L}}})$Ahuja et al. 1990 $\mathbb {N} $Thorup$O(E+V\log {\log {V}})$Thorup 2004 Directed graphs with arbitrary weights without negative cycles WeightsAlgorithmTime complexityAuthor $\mathbb {R} $O(V 2EL)Ford 1956 $\mathbb {R} $Bellman–Ford algorithmO(VE)Shimbel 1955, Bellman 1958, Moore 1959 $\mathbb {R} $Johnson-Dijkstra with binary heapO(V (E + log V))Johnson 1977 $\mathbb {R} $Johnson-Dijkstra with Fibonacci heapO(V (E + log V))Fredman & Tarjan 1984, Fredman & Tarjan 1987, adapted after Johnson 1977 $\mathbb {N} $Johnson's technique applied to Dial's algorithm[2]O(V (E + L))Dial 1969, adapted after Johnson 1977 Directed graphs with arbitrary weights with negative cycles Finds a negative cycle or calculates distances to all vertices. WeightsAlgorithmTime complexityAuthor $\mathbb {Z} $Andrew V. Goldberg$O(E{\sqrt {V}}\log {N})$ Planar graphs with nonnegative weights WeightsAlgorithmTime complexityAuthor $\mathbb {R} _{\geq 0}$$O(V)$Henzinger et al. 1997 All-pairs shortest paths The all-pairs shortest path problem finds the shortest paths between every pair of vertices v, v' in the graph. The all-pairs shortest paths problem for unweighted directed graphs was introduced by Shimbel (1953), who observed that it could be solved by a linear number of matrix multiplications that takes a total time of O(V4). Undirected graph WeightsTime complexityAlgorithm $\mathbb {R} $+O(V3)Floyd–Warshall algorithm $\{1,\infty \}$$O(V^{\omega }\log V)$Seidel's algorithm (expected running time) $\mathbb {N} $$O(V^{3}/2^{\Omega (\log n)^{1/2}})$Williams 2014 $\mathbb {R} $+O(EV log α(E,V))Pettie & Ramachandran 2002 $\mathbb {N} $O(EV)Thorup 1999 applied to every vertex (requires constant-time multiplication). Directed graph WeightsTime complexityAlgorithm $\mathbb {R} $ (no negative cycles)O(V3)Floyd–Warshall algorithm $\mathbb {N} $$O(V^{3}/2^{\Omega (\log n)^{1/2}})$Williams 2014 $\mathbb {R} $ (no negative cycles)O(EV + V2 log V)Johnson–Dijkstra $\mathbb {R} $ (no negative cycles)O(EV + V2 log log V)Pettie 2004 $\mathbb {N} $O(EV + V2 log log V)Hagerup 2000 Applications Shortest path algorithms are applied to automatically find directions between physical locations, such as driving directions on web mapping websites like MapQuest or Google Maps. For this application fast specialized algorithms are available.[3] If one represents a nondeterministic abstract machine as a graph where vertices describe states and edges describe possible transitions, shortest path algorithms can be used to find an optimal sequence of choices to reach a certain goal state, or to establish lower bounds on the time needed to reach a given state. For example, if vertices represent the states of a puzzle like a Rubik's Cube and each directed edge corresponds to a single move or turn, shortest path algorithms can be used to find a solution that uses the minimum possible number of moves. In a networking or telecommunications mindset, this shortest path problem is sometimes called the min-delay path problem and usually tied with a widest path problem. For example, the algorithm may seek the shortest (min-delay) widest path, or widest shortest (min-delay) path. A more lighthearted application is the games of "six degrees of separation" that try to find the shortest path in graphs like movie stars appearing in the same film. Other applications, often studied in operations research, include plant and facility layout, robotics, transportation, and VLSI design.[4] Road networks A road network can be considered as a graph with positive weights. The nodes represent road junctions and each edge of the graph is associated with a road segment between two junctions. The weight of an edge may correspond to the length of the associated road segment, the time needed to traverse the segment, or the cost of traversing the segment. Using directed edges it is also possible to model one-way streets. Such graphs are special in the sense that some edges are more important than others for long-distance travel (e.g. highways). This property has been formalized using the notion of highway dimension.[5] There are a great number of algorithms that exploit this property and are therefore able to compute the shortest path a lot quicker than would be possible on general graphs. All of these algorithms work in two phases. In the first phase, the graph is preprocessed without knowing the source or target node. The second phase is the query phase. In this phase, source and target node are known. The idea is that the road network is static, so the preprocessing phase can be done once and used for a large number of queries on the same road network. The algorithm with the fastest known query time is called hub labeling and is able to compute shortest path on the road networks of Europe or the US in a fraction of a microsecond.[6] Other techniques that have been used are: • ALT (A* search, landmarks, and triangle inequality) • Arc flags • Contraction hierarchies • Transit node routing • Reach-based pruning • Labeling • Hub labels Related problems For shortest path problems in computational geometry, see Euclidean shortest path. The shortest multiple disconnected path [7] is a representation of the primitive path network within the framework of Reptation theory. The widest path problem seeks a path so that the minimum label of any edge is as large as possible. Other related problems may be classified into the following categories. Paths with constraints Unlike the shortest path problem, which can be solved in polynomial time in graphs without negative cycles, shortest path problems which include additional constraints on the desired solution path are called Constrained Shortest Path First, and are harder to solve. One example is the constrained shortest path problem,[8] which attempts to minimize the total cost of the path while at the same time maintaining another metric below a given threshold. This makes the problem NP-complete (such problems are not believed to be efficiently solvable for large sets of data, see P = NP problem). Another NP-complete example requires a specific set of vertices to be included in the path,[9] which makes the problem similar to the Traveling Salesman Problem (TSP). The TSP is the problem of finding the shortest path that goes through every vertex exactly once, and returns to the start. The problem of finding the longest path in a graph is also NP-complete. Partial observability The Canadian traveller problem and the stochastic shortest path problem are generalizations where either the graph isn't completely known to the mover, changes over time, or where actions (traversals) are probabilistic. [10] [11] Strategic shortest paths Sometimes, the edges in a graph have personalities: each edge has its own selfish interest. An example is a communication network, in which each edge is a computer that possibly belongs to a different person. Different computers have different transmission speeds, so every edge in the network has a numeric weight equal to the number of milliseconds it takes to transmit a message. Our goal is to send a message between two points in the network in the shortest time possible. If we know the transmission-time of each computer (the weight of each edge), then we can use a standard shortest-paths algorithm. If we do not know the transmission times, then we have to ask each computer to tell us its transmission-time. But, the computers may be selfish: a computer might tell us that its transmission time is very long, so that we will not bother it with our messages. A possible solution to this problem is to use a variant of the VCG mechanism, which gives the computers an incentive to reveal their true weights. Negative cycle detection In some cases, the main goal is not to find the shortest path, but only to detect if the graph contains a negative cycle. Some shortest-paths algorithms can be used for this purpose: • The Bellman–Ford algorithm can be used to detect a negative cycle in time $O(\|V\|\|E\|)$. • Cherkassky and Goldberg[12] survey several other algorithms for negative cycle detection. General algebraic framework on semirings: the algebraic path problem Many problems can be framed as a form of the shortest path for some suitably substituted notions of addition along a path and taking the minimum. The general approach to these is to consider the two operations to be those of a semiring. Semiring multiplication is done along the path, and the addition is between paths. This general framework is known as the algebraic path problem.[13][14][15] Most of the classic shortest-path algorithms (and new ones) can be formulated as solving linear systems over such algebraic structures.[16] More recently, an even more general framework for solving these (and much less obviously related problems) has been developed under the banner of valuation algebras.[17] Shortest path in stochastic time-dependent networks In real-life situations, the transportation network is usually stochastic and time-dependent. In fact, a traveler traversing a link daily may experiences different travel times on that link due not only to the fluctuations in travel demand (origin-destination matrix) but also due to such incidents as work zones, bad weather conditions, accidents and vehicle breakdowns. As a result, a stochastic time-dependent (STD) network is a more realistic representation of an actual road network compared with the deterministic one.[18][19] Despite considerable progress during the course of the past decade, it remains a controversial question how an optimal path should be defined and identified in stochastic road networks. In other words, there is no unique definition of an optimal path under uncertainty. One possible and common answer to this question is to find a path with the minimum expected travel time. The main advantage of using this approach is that efficient shortest path algorithms introduced for the deterministic networks can be readily employed to identify the path with the minimum expected travel time in a stochastic network. However, the resulting optimal path identified by this approach may not be reliable, because this approach fails to address travel time variability. To tackle this issue some researchers use distribution of travel time instead of expected value of it so they find the probability distribution of total travelling time using different optimization methods such as dynamic programming and Dijkstra's algorithm .[20] These methods use stochastic optimization, specifically stochastic dynamic programming to find the shortest path in networks with probabilistic arc length.[21] The concept of travel time reliability is used interchangeably with travel time variability in the transportation research literature, so that, in general, one can say that the higher the variability in travel time, the lower the reliability would be, and vice versa. In order to account for travel time reliability more accurately, two common alternative definitions for an optimal path under uncertainty have been suggested. Some have introduced the concept of the most reliable path, aiming to maximize the probability of arriving on time or earlier than a given travel time budget. Others, alternatively, have put forward the concept of an α-reliable path based on which they intended to minimize the travel time budget required to ensure a pre-specified on-time arrival probability. See also • Bidirectional search, an algorithm that finds the shortest path between two vertices on a directed graph • Euclidean shortest path • Flow network • K shortest path routing • Min-plus matrix multiplication • Pathfinding • Shortest Path Bridging • Shortest path tree • TRILL (TRansparent Interconnection of Lots of Links) References Notes 1. Cormen et al. 2001, p. 655 2. Dial, Robert B. (1969). "Algorithm 360: Shortest-Path Forest with Topological Ordering [H]". Communications of the ACM. 12 (11): 632–633. doi:10.1145/363269.363610. S2CID 6754003. 3. Sanders, Peter (March 23, 2009). "Fast route planning". Google Tech Talk. Archived from the original on 2021-12-11. 4. Chen, Danny Z. (December 1996). "Developing algorithms and software for geometric path planning problems". ACM Computing Surveys. 28 (4es). Article 18. doi:10.1145/242224.242246. S2CID 11761485. 5. Abraham, Ittai; Fiat, Amos; Goldberg, Andrew V.; Werneck, Renato F. "Highway Dimension, Shortest Paths, and Provably Efficient Algorithms". ACM-SIAM Symposium on Discrete Algorithms, pages 782–793, 2010. 6. Abraham, Ittai; Delling, Daniel; Goldberg, Andrew V.; Werneck, Renato F. research.microsoft.com/pubs/142356/HL-TR.pdf "A Hub-Based Labeling Algorithm for Shortest Paths on Road Networks". Symposium on Experimental Algorithms, pages 230–241, 2011. 7. Kroger, Martin (2005). "Shortest multiple disconnected path for the analysis of entanglements in two- and three-dimensional polymeric systems". Computer Physics Communications. 168 (3): 209–232. Bibcode:2005CoPhC.168..209K. doi:10.1016/j.cpc.2005.01.020. 8. Lozano, Leonardo; Medaglia, Andrés L (2013). "On an exact method for the constrained shortest path problem". Computers & Operations Research. 40 (1): 378--384. doi:10.1016/j.cor.2012.07.008. 9. Osanlou, Kevin; Bursuc, Andrei; Guettier, Christophe; Cazenave, Tristan; Jacopin, Eric (2019). "Optimal Solving of Constrained Path-Planning Problems with Graph Convolutional Networks and Optimized Tree Search". 2019 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS). p. 3519--3525. arXiv:2108.01036. doi:10.1109/IROS40897.2019.8968113. ISBN 978-1-7281-4004-9. S2CID 210706773. 10. Bar-Noy, Amotz; Schieber, Baruch (1991). "The canadian traveller problem". Proceedings of the Second Annual ACM-SIAM Symposium on Discrete Algorithms: 261–270. CiteSeerX 10.1.1.1088.3015. 11. Nikolova, Evdokia; Karger, David R. "Route planning under uncertainty: the Canadian traveller problem" (PDF). Proceedings of the 23rd National Conference on Artificial Intelligence (AAAI): 969--974. Archived (PDF) from the original on 2022-10-09. 12. Cherkassky, Boris V.; Goldberg, Andrew V. (1999-06-01). "Negative-cycle detection algorithms". Mathematical Programming. 85 (2): 277–311. doi:10.1007/s101070050058. ISSN 1436-4646. S2CID 79739. 13. Pair, Claude (1967). "Sur des algorithmes pour des problèmes de cheminement dans les graphes finis (On algorithms for path problems in finite graphs)". In Rosentiehl, Pierre (ed.). Théorie des graphes (journées internationales d'études) -- Theory of Graphs (international symposium). Rome (Italy), July 1966: Dunod (Paris) et Gordon and Breach (New York). p. 271.{{cite conference}}: CS1 maint: location (link) 14. Derniame, Jean Claude; Pair, Claude (1971). Problèmes de cheminement dans les graphes (Path Problems in Graphs). Dunod (Paris). 15. Baras, John; Theodorakopoulos, George (4 April 2010). Path Problems in Networks. Morgan & Claypool Publishers. pp. 9–. ISBN 978-1-59829-924-3. 16. Gondran, Michel; Minoux, Michel (2008). Graphs, Dioids and Semirings: New Models and Algorithms. Springer Science & Business Media. chapter 4. ISBN 978-0-387-75450-5. 17. Pouly, Marc; Kohlas, Jürg (2011). Generic Inference: A Unifying Theory for Automated Reasoning. John Wiley & Sons. Chapter 6. Valuation Algebras for Path Problems. ISBN 978-1-118-01086-0. 18. Loui, R.P., 1983. Optimal paths in graphs with stochastic or multidimensional weights. Communications of the ACM, 26(9), pp.670-676. 19. Rajabi-Bahaabadi, Mojtaba; Shariat-Mohaymany, Afshin; Babaei, Mohsen; Ahn, Chang Wook (2015). "Multi-objective path finding in stochastic time-dependent road networks using non-dominated sorting genetic algorithm". Expert Systems with Applications. 42 (12): 5056–5064. doi:10.1016/j.eswa.2015.02.046. 20. Olya, Mohammad Hessam (2014). "Finding shortest path in a combined exponential – gamma probability distribution arc length". International Journal of Operational Research. 21 (1): 25–37. doi:10.1504/IJOR.2014.064020. 21. Olya, Mohammad Hessam (2014). "Applying Dijkstra's algorithm for general shortest path problem with normal probability distribution arc length". International Journal of Operational Research. 21 (2): 143–154. doi:10.1504/IJOR.2014.064541. Bibliography • Ahuja, Ravindra K.; Mehlhorn, Kurt; Orlin, James; Tarjan, Robert E. (April 1990). "Faster algorithms for the shortest path problem" (PDF). Journal of the ACM. ACM. 37 (2): 213–223. doi:10.1145/77600.77615. hdl:1721.1/47994. S2CID 5499589. Archived (PDF) from the original on 2022-10-09. • Bellman, Richard (1958). "On a routing problem". Quarterly of Applied Mathematics. 16: 87–90. doi:10.1090/qam/102435. MR 0102435. • Cherkassky, Boris V.; Goldberg, Andrew V.; Radzik, Tomasz (1996). "Shortest paths algorithms: theory and experimental evaluation". Mathematical Programming. Ser. A. 73 (2): 129–174. doi:10.1016/0025-5610(95)00021-6. MR 1392160. • Cormen, Thomas H.; Leiserson, Charles E.; Rivest, Ronald L.; Stein, Clifford (2001) [1990]. "Single-Source Shortest Paths and All-Pairs Shortest Paths". Introduction to Algorithms (2nd ed.). MIT Press and McGraw-Hill. pp. 580–642. ISBN 0-262-03293-7. • Dantzig, G. B. (January 1960). "On the Shortest Route through a Network". Management Science. 6 (2): 187–190. doi:10.1287/mnsc.6.2.187. • Dijkstra, E. W. (1959). "A note on two problems in connexion with graphs". Numerische Mathematik. 1: 269–271. doi:10.1007/BF01386390. S2CID 123284777. • Ford, L. R. (1956). "Network Flow Theory". Rand Corporation. P-923. {{cite journal}}: Cite journal requires \|journal= (help) • Fredman, Michael Lawrence; Tarjan, Robert E. (1984). Fibonacci heaps and their uses in improved network optimization algorithms. 25th Annual Symposium on Foundations of Computer Science. IEEE. pp. 338–346. doi:10.1109/SFCS.1984.715934. ISBN 0-8186-0591-X. • Fredman, Michael Lawrence; Tarjan, Robert E. (1987). "Fibonacci heaps and their uses in improved network optimization algorithms". Journal of the Association for Computing Machinery. 34 (3): 596–615. doi:10.1145/28869.28874. S2CID 7904683. • Gabow, H. N. (1983). "Scaling algorithms for network problems". Proceedings of the 24th Annual Symposium on Foundations of Computer Science (FOCS 1983) (PDF). pp. 248–258. doi:10.1109/SFCS.1983.68. • Gabow, Harold N. (1985). "Scaling algorithms for network problems". Journal of Computer and System Sciences. 31 (2): 148–168. doi:10.1016/0022-0000(85)90039-X. MR 0828519. • Hagerup, Torben (2000). Montanari, Ugo; Rolim, José D. P.; Welzl, Emo (eds.). Improved Shortest Paths on the Word RAM. pp. 61–72. ISBN 978-3-540-67715-4. {{cite book}}: \|journal= ignored (help) • Henzinger, Monika R.; Klein, Philip; Rao, Satish; Subramanian, Sairam (1997). "Faster Shortest-Path Algorithms for Planar Graphs". Journal of Computer and System Sciences. 55 (1): 3–23. doi:10.1006/jcss.1997.1493. • Johnson, Donald B. (1977). "Efficient algorithms for shortest paths in sparse networks". Journal of the ACM. 24 (1): 1–13. doi:10.1145/321992.321993. S2CID 207678246. • Altıntaş, Gökhan (2020). Exact Solutions of Shortest-Path Problems Based on Mechanical Analogies: In Connection with Labyrinths. Amazon Digital Services LLC. p. 97. ISBN 9798655831896. • Johnson, Donald B. (December 1981). "A priority queue in which initialization and queue operations take O(log log D) time". Mathematical Systems Theory. 15 (1): 295–309. doi:10.1007/BF01786986. MR 0683047. S2CID 35703411. • Karlsson, Rolf G.; Poblete, Patricio V. (1983). "An O(m log log D) algorithm for shortest paths". Discrete Applied Mathematics. 6 (1): 91–93. doi:10.1016/0166-218X(83)90104-X. MR 0700028. • Leyzorek, M.; Gray, R. S.; Johnson, A. A.; Ladew, W. C.; Meaker, S. R., Jr.; Petry, R. M.; Seitz, R. N. (1957). Investigation of Model Techniques — First Annual Report — 6 June 1956 — 1 July 1957 — A Study of Model Techniques for Communication Systems. Cleveland, Ohio: Case Institute of Technology.{{cite book}}: CS1 maint: multiple names: authors list (link) • Moore, E. F. (1959). "The shortest path through a maze". Proceedings of an International Symposium on the Theory of Switching (Cambridge, Massachusetts, 2–5 April 1957). Cambridge: Harvard University Press. pp. 285–292. • Pettie, Seth; Ramachandran, Vijaya (2002). Computing shortest paths with comparisons and additions. pp. 267–276. ISBN 978-0-89871-513-2. {{cite book}}: \|journal= ignored (help) • Pettie, Seth (26 January 2004). "A new approach to all-pairs shortest paths on real-weighted graphs". Theoretical Computer Science. 312 (1): 47–74. doi:10.1016/s0304-3975(03)00402-x. • Pollack, Maurice; Wiebenson, Walter (March–April 1960). "Solution of the Shortest-Route Problem—A Review". Oper. Res. 8 (2): 224–230. doi:10.1287/opre.8.2.224. Attributes Dijkstra's algorithm to Minty ("private communication") on p. 225. • Schrijver, Alexander (2004). Combinatorial Optimization — Polyhedra and Efficiency. Algorithms and Combinatorics. Vol. 24. Springer. ISBN 978-3-540-20456-5. Here: vol.A, sect.7.5b, p. 103 • Shimbel, Alfonso (1953). "Structural parameters of communication networks". Bulletin of Mathematical Biophysics. 15 (4): 501–507. doi:10.1007/BF02476438. • Shimbel, A. (1955). Structure in communication nets. Proceedings of the Symposium on Information Networks. New York, NY: Polytechnic Press of the Polytechnic Institute of Brooklyn. pp. 199–203. • Thorup, Mikkel (1999). "Undirected single-source shortest paths with positive integer weights in linear time". Journal of the ACM. 46 (3): 362–394. doi:10.1145/316542.316548. S2CID 207654795. • Thorup, Mikkel (2004). "Integer priority queues with decrease key in constant time and the single source shortest paths problem". Journal of Computer and System Sciences. 69 (3): 330–353. doi:10.1016/j.jcss.2004.04.003. • Whiting, P. D.; Hillier, J. A. (March–June 1960). "A Method for Finding the Shortest Route through a Road Network". Operational Research Quarterly. 11 (1/2): 37–40. doi:10.1057/jors.1960.32. • Williams, Ryan (2014). "Faster all-pairs shortest paths via circuit complexity". Proceedings of the 46th Annual ACM Symposium on Theory of Computing (STOC '14). New York: ACM. pp. 664–673. arXiv:1312.6680. doi:10.1145/2591796.2591811. MR 3238994. Further reading • Frigioni, D.; Marchetti-Spaccamela, A.; Nanni, U. (1998). "Fully dynamic output bounded single source shortest path problem". Proc. 7th Annu. ACM-SIAM Symp. Discrete Algorithms. Atlanta, GA. pp. 212–221. CiteSeerX 10.1.1.32.9856. • Dreyfus, S. E. (October 1967). An Appraisal of Some Shortest Path Algorithms (PDF) (Report). Project Rand. United States Air Force. RM-5433-PR. Archived (PDF) from the original on November 17, 2015. DTIC AD-661265. Graph and tree traversal algorithms • α–β pruning • A* • IDA* • LPA* • SMA* • Best-first search • Beam search • Bidirectional search • Breadth-first search • Lexicographic • Parallel • B* • Depth-first search • Iterative Deepening • D* • Fringe search • Jump point search • Monte Carlo tree search • SSS* Shortest path • Bellman–Ford • Dijkstra's • Floyd–Warshall • Johnson's • Shortest path faster • Yen's Minimum spanning tree • Borůvka's • Kruskal's • Prim's • Reverse-delete List of graph search algorithms Authority control: National • Germany	Wikipedia
Simplicial complex In mathematics, a simplicial complex is a set composed of points, line segments, triangles, and their n-dimensional counterparts (see illustration). Simplicial complexes should not be confused with the more abstract notion of a simplicial set appearing in modern simplicial homotopy theory. The purely combinatorial counterpart to a simplicial complex is an abstract simplicial complex. To distinguish a simplicial complex from an abstract simplicial complex, the former is often called a geometric simplicial complex.[1]: 7 Definitions A simplicial complex ${\mathcal {K}}$ is a set of simplices that satisfies the following conditions: 1. Every face of a simplex from ${\mathcal {K}}$ is also in ${\mathcal {K}}$. 2. The non-empty intersection of any two simplices $\sigma _{1},\sigma _{2}\in {\mathcal {K}}$ is a face of both $\sigma _{1}$ and $\sigma _{2}$. See also the definition of an abstract simplicial complex, which loosely speaking is a simplicial complex without an associated geometry. A simplicial k-complex ${\mathcal {K}}$ is a simplicial complex where the largest dimension of any simplex in ${\mathcal {K}}$ equals k. For instance, a simplicial 2-complex must contain at least one triangle, and must not contain any tetrahedra or higher-dimensional simplices. A pure or homogeneous simplicial k-complex ${\mathcal {K}}$ is a simplicial complex where every simplex of dimension less than k is a face of some simplex $\sigma \in {\mathcal {K}}$ of dimension exactly k. Informally, a pure 1-complex "looks" like it's made of a bunch of lines, a 2-complex "looks" like it's made of a bunch of triangles, etc. An example of a non-homogeneous complex is a triangle with a line segment attached to one of its vertices. Pure simplicial complexes can be thought of as triangulations and provide a definition of polytopes. A facet is a maximal simplex, i.e., any simplex in a complex that is not a face of any larger simplex.[2] (Note the difference from a "face" of a simplex). A pure simplicial complex can be thought of as a complex where all facets have the same dimension. For (boundary complexes of) simplicial polytopes this coincides with the meaning from polyhedral combinatorics. Sometimes the term face is used to refer to a simplex of a complex, not to be confused with a face of a simplex. For a simplicial complex embedded in a k-dimensional space, the k-faces are sometimes referred to as its cells. The term cell is sometimes used in a broader sense to denote a set homeomorphic to a simplex, leading to the definition of cell complex. The underlying space, sometimes called the carrier of a simplicial complex is the union of its simplices. It is usually denoted by $\|{\mathcal {K}}\|$ or $\|\|{\mathcal {K}}\|\|$. Support The relative interiors of all simplices in ${\mathcal {K}}$ form a partition of its underlying space $\|{\mathcal {K}}\|$: for each point $x\in \|{\mathcal {K}}\|$, there is exactly one simplex in ${\mathcal {K}}$ containing $x$ in its relative interior. This simplex is called the support of x and denoted $\operatorname {supp} (x)$.[3]: 9 Closure, star, and link • Two simplices and their closure. • A vertex and its star. • A vertex and its link. Let K be a simplicial complex and let S be a collection of simplices in K. The closure of S (denoted $\mathrm {Cl} \ S$) is the smallest simplicial subcomplex of K that contains each simplex in S. $\mathrm {Cl} \ S$ is obtained by repeatedly adding to S each face of every simplex in S. The star of S (denoted $\mathrm {st} \ S$) is the union of the stars of each simplex in S. For a single simplex s, the star of s is the set of simplices having s as a face. The star of S is generally not a simplicial complex itself, so some authors define the closed star of S (denoted $\mathrm {St} \ S$) as $\mathrm {Cl} \ \mathrm {st} \ S$ the closure of the star of S. The link of S (denoted $\mathrm {Lk} \ S$) equals $\mathrm {Cl} {\big (}\mathrm {st} (S){\big )}\setminus \mathrm {st} {\big (}\mathrm {Cl} (S){\big )}$. It is the closed star of S minus the stars of all faces of S. Algebraic topology Main article: Simplicial homology In algebraic topology, simplicial complexes are often useful for concrete calculations. For the definition of homology groups of a simplicial complex, one can read the corresponding chain complex directly, provided that consistent orientations are made of all simplices. The requirements of homotopy theory lead to the use of more general spaces, the CW complexes. Infinite complexes are a technical tool basic in algebraic topology. See also the discussion at Polytope of simplicial complexes as subspaces of Euclidean space made up of subsets, each of which is a simplex. That somewhat more concrete concept is there attributed to Alexandrov. Any finite simplicial complex in the sense talked about here can be embedded as a polytope in that sense, in some large number of dimensions. In algebraic topology, a compact topological space which is homeomorphic to the geometric realization of a finite simplicial complex is usually called a polyhedron (see Spanier 1966, Maunder 1996, Hilton & Wylie 1967). Combinatorics Combinatorialists often study the f-vector of a simplicial d-complex Δ, which is the integer sequence $(f_{0},f_{1},f_{2},\ldots ,f_{d+1})$, where fi is the number of (i−1)-dimensional faces of Δ (by convention, f0 = 1 unless Δ is the empty complex). For instance, if Δ is the boundary of the octahedron, then its f-vector is (1, 6, 12, 8), and if Δ is the first simplicial complex pictured above, its f-vector is (1, 18, 23, 8, 1). A complete characterization of the possible f-vectors of simplicial complexes is given by the Kruskal–Katona theorem. By using the f-vector of a simplicial d-complex Δ as coefficients of a polynomial (written in decreasing order of exponents), we obtain the f-polynomial of Δ. In our two examples above, the f-polynomials would be $x^{3}+6x^{2}+12x+8$ and $x^{4}+18x^{3}+23x^{2}+8x+1$, respectively. Combinatorists are often quite interested in the h-vector of a simplicial complex Δ, which is the sequence of coefficients of the polynomial that results from plugging x − 1 into the f-polynomial of Δ. Formally, if we write FΔ(x) to mean the f-polynomial of Δ, then the h-polynomial of Δ is $F_{\Delta }(x-1)=h_{0}x^{d+1}+h_{1}x^{d}+h_{2}x^{d-1}+\cdots +h_{d}x+h_{d+1}$ and the h-vector of Δ is $(h_{0},h_{1},h_{2},\cdots ,h_{d+1}).$ We calculate the h-vector of the octahedron boundary (our first example) as follows: $F(x-1)=(x-1)^{3}+6(x-1)^{2}+12(x-1)+8=x^{3}+3x^{2}+3x+1.$ So the h-vector of the boundary of the octahedron is (1, 3, 3, 1). It is not an accident this h-vector is symmetric. In fact, this happens whenever Δ is the boundary of a simplicial polytope (these are the Dehn–Sommerville equations). In general, however, the h-vector of a simplicial complex is not even necessarily positive. For instance, if we take Δ to be the 2-complex given by two triangles intersecting only at a common vertex, the resulting h-vector is (1, 3, −2). A complete characterization of all simplicial polytope h-vectors is given by the celebrated g-theorem of Stanley, Billera, and Lee. Simplicial complexes can be seen to have the same geometric structure as the contact graph of a sphere packing (a graph where vertices are the centers of spheres and edges exist if the corresponding packing elements touch each other) and as such can be used to determine the combinatorics of sphere packings, such as the number of touching pairs (1-simplices), touching triplets (2-simplices), and touching quadruples (3-simplices) in a sphere packing. Computational problems Main article: Simplicial complex recognition problem The simplicial complex recognition problem is: given a finite simplicial complex, decide whether it is homeomorphic to a given geometric object. This problem is undecidable for any d-dimensional manifolds for d ≥ 5. See also • Abstract simplicial complex • Barycentric subdivision • Causal dynamical triangulation • Delta set • Polygonal chain – 1 dimensional simplicial complex • Tucker's lemma • Simplex tree References 1. Matoušek, Jiří (2007). Using the Borsuk-Ulam Theorem: Lectures on Topological Methods in Combinatorics and Geometry (2nd ed.). Berlin-Heidelberg: Springer-Verlag. ISBN 978-3-540-00362-5. Written in cooperation with Anders Björner and Günter M. Ziegler , Section 4.3 2. De Loera, Jesús A.; Rambau, Jörg; Santos, Francisco (2010), Triangulations: Structures for Algorithms and Applications, Algorithms and Computation in Mathematics, vol. 25, Springer, p. 493, ISBN 9783642129711. 3. Matoušek, Jiří (2007). Using the Borsuk-Ulam Theorem: Lectures on Topological Methods in Combinatorics and Geometry (2nd ed.). Berlin-Heidelberg: Springer-Verlag. ISBN 978-3-540-00362-5. Written in cooperation with Anders Björner and Günter M. Ziegler , Section 4.3 • Spanier, Edwin H. (1966), Algebraic Topology, Springer, ISBN 0-387-94426-5 • Maunder, Charles R.F. (1996), Algebraic Topology (Reprint of the 1980 ed.), Mineola, NY: Dover, ISBN 0-486-69131-4, MR 1402473 • Hilton, Peter J.; Wylie, Shaun (1967), Homology Theory, New York: Cambridge University Press, ISBN 0-521-09422-4, MR 0115161 External links • Weisstein, Eric W. "Simplicial complex". MathWorld. • Norman J. Wildberger. "Simplices and simplicial complexes". A Youtube talk.. 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Aristotle Aristotle (/ˈærɪstɒtəl/;[1] Greek: Ἀριστοτέλης Aristotélēs, pronounced [aristotélɛːs]; 384–322 BC) was an Ancient Greek philosopher and polymath. His writings cover a broad range of subjects spanning the natural sciences, philosophy, linguistics, economics, politics, psychology and the arts. As the founder of the Peripatetic school of philosophy in the Lyceum in Athens, he began the wider Aristotelian tradition that followed, which set the groundwork for the development of modern science. Aristotle Roman copy (in marble) of a Greek bronze bust of Aristotle by Lysippos (c. 330 BC), with modern alabaster mantle Born384 BC Stagira, Chalcidian League Died322 BC (aged 61–62) Chalcis, Euboea, Macedonian Empire EducationPlatonic Academy Notable work • Organon • Physics • Metaphysics • Nicomachean Ethics • Politics • Rhetoric • Poetics EraAncient Greek philosophy RegionWestern philosophy School • Peripatetic school Notable studentsAlexander the Great, Theophrastus, Aristoxenus Main interests • Logic • Natural philosophy • Metaphysics • Ethics • Politics • Rhetoric • Poetics Notable ideas Aristotelianism Theoretical philosophy • Aristotelian logic, syllogism • Four causes • Genus and differentia • Hylomorphism, substance, essence, accident • Hypokeimenon • Potentiality and actuality • Theory of universals • Unmoved mover Natural philosophy • Aristotelian biology • Aristotelian physics • Common sense • Eternity of the world • Five wits • Horror vacui • Theory of elements, aether • Rational animal Practical philosophy • Aristotelian ethics • Catharsis • Deliberative, epideictic and forensic rhetoric • Enthymeme and Paradeigma • Family as a model for the state • Golden mean • Kyklos • Magnanimity • Mimesis • Natural slavery • Intellectual virtues: sophia, episteme, nous, phronesis, techne • Three appeals: ethos, logos, pathos • Views on women Influences • Plato • Socrates • Heraclitus • Parmenides • Empedocles • Phaleas • Hippodamus • Hippias Influenced • Averroism, Avicennism, Literary Neo-Aristotelianism, Maimonideanism, Objectivism, Peripatetics, Scholasticism (Lullism, Neo, Scotism, Second, Thomism, etc.), additionally Middle Platonism and Neoplatonism. See: List of writers influenced by Aristotle, Commentaries on Aristotle, Pseudo-Aristotle Little is known about Aristotle's life. He was born in the city of Stagira in northern Greece during the Classical period. His father, Nicomachus, died when Aristotle was a child, and he was brought up by a guardian. At 17 or 18 he joined Plato's Academy in Athens and remained there till the age of 37 (c. 347 BC). Shortly after Plato died, Aristotle left Athens and, at the request of Philip II of Macedon, tutored his son Alexander the Great beginning in 343 BC. He established a library in the Lyceum which helped him to produce many of his hundreds of books on papyrus scrolls. Though Aristotle wrote many elegant treatises and dialogues for publication, only around a third of his original output has survived, none of it intended for publication. Aristotle provided a complex synthesis of the various philosophies existing prior to him. It was above all from his teachings that the West inherited its intellectual lexicon, as well as problems and methods of inquiry. As a result, his philosophy has exerted a unique influence on almost every form of knowledge in the West and it continues to be a subject of contemporary philosophical discussion. Aristotle's views profoundly shaped medieval scholarship. The influence of physical science extended from late antiquity and the Early Middle Ages into the Renaissance, and were not replaced systematically until the Enlightenment and theories such as classical mechanics were developed. Some of Aristotle's zoological observations found in his biology, such as on the hectocotyl (reproductive) arm of the octopus, were disbelieved until the 19th century. He influenced Judeo-Islamic philosophies during the Middle Ages, as well as Christian theology, especially the Neoplatonism of the Early Church and the scholastic tradition of the Catholic Church. Aristotle was revered among medieval Muslim scholars as "The First Teacher", and among medieval Christians like Thomas Aquinas as simply "The Philosopher", while the poet Dante called him "the master of those who know". His works contain the earliest known formal study of logic, and were studied by medieval scholars such as Peter Abelard and Jean Buridan. Aristotle's influence on logic continued well into the 19th century. In addition, his ethics, though always influential, gained renewed interest with the modern advent of virtue ethics. Life In general, the details of Aristotle's life are not well-established. The biographies written in ancient times are often speculative and historians only agree on a few salient points.[upper-alpha 1] Aristotle was born in 384 BC[upper-alpha 2] in Stagira, Chalcidice,[2] about 55 km (34 miles) east of modern-day Thessaloniki.[3][4] His father, Nicomachus, was the personal physician to King Amyntas of Macedon. While he was young, Aristotle learned about biology and medical information, which was taught by his father.[5] Both of Aristotle's parents died when he was about thirteen, and Proxenus of Atarneus became his guardian.[6] Although little information about Aristotle's childhood has survived, he probably spent some time within the Macedonian palace, making his first connections with the Macedonian monarchy.[7] At the age of seventeen or eighteen, Aristotle moved to Athens to continue his education at Plato's Academy.[8] He probably experienced the Eleusinian Mysteries as he wrote when describing the sights one viewed at the Eleusinian Mysteries, "to experience is to learn" [παθείν μαθεĩν].[9] Aristotle remained in Athens for nearly twenty years before leaving in 348/47 BC. The traditional story about his departure records that he was disappointed with the Academy's direction after control passed to Plato's nephew Speusippus, although it is possible that he feared the anti-Macedonian sentiments in Athens at that time and left before Plato died.[10] Aristotle then accompanied Xenocrates to the court of his friend Hermias of Atarneus in Asia Minor. After the death of Hermias, Aristotle travelled with his pupil Theophrastus to the island of Lesbos, where together they researched the botany and zoology of the island and its sheltered lagoon. While in Lesbos, Aristotle married Pythias, either Hermias's adoptive daughter or niece. They had a daughter, whom they also named Pythias. In 343 BC, Aristotle was invited by Philip II of Macedon to become the tutor to his son Alexander.[11][12] Aristotle was appointed as the head of the royal academy of Macedon. During Aristotle's time in the Macedonian court, he gave lessons not only to Alexander but also to two other future kings: Ptolemy and Cassander.[13] Aristotle encouraged Alexander toward eastern conquest, and Aristotle's own attitude towards Persia was unabashedly ethnocentric. In one famous example, he counsels Alexander to be "a leader to the Greeks and a despot to the barbarians, to look after the former as after friends and relatives, and to deal with the latter as with beasts or plants".[13] By 335 BC, Aristotle had returned to Athens, establishing his own school there known as the Lyceum. Aristotle conducted courses at the school for the next twelve years. While in Athens, his wife Pythias died and Aristotle became involved with Herpyllis of Stagira. They had a son whom Aristotle named after his father, Nicomachus. If the Suda – an uncritical compilation from the Middle Ages – is accurate, he may also have had an erômenos, Palaephatus of Abydus.[14] This period in Athens, between 335 and 323 BC, is when Aristotle is believed to have composed many of his works.[12] He wrote many dialogues, of which only fragments have survived. Those works that have survived are in treatise form and were not, for the most part, intended for widespread publication; they are generally thought to be lecture aids for his students. His most important treatises include Physics, Metaphysics, Nicomachean Ethics, Politics, On the Soul and Poetics. Aristotle studied and made significant contributions to "logic, metaphysics, mathematics, physics, biology, botany, ethics, politics, agriculture, medicine, dance, and theatre."[15] Near the end of his life, Alexander and Aristotle became estranged over Alexander's relationship with Persia and Persians. A widespread tradition in antiquity suspected Aristotle of playing a role in Alexander's death, but the only evidence of this is an unlikely claim made some six years after the death.[16] Following Alexander's death, anti-Macedonian sentiment in Athens was rekindled. In 322 BC, Demophilus and Eurymedon the Hierophant reportedly denounced Aristotle for impiety,[17] prompting him to flee to his mother's family estate in Chalcis, on Euboea, at which occasion he was said to have stated: "I will not allow the Athenians to sin twice against philosophy"[18][19][20] – a reference to Athens's trial and execution of Socrates. He died in Chalcis, Euboea[2][21][15] of natural causes later that same year, having named his student Antipater as his chief executor and leaving a will in which he asked to be buried next to his wife.[22] Theoretical philosophy Logic With the Prior Analytics, Aristotle is credited with the earliest study of formal logic,[23] and his conception of it was the dominant form of Western logic until 19th-century advances in mathematical logic.[24] Kant stated in the Critique of Pure Reason that with Aristotle logic reached its completion.[25] Organon One of Aristotle's types of syllogism[upper-alpha 3] In wordsIn terms[upper-alpha 4] In equations[upper-alpha 5] All men are mortal. All Greeks are men. ∴ All Greeks are mortal. M a P S a M S a P What is today called Aristotelian logic with its types of syllogism (methods of logical argument),[26] Aristotle himself would have labelled "analytics". The term "logic" he reserved to mean dialectics. Most of Aristotle's work is probably not in its original form, because it was most likely edited by students and later lecturers. The logical works of Aristotle were compiled into a set of six books called the Organon around 40 BC by Andronicus of Rhodes or others among his followers.[28] The books are: 1. Categories 2. On Interpretation 3. Prior Analytics 4. Posterior Analytics 5. Topics 6. On Sophistical Refutations The order of the books (or the teachings from which they are composed) is not certain, but this list was derived from analysis of Aristotle's writings. It goes from the basics, the analysis of simple terms in the Categories, the analysis of propositions and their elementary relations in On Interpretation, to the study of more complex forms, namely, syllogisms (in the Analytics)[31][32] and dialectics (in the Topics and Sophistical Refutations). The first three treatises form the core of the logical theory stricto sensu: the grammar of the language of logic and the correct rules of reasoning. The Rhetoric is not conventionally included, but it states that it relies on the Topics.[33] Metaphysics The word "metaphysics" appears to have been coined by the first century AD editor who assembled various small selections of Aristotle's works to the treatise we know by the name Metaphysics.[34] Aristotle called it "first philosophy", and distinguished it from mathematics and natural science (physics) as the contemplative (theoretikē) philosophy which is "theological" and studies the divine. He wrote in his Metaphysics (1026a16): if there were no other independent things besides the composite natural ones, the study of nature would be the primary kind of knowledge; but if there is some motionless independent thing, the knowledge of this precedes it and is first philosophy, and it is universal in just this way, because it is first. And it belongs to this sort of philosophy to study being as being, both what it is and what belongs to it just by virtue of being.[35] Substance Aristotle examines the concepts of substance (ousia) and essence (to ti ên einai, "the what it was to be") in his Metaphysics (Book VII), and he concludes that a particular substance is a combination of both matter and form, a philosophical theory called hylomorphism. In Book VIII, he distinguishes the matter of the substance as the substratum, or the stuff of which it is composed. For example, the matter of a house is the bricks, stones, timbers, etc., or whatever constitutes the potential house, while the form of the substance is the actual house, namely 'covering for bodies and chattels' or any other differentia that let us define something as a house. The formula that gives the components is the account of the matter, and the formula that gives the differentia is the account of the form.[36][34] Immanent realism Like his teacher Plato, Aristotle's philosophy aims at the universal. Aristotle's ontology places the universal (katholou) in particulars (kath' hekaston), things in the world, whereas for Plato the universal is a separately existing form which actual things imitate. For Aristotle, "form" is still what phenomena are based on, but is "instantiated" in a particular substance.[34] Plato argued that all things have a universal form, which could be either a property or a relation to other things. When one looks at an apple, for example, one sees an apple, and one can also analyse a form of an apple. In this distinction, there is a particular apple and a universal form of an apple. Moreover, one can place an apple next to a book, so that one can speak of both the book and apple as being next to each other. Plato argued that there are some universal forms that are not a part of particular things. For example, it is possible that there is no particular good in existence, but "good" is still a proper universal form. Aristotle disagreed with Plato on this point, arguing that all universals are instantiated at some period of time, and that there are no universals that are unattached to existing things. In addition, Aristotle disagreed with Plato about the location of universals. Where Plato spoke of the forms as existing separately from the things that participate in them, Aristotle maintained that universals exist within each thing on which each universal is predicated. So, according to Aristotle, the form of apple exists within each apple, rather than in the world of the forms.[34][37] Potentiality and actuality Concerning the nature of change (kinesis) and its causes, as he outlines in his Physics and On Generation and Corruption (319b–320a), he distinguishes coming-to-be (genesis, also translated as 'generation') from: 1. growth and diminution, which is change in quantity; 2. locomotion, which is change in space; and 3. alteration, which is change in quality. Coming-to-be is a change where the substrate of the thing that has undergone the change has itself changed. In that particular change he introduces the concept of potentiality (dynamis) and actuality (entelecheia) in association with the matter and the form. Referring to potentiality, this is what a thing is capable of doing or being acted upon if the conditions are right and it is not prevented by something else. For example, the seed of a plant in the soil is potentially (dynamei) a plant, and if it is not prevented by something, it will become a plant. Potentially beings can either 'act' (poiein) or 'be acted upon' (paschein), which can be either innate or learned. For example, the eyes possess the potentiality of sight (innate – being acted upon), while the capability of playing the flute can be possessed by learning (exercise – acting). Actuality is the fulfilment of the end of the potentiality. Because the end (telos) is the principle of every change, and potentiality exists for the sake of the end, actuality, accordingly, is the end. Referring then to the previous example, it can be said that an actuality is when a plant does one of the activities that plants do.[34] For that for the sake of which (to hou heneka) a thing is, is its principle, and the becoming is for the sake of the end; and the actuality is the end, and it is for the sake of this that the potentiality is acquired. For animals do not see in order that they may have sight, but they have sight that they may see.[38] In summary, the matter used to make a house has potentiality to be a house and both the activity of building and the form of the final house are actualities, which is also a final cause or end. Then Aristotle proceeds and concludes that the actuality is prior to potentiality in formula, in time and in substantiality. With this definition of the particular substance (i.e., matter and form), Aristotle tries to solve the problem of the unity of the beings, for example, "what is it that makes a man one"? Since, according to Plato there are two Ideas: animal and biped, how then is man a unity? However, according to Aristotle, the potential being (matter) and the actual one (form) are one and the same.[34][39] Epistemology Aristotle's immanent realism means his epistemology is based on the study of things that exist or happen in the world, and rises to knowledge of the universal, whereas for Plato epistemology begins with knowledge of universal Forms (or ideas) and descends to knowledge of particular imitations of these.[33] Aristotle uses induction from examples alongside deduction, whereas Plato relies on deduction from a priori principles.[33] Natural philosophy Aristotle's "natural philosophy" spans a wide range of natural phenomena including those now covered by physics, biology and other natural sciences.[40] In Aristotle's terminology, "natural philosophy" is a branch of philosophy examining the phenomena of the natural world, and includes fields that would be regarded today as physics, biology and other natural sciences. Aristotle's work encompassed virtually all facets of intellectual inquiry. Aristotle makes philosophy in the broad sense coextensive with reasoning, which he also would describe as "science". However, his use of the term science carries a different meaning than that covered by the term "scientific method". For Aristotle, "all science (dianoia) is either practical, poetical or theoretical" (Metaphysics 1025b25). His practical science includes ethics and politics; his poetical science means the study of fine arts including poetry; his theoretical science covers physics, mathematics and metaphysics.[40] Physics Five elements In his On Generation and Corruption, Aristotle related each of the four elements proposed earlier by Empedocles, earth, water, air, and fire, to two of the four sensible qualities, hot, cold, wet, and dry. In the Empedoclean scheme, all matter was made of the four elements, in differing proportions. Aristotle's scheme added the heavenly aether, the divine substance of the heavenly spheres, stars and planets.[41] Aristotle's elements[41] ElementHot/ColdWet/DryMotionModern state of matter EarthColdDryDownSolid WaterColdWetDownLiquid AirHotWetUpGas FireHotDryUpPlasma Aether(divine substance) —Circular (in heavens) Vacuum Motion Aristotle describes two kinds of motion: "violent" or "unnatural motion", such as that of a thrown stone, in the Physics (254b10), and "natural motion", such as of a falling object, in On the Heavens (300a20). In violent motion, as soon as the agent stops causing it, the motion stops also: in other words, the natural state of an object is to be at rest,[42][upper-alpha 6] since Aristotle does not address friction.[43] With this understanding, it can be observed that, as Aristotle stated, heavy objects (on the ground, say) require more force to make them move; and objects pushed with greater force move faster.[44][upper-alpha 7] This would imply the equation[44] $F=mv$, incorrect in modern physics.[44] Natural motion depends on the element concerned: the aether naturally moves in a circle around the heavens,[upper-alpha 8] while the 4 Empedoclean elements move vertically up (like fire, as is observed) or down (like earth) towards their natural resting places.[45][43][upper-alpha 9] In the Physics (215a25), Aristotle effectively states a quantitative law, that the speed, v, of a falling body is proportional (say, with constant c) to its weight, W, and inversely proportional to the density,[upper-alpha 10] ρ, of the fluid in which it is falling:;[45][43] $v=c{\frac {W}{\rho }}$ Aristotle implies that in a vacuum the speed of fall would become infinite, and concludes from this apparent absurdity that a vacuum is not possible.[45][43] Opinions have varied on whether Aristotle intended to state quantitative laws. Henri Carteron held the "extreme view"[43] that Aristotle's concept of force was basically qualitative,[46] but other authors reject this.[43] Archimedes corrected Aristotle's theory that bodies move towards their natural resting places; metal boats can float if they displace enough water; floating depends in Archimedes' scheme on the mass and volume of the object, not, as Aristotle thought, its elementary composition.[45] Aristotle's writings on motion remained influential until the Early Modern period. John Philoponus (in Late antiquity) and Galileo (in Early modern period) are said to have shown by experiment that Aristotle's claim that a heavier object falls faster than a lighter object is incorrect.[40] A contrary opinion is given by Carlo Rovelli, who argues that Aristotle's physics of motion is correct within its domain of validity, that of objects in the Earth's gravitational field immersed in a fluid such as air. In this system, heavy bodies in steady fall indeed travel faster than light ones (whether friction is ignored, or not[45]), and they do fall more slowly in a denser medium.[44][upper-alpha 11] Newton's "forced" motion corresponds to Aristotle's "violent" motion with its external agent, but Aristotle's assumption that the agent's effect stops immediately it stops acting (e.g., the ball leaves the thrower's hand) has awkward consequences: he has to suppose that surrounding fluid helps to push the ball along to make it continue to rise even though the hand is no longer acting on it, resulting in the Medieval theory of impetus.[45] Four causes Aristotle suggested that the reason for anything coming about can be attributed to four different types of simultaneously active factors. His term aitia is traditionally translated as "cause", but it does not always refer to temporal sequence; it might be better translated as "explanation", but the traditional rendering will be employed here.[48][49] • Material cause describes the material out of which something is composed. Thus the material cause of a table is wood. It is not about action. It does not mean that one domino knocks over another domino.[48] • The formal cause is its form, i.e., the arrangement of that matter. It tells one what a thing is, that a thing is determined by the definition, form, pattern, essence, whole, synthesis or archetype. It embraces the account of causes in terms of fundamental principles or general laws, as the whole (i.e., macrostructure) is the cause of its parts, a relationship known as the whole-part causation. Plainly put, the formal cause is the idea in the mind of the sculptor that brings the sculpture into being. A simple example of the formal cause is the mental image or idea that allows an artist, architect, or engineer to create a drawing.[48] • The efficient cause is "the primary source", or that from which the change under consideration proceeds. It identifies 'what makes of what is made and what causes change of what is changed' and so suggests all sorts of agents, non-living or living, acting as the sources of change or movement or rest. Representing the current understanding of causality as the relation of cause and effect, this covers the modern definitions of "cause" as either the agent or agency or particular events or states of affairs. In the case of two dominoes, when the first is knocked over it causes the second also to fall over.[48] In the case of animals, this agency is a combination of how it develops from the egg, and how its body functions.[50] • The final cause (telos) is its purpose, the reason why a thing exists or is done, including both purposeful and instrumental actions and activities. The final cause is the purpose or function that something is supposed to serve. This covers modern ideas of motivating causes, such as volition.[48] In the case of living things, it implies adaptation to a particular way of life.[50] Optics Aristotle describes experiments in optics using a camera obscura in Problems, book 15. The apparatus consisted of a dark chamber with a small aperture that let light in. With it, he saw that whatever shape he made the hole, the sun's image always remained circular. He also noted that increasing the distance between the aperture and the image surface magnified the image.[51] Chance and spontaneity According to Aristotle, spontaneity and chance are causes of some things, distinguishable from other types of cause such as simple necessity. Chance as an incidental cause lies in the realm of accidental things, "from what is spontaneous". There is also more a specific kind of chance, which Aristotle names "luck", that only applies to people's moral choices.[52][53] Astronomy In astronomy, Aristotle refuted Democritus's claim that the Milky Way was made up of "those stars which are shaded by the earth from the sun's rays," pointing out partly correctly that if "the size of the sun is greater than that of the earth and the distance of the stars from the earth many times greater than that of the sun, then... the sun shines on all the stars and the earth screens none of them."[54] He also wrote descriptions of comets, including the Great Comet of 371 BC.[55] Geology and natural sciences Aristotle was one of the first people to record any geological observations. He stated that geological change was too slow to be observed in one person's lifetime.[56][57] The geologist Charles Lyell noted that Aristotle described such change, including "lakes that had dried up" and "deserts that had become watered by rivers", giving as examples the growth of the Nile delta since the time of Homer, and "the upheaving of one of the Aeolian islands, previous to a volcanic eruption."'[58] Aristotle also made many observations about the hydrologic cycle and meteorology (including his major writings "Meteorologica"). For example, he made some of the earliest observations about desalination: he observed early – and correctly – that when seawater is heated, freshwater evaporates and that the oceans are then replenished by the cycle of rainfall and river runoff ("I have proved by experiment that salt water evaporated forms fresh and the vapor does not when it condenses condense into sea water again.")[59] Biology Empirical research Aristotle was the first person to study biology systematically,[60] and biology forms a large part of his writings. He spent two years observing and describing the zoology of Lesbos and the surrounding seas, including in particular the Pyrrha lagoon in the centre of Lesbos.[61][62] His data in History of Animals, Generation of Animals, Movement of Animals, and Parts of Animals are assembled from his own observations,[63] statements given by people with specialized knowledge such as beekeepers and fishermen, and less accurate accounts provided by travellers from overseas.[64] His apparent emphasis on animals rather than plants is a historical accident: his works on botany have been lost, but two books on plants by his pupil Theophrastus have survived.[65] Aristotle reports on the sea-life visible from observation on Lesbos and the catches of fishermen. He describes the catfish, electric ray, and frogfish in detail, as well as cephalopods such as the octopus and paper nautilus. His description of the hectocotyl arm of cephalopods, used in sexual reproduction, was widely disbelieved until the 19th century.[66] He gives accurate descriptions of the four-chambered fore-stomachs of ruminants,[67] and of the ovoviviparous embryological development of the hound shark.[68] He notes that an animal's structure is well matched to function, so, among birds, the heron, which lives in marshes with soft mud and lives by catching fish, has a long neck and long legs, and a sharp spear-like beak, whereas ducks that swim have short legs and webbed feet.[69] Darwin, too, noted these sorts of differences between similar kinds of animal, but unlike Aristotle used the data to come to the theory of evolution.[70] Aristotle's writings can seem to modern readers close to implying evolution, but while Aristotle was aware that new mutations or hybridizations could occur, he saw these as rare accidents. For Aristotle, accidents, like heat waves in winter, must be considered distinct from natural causes. He was thus critical of Empedocles's materialist theory of a "survival of the fittest" origin of living things and their organs, and ridiculed the idea that accidents could lead to orderly results.[71] To put his views into modern terms, he nowhere says that different species can have a common ancestor, or that one kind can change into another, or that kinds can become extinct.[72] Scientific style Aristotle did not do experiments in the modern sense.[73] He used the ancient Greek term pepeiramenoi to mean observations, or at most investigative procedures like dissection.[74] In Generation of Animals, he finds a fertilized hen's egg of a suitable stage and opens it to see the embryo's heart beating inside.[75][76] Instead, he practiced a different style of science: systematically gathering data, discovering patterns common to whole groups of animals, and inferring possible causal explanations from these.[77][78] This style is common in modern biology when large amounts of data become available in a new field, such as genomics. It does not result in the same certainty as experimental science, but it sets out testable hypotheses and constructs a narrative explanation of what is observed. In this sense, Aristotle's biology is scientific.[77] From the data he collected and documented, Aristotle inferred quite a number of rules relating the life-history features of the live-bearing tetrapods (terrestrial placental mammals) that he studied. Among these correct predictions are the following. Brood size decreases with (adult) body mass, so that an elephant has fewer young (usually just one) per brood than a mouse. Lifespan increases with gestation period, and also with body mass, so that elephants live longer than mice, have a longer period of gestation, and are heavier. As a final example, fecundity decreases with lifespan, so long-lived kinds like elephants have fewer young in total than short-lived kinds like mice.[79] Classification of living things Aristotle distinguished about 500 species of animals,[81][82] arranging these in the History of Animals in a graded scale of perfection, a nonreligious version of the scala naturae, with man at the top. His system had eleven grades of animal, from highest potential to lowest, expressed in their form at birth: the highest gave live birth to hot and wet creatures, the lowest laid cold, dry mineral-like eggs. Animals came above plants, and these in turn were above minerals.[83][84] He grouped what the modern zoologist would call vertebrates as the hotter "animals with blood", and below them the colder invertebrates as "animals without blood". Those with blood were divided into the live-bearing (mammals), and the egg-laying (birds, reptiles, fish). Those without blood were insects, crustacea (non-shelled – cephalopods, and shelled) and the hard-shelled molluscs (bivalves and gastropods). He recognised that animals did not exactly fit into a linear scale, and noted various exceptions, such as that sharks had a placenta like the tetrapods. To a modern biologist, the explanation, not available to Aristotle, is convergent evolution.[85] Philosophers of science have generally concluded that Aristotle was not interested in taxonomy,[86][87] but zoologists who studied this question in the early 21st century think otherwise.[88][89][90] He believed that purposive final causes guided all natural processes; this teleological view justified his observed data as an expression of formal design.[91] Aristotle's Scala naturae (highest to lowest) GroupExamples (given by Aristotle) BloodLegsSouls (Rational, Sensitive, Vegetative) Qualities (Hot–Cold, Wet–Dry) ManManwith blood2 legsR, S, VHot, Wet Live-bearing tetrapodsCat, harewith blood4 legsS, VHot, Wet CetaceansDolphin, whalewith bloodnoneS, VHot, Wet BirdsBee-eater, nightjarwith blood2 legsS, VHot, Wet, except Dry eggs Egg-laying tetrapodsChameleon, crocodilewith blood4 legsS, VCold, Wet except scales, eggs SnakesWater snake, Ottoman viperwith bloodnoneS, VCold, Wet except scales, eggs Egg-laying fishesSea bass, parrotfishwith bloodnoneS, VCold, Wet, including eggs (Among the egg-laying fishes): placental selachians Shark, skatewith bloodnoneS, VCold, Wet, but placenta like tetrapods CrustaceansShrimp, crabwithoutmany legsS, VCold, Wet except shell CephalopodsSquid, octopuswithouttentaclesS, VCold, Wet Hard-shelled animalsCockle, trumpet snailwithoutnoneS, VCold, Dry (mineral shell) Larva-bearing insectsAnt, cicadawithout6 legsS, VCold, Dry Spontaneously generatingSponges, wormswithoutnoneS, VCold, Wet or Dry, from earth PlantsFigwithoutnoneVCold, Dry MineralsIronwithoutnonenoneCold, Dry Soul Aristotle's psychology, given in his treatise On the Soul (peri psychēs), posits three kinds of soul ("psyches"): the vegetative soul, the sensitive soul, and the rational soul. Humans have a rational soul. The human soul incorporates the powers of the other kinds: Like the vegetative soul it can grow and nourish itself; like the sensitive soul it can experience sensations and move locally. The unique part of the human, rational soul is its ability to receive forms of other things and to compare them using the nous (intellect) and logos (reason).[92] For Aristotle, the soul is the form of a living being. Because all beings are composites of form and matter, the form of living beings is that which endows them with what is specific to living beings, e.g. the ability to initiate movement (or in the case of plants, growth and chemical transformations, which Aristotle considers types of movement).[11] In contrast to earlier philosophers, but in accordance with the Egyptians, he placed the rational soul in the heart, rather than the brain.[93] Notable is Aristotle's division of sensation and thought, which generally differed from the concepts of previous philosophers, with the exception of Alcmaeon.[94] In On the Soul, Aristotle famously criticizes Plato's theory of the soul and develops his own in response to Plato's. The first criticism is against Plato's view of the soul in the Timaeus that the soul takes up space and is able to come into physical contact with bodies.[95] 20th-century scholarship overwhelmingly opposed Aristotle's interpretation of Plato and maintained that he had misunderstood Plato.[96] Today's scholars have tended to re-assess Aristotle's interpretation and have warmed up to it.[97] Aristotle's other criticism is that Plato's view of reincarnation entails that it is possible for a soul and its body to be mis-matched; in principle, Aristotle alleges, any soul can go with any body, according to Plato's theory.[98] Aristotle's claim that the soul is the form of a living being is meant to eliminate that possibility and thus rule out reincarnation.[99] Memory According to Aristotle in On the Soul, memory is the ability to hold a perceived experience in the mind and to distinguish between the internal "appearance" and an occurrence in the past.[100] In other words, a memory is a mental picture (phantasm) that can be recovered. Aristotle believed an impression is left on a semi-fluid bodily organ that undergoes several changes in order to make a memory. A memory occurs when stimuli such as sights or sounds are so complex that the nervous system cannot receive all the impressions at once. These changes are the same as those involved in the operations of sensation, Aristotelian 'common sense', and thinking.[101][102] Aristotle uses the term 'memory' for the actual retaining of an experience in the impression that can develop from sensation, and for the intellectual anxiety that comes with the impression because it is formed at a particular time and processing specific contents. Memory is of the past, prediction is of the future, and sensation is of the present. Retrieval of impressions cannot be performed suddenly. A transitional channel is needed and located in past experiences, both for previous experience and present experience.[103] Because Aristotle believes people receive all kinds of sense perceptions and perceive them as impressions, people are continually weaving together new impressions of experiences. To search for these impressions, people search the memory itself.[104] Within the memory, if one experience is offered instead of a specific memory, that person will reject this experience until they find what they are looking for. Recollection occurs when one retrieved experience naturally follows another. If the chain of "images" is needed, one memory will stimulate the next. When people recall experiences, they stimulate certain previous experiences until they reach the one that is needed.[105] Recollection is thus the self-directed activity of retrieving the information stored in a memory impression.[106] Only humans can remember impressions of intellectual activity, such as numbers and words. Animals that have perception of time can retrieve memories of their past observations. Remembering involves only perception of the things remembered and of the time passed.[107] Aristotle believed the chain of thought, which ends in recollection of certain impressions, was connected systematically in relationships such as similarity, contrast, and contiguity, described in his laws of association. Aristotle believed that past experiences are hidden within the mind. A force operates to awaken the hidden material to bring up the actual experience. According to Aristotle, association is the power innate in a mental state, which operates upon the unexpressed remains of former experiences, allowing them to rise and be recalled.[108][109] Dreams Aristotle describes sleep in On Sleep and Wakefulness.[110] Sleep takes place as a result of overuse of the senses[111] or of digestion,[112] so it is vital to the body.[111] While a person is asleep, the critical activities, which include thinking, sensing, recalling and remembering, do not function as they do during wakefulness. Since a person cannot sense during sleep they cannot have desire, which is the result of sensation. However, the senses are able to work during sleep,[113] albeit differently,[110] unless they are weary.[111] Dreams do not involve actually sensing a stimulus. In dreams, sensation is still involved, but in an altered manner.[111] Aristotle explains that when a person stares at a moving stimulus such as the waves in a body of water, and then looks away, the next thing they look at appears to have a wavelike motion. When a person perceives a stimulus and the stimulus is no longer the focus of their attention, it leaves an impression.[110] When the body is awake and the senses are functioning properly, a person constantly encounters new stimuli to sense and so the impressions of previously perceived stimuli are ignored.[111] However, during sleep the impressions made throughout the day are noticed as there are no new distracting sensory experiences.[110] So, dreams result from these lasting impressions. Since impressions are all that are left and not the exact stimuli, dreams do not resemble the actual waking experience.[114] During sleep, a person is in an altered state of mind. Aristotle compares a sleeping person to a person who is overtaken by strong feelings toward a stimulus. For example, a person who has a strong infatuation with someone may begin to think they see that person everywhere because they are so overtaken by their feelings. Since a person sleeping is in a suggestible state and unable to make judgements, they become easily deceived by what appears in their dreams, like the infatuated person.[110] This leads the person to believe the dream is real, even when the dreams are absurd in nature.[110] In De Anima iii 3, Aristotle ascribes the ability to create, to store, and to recall images in the absence of perception to the faculty of imagination, phantasia.[11] One component of Aristotle's theory of dreams disagrees with previously held beliefs. He claimed that dreams are not foretelling and not sent by a divine being. Aristotle reasoned naturalistically that instances in which dreams do resemble future events are simply coincidences.[115] Aristotle claimed that a dream is first established by the fact that the person is asleep when they experience it. If a person had an image appear for a moment after waking up or if they see something in the dark it is not considered a dream because they were awake when it occurred. Secondly, any sensory experience that is perceived while a person is asleep does not qualify as part of a dream. For example, if, while a person is sleeping, a door shuts and in their dream they hear a door is shut, this sensory experience is not part of the dream. Lastly, the images of dreams must be a result of lasting impressions of waking sensory experiences.[114] Practical philosophy Aristotle's practical philosophy covers areas such as ethics, politics, economics, and rhetoric.[40] Virtues and their accompanying vices[15] Too littleVirtuous meanToo much HumblenessHigh-mindednessVainglory Lack of purposeRight ambitionOver-ambition SpiritlessnessGood temperIrascibility RudenessCivilityObsequiousness CowardiceCourageRashness InsensibilitySelf-controlIntemperance SarcasmSincerityBoastfulness BoorishnessWitBuffoonery ShamelessnessModestyShyness CallousnessJust resentmentSpitefulness PettinessGenerosityVulgarity MeannessLiberalityWastefulness Ethics Aristotle considered ethics to be a practical rather than theoretical study, i.e., one aimed at becoming good and doing good rather than knowing for its own sake. He wrote several treatises on ethics, most notably including the Nicomachean Ethics.[116] Aristotle taught that virtue has to do with the proper function (ergon) of a thing. An eye is only a good eye in so much as it can see, because the proper function of an eye is sight. Aristotle reasoned that humans must have a function specific to humans, and that this function must be an activity of the psuchē (soul) in accordance with reason (logos). Aristotle identified such an optimum activity (the virtuous mean, between the accompanying vices of excess or deficiency[15]) of the soul as the aim of all human deliberate action, eudaimonia, generally translated as "happiness" or sometimes "well-being". To have the potential of ever being happy in this way necessarily requires a good character (ēthikē aretē), often translated as moral or ethical virtue or excellence.[117] Aristotle taught that to achieve a virtuous and potentially happy character requires a first stage of having the fortune to be habituated not deliberately, but by teachers, and experience, leading to a later stage in which one consciously chooses to do the best things. When the best people come to live life this way their practical wisdom (phronesis) and their intellect (nous) can develop with each other towards the highest possible human virtue, the wisdom of an accomplished theoretical or speculative thinker, or in other words, a philosopher.[118] Politics In addition to his works on ethics, which address the individual, Aristotle addressed the city in his work titled Politics. Aristotle considered the city to be a natural community. Moreover, he considered the city to be prior in importance to the family, which in turn is prior to the individual, "for the whole must of necessity be prior to the part".[119] He famously stated that "man is by nature a political animal" and argued that humanity's defining factor among others in the animal kingdom is its rationality.[120] Aristotle conceived of politics as being like an organism rather than like a machine, and as a collection of parts none of which can exist without the others. Aristotle's conception of the city is organic, and he is considered one of the first to conceive of the city in this manner.[121] The common modern understanding of a political community as a modern state is quite different from Aristotle's understanding. Although he was aware of the existence and potential of larger empires, the natural community according to Aristotle was the city (polis) which functions as a political "community" or "partnership" (koinōnia). The aim of the city is not just to avoid injustice or for economic stability, but rather to allow at least some citizens the possibility to live a good life, and to perform beautiful acts: "The political partnership must be regarded, therefore, as being for the sake of noble actions, not for the sake of living together." This is distinguished from modern approaches, beginning with social contract theory, according to which individuals leave the state of nature because of "fear of violent death" or its "inconveniences".[upper-alpha 12] In Protrepticus, the character 'Aristotle' states:[122] For we all agree that the most excellent man should rule, i.e., the supreme by nature, and that the law rules and alone is authoritative; but the law is a kind of intelligence, i.e. a discourse based on intelligence. And again, what standard do we have, what criterion of good things, that is more precise than the intelligent man? For all that this man will choose, if the choice is based on his knowledge, are good things and their contraries are bad. And since everybody chooses most of all what conforms to their own proper dispositions (a just man choosing to live justly, a man with bravery to live bravely, likewise a self-controlled man to live with self-control), it is clear that the intelligent man will choose most of all to be intelligent; for this is the function of that capacity. Hence it's evident that, according to the most authoritative judgment, intelligence is supreme among goods.[122] As Plato's disciple Aristotle was rather critical concerning democracy and, following the outline of certain ideas from Plato's Statesman, he developed a coherent theory of integrating various forms of power into a so-called mixed state: It is … constitutional to take … from oligarchy that offices are to be elected, and from democracy that this is not to be on a property-qualification. This then is the mode of the mixture; and the mark of a good mixture of democracy and oligarchy is when it is possible to speak of the same constitution as a democracy and as an oligarchy. — Aristotle. Politics, Book 4, 1294b.10–18 To illustrate this approach, Aristotle proposed a first-of-its-kind mathematical model of voting, albeit textually described, where the democratic principle of "one voter–one vote" is combined with the oligarchic "merit-weighted voting"; for relevant quotes and their translation into mathematical formulas see.[123] Economics Aristotle made substantial contributions to economic thought, especially to thought in the Middle Ages.[124] In Politics, Aristotle addresses the city, property, and trade. His response to criticisms of private property, in Lionel Robbins's view, anticipated later proponents of private property among philosophers and economists, as it related to the overall utility of social arrangements.[124] Aristotle believed that although communal arrangements may seem beneficial to society, and that although private property is often blamed for social strife, such evils in fact come from human nature. In Politics, Aristotle offers one of the earliest accounts of the origin of money.[124] Money came into use because people became dependent on one another, importing what they needed and exporting the surplus. For the sake of convenience, people then agreed to deal in something that is intrinsically useful and easily applicable, such as iron or silver.[125] Aristotle's discussions on retail and interest was a major influence on economic thought in the Middle Ages. He had a low opinion of retail, believing that contrary to using money to procure things one needs in managing the household, retail trade seeks to make a profit. It thus uses goods as a means to an end, rather than as an end unto itself. He believed that retail trade was in this way unnatural. Similarly, Aristotle considered making a profit through interest unnatural, as it makes a gain out of the money itself, and not from its use.[125] Aristotle gave a summary of the function of money that was perhaps remarkably precocious for his time. He wrote that because it is impossible to determine the value of every good through a count of the number of other goods it is worth, the necessity arises of a single universal standard of measurement. Money thus allows for the association of different goods and makes them "commensurable".[125] He goes on to state that money is also useful for future exchange, making it a sort of security. That is, "if we do not want a thing now, we shall be able to get it when we do want it".[125] Rhetoric and poetics Aristotle's Rhetoric proposes that a speaker can use three basic kinds of appeals to persuade his audience: ethos (an appeal to the speaker's character), pathos (an appeal to the audience's emotion), and logos (an appeal to logical reasoning).[127] He also categorizes rhetoric into three genres: epideictic (ceremonial speeches dealing with praise or blame), forensic (judicial speeches over guilt or innocence), and deliberative (speeches calling on an audience to make a decision on an issue).[128] Aristotle also outlines two kinds of rhetorical proofs: enthymeme (proof by syllogism) and paradeigma (proof by example).[129] Aristotle writes in his Poetics that epic poetry, tragedy, comedy, dithyrambic poetry, painting, sculpture, music, and dance are all fundamentally acts of mimesis ("imitation"), each varying in imitation by medium, object, and manner.[130][131] He applies the term mimesis both as a property of a work of art and also as the product of the artist's intention[130] and contends that the audience's realisation of the mimesis is vital to understanding the work itself.[130] Aristotle states that mimesis is a natural instinct of humanity that separates humans from animals[130][132] and that all human artistry "follows the pattern of nature".[130] Because of this, Aristotle believed that each of the mimetic arts possesses what Stephen Halliwell calls "highly structured procedures for the achievement of their purposes."[133] For example, music imitates with the media of rhythm and harmony, whereas dance imitates with rhythm alone, and poetry with language. The forms also differ in their object of imitation. Comedy, for instance, is a dramatic imitation of men worse than average; whereas tragedy imitates men slightly better than average. Lastly, the forms differ in their manner of imitation – through narrative or character, through change or no change, and through drama or no drama.[134] While it is believed that Aristotle's Poetics originally comprised two books – one on comedy and one on tragedy – only the portion that focuses on tragedy has survived. Aristotle taught that tragedy is composed of six elements: plot-structure, character, style, thought, spectacle, and lyric poetry.[135] The characters in a tragedy are merely a means of driving the story; and the plot, not the characters, is the chief focus of tragedy. Tragedy is the imitation of action arousing pity and fear, and is meant to effect the catharsis of those same emotions. Aristotle concludes Poetics with a discussion on which, if either, is superior: epic or tragic mimesis. He suggests that because tragedy possesses all the attributes of an epic, possibly possesses additional attributes such as spectacle and music, is more unified, and achieves the aim of its mimesis in shorter scope, it can be considered superior to epic.[136] Aristotle was a keen systematic collector of riddles, folklore, and proverbs; he and his school had a special interest in the riddles of the Delphic Oracle and studied the fables of Aesop.[137] Views on women Aristotle's analysis of procreation describes an active, ensouling masculine element bringing life to an inert, passive female element. The biological differences are a result of the fact that the female body is well-suited for reproduction, which changes her body temperature, which in turn makes her, in Aristotle's view, incapable of participating in political life.[138] On this ground, proponents of feminist metaphysics have accused Aristotle of misogyny[139] and sexism.[140] However, Aristotle gave equal weight to women's happiness as he did to men's, and commented in his Rhetoric that the things that lead to happiness need to be in women as well as men.[upper-alpha 13] Influence More than 2300 years after his death, Aristotle remains one of the most influential people who ever lived.[142][143][144] He contributed to almost every field of human knowledge then in existence, and he was the founder of many new fields. According to the philosopher Bryan Magee, "it is doubtful whether any human being has ever known as much as he did".[145] Among countless other achievements, Aristotle was the founder of formal logic,[146] pioneered the study of zoology, and left every future scientist and philosopher in his debt through his contributions to the scientific method.[2][147][148] Taneli Kukkonen, writing in The Classical Tradition, observes that his achievement in founding two sciences is unmatched, and his reach in influencing "every branch of intellectual enterprise" including Western ethical and political theory, theology, rhetoric and literary analysis is equally long. As a result, Kukkonen argues, any analysis of reality today "will almost certainly carry Aristotelian overtones ... evidence of an exceptionally forceful mind."[148] Jonathan Barnes wrote that "an account of Aristotle's intellectual afterlife would be little less than a history of European thought".[149] Aristotle has been called the father of logic, biology, political science, zoology, embryology, natural law, scientific method, rhetoric, psychology, realism, criticism, individualism, teleology, and meteorology.[151] On his successor, Theophrastus Aristotle's pupil and successor, Theophrastus, wrote the History of Plants, a pioneering work in botany. Some of his technical terms remain in use, such as carpel from carpos, fruit, and pericarp, from pericarpion, seed chamber.[152] Theophrastus was much less concerned with formal causes than Aristotle was, instead pragmatically describing how plants functioned.[153][154] On later Greek philosophers The immediate influence of Aristotle's work was felt as the Lyceum grew into the Peripatetic school. Aristotle's students included Aristoxenus, Dicaearchus, Demetrius of Phalerum, Eudemos of Rhodes, Harpalus, Hephaestion, Mnason of Phocis, Nicomachus, and Theophrastus. Aristotle's influence over Alexander the Great is seen in the latter's bringing with him on his expedition a host of zoologists, botanists, and researchers. He had also learned a great deal about Persian customs and traditions from his teacher. Although his respect for Aristotle was diminished as his travels made it clear that much of Aristotle's geography was clearly wrong, when the old philosopher released his works to the public, Alexander complained "Thou hast not done well to publish thy acroamatic doctrines; for in what shall I surpass other men if those doctrines wherein I have been trained are to be all men's common property?"[155] On Hellenistic science After Theophrastus, the Lyceum failed to produce any original work. Though interest in Aristotle's ideas survived, they were generally taken unquestioningly.[156] It is not until the age of Alexandria under the Ptolemies that advances in biology can be again found. The first medical teacher at Alexandria, Herophilus of Chalcedon, corrected Aristotle, placing intelligence in the brain, and connected the nervous system to motion and sensation. Herophilus also distinguished between veins and arteries, noting that the latter pulse while the former do not.[157] Though a few ancient atomists such as Lucretius challenged the teleological viewpoint of Aristotelian ideas about life, teleology (and after the rise of Christianity, natural theology) would remain central to biological thought essentially until the 18th and 19th centuries. Ernst Mayr states that there was "nothing of any real consequence in biology after Lucretius and Galen until the Renaissance."[158] On Byzantine scholars Greek Christian scribes played a crucial role in the preservation of Aristotle by copying all the extant Greek language manuscripts of the corpus. The first Greek Christians to comment extensively on Aristotle were Philoponus, Elias, and David in the sixth century, and Stephen of Alexandria in the early seventh century.[159] John Philoponus stands out for having attempted a fundamental critique of Aristotle's views on the eternity of the world, movement, and other elements of Aristotelian thought.[160] Philoponus questioned Aristotle's teaching of physics, noting its flaws and introducing the theory of impetus to explain his observations.[161] After a hiatus of several centuries, formal commentary by Eustratius and Michael of Ephesus reappeared in the late eleventh and early twelfth centuries, apparently sponsored by Anna Comnena.[162] On the medieval Islamic world Aristotle was one of the most revered Western thinkers in early Islamic theology. Most of the still extant works of Aristotle,[163] as well as a number of the original Greek commentaries, were translated into Arabic and studied by Muslim philosophers, scientists and scholars. Averroes, Avicenna and Alpharabius, who wrote on Aristotle in great depth, also influenced Thomas Aquinas and other Western Christian scholastic philosophers. Alkindus greatly admired Aristotle's philosophy,[164] and Averroes spoke of Aristotle as the "exemplar" for all future philosophers.[165] Medieval Muslim scholars regularly described Aristotle as the "First Teacher".[163] The title was later used by Western philosophers (as in the famous poem of Dante) who were influenced by the tradition of Islamic philosophy.[166] On medieval Europe With the loss of the study of ancient Greek in the early medieval Latin West, Aristotle was practically unknown there from c. AD 600 to c. 1100 except through the Latin translation of the Organon made by Boethius. In the twelfth and thirteenth centuries, interest in Aristotle revived and Latin Christians had translations made, both from Arabic translations, such as those by Gerard of Cremona,[168] and from the original Greek, such as those by James of Venice and William of Moerbeke. After the Scholastic Thomas Aquinas wrote his Summa Theologica, working from Moerbeke's translations and calling Aristotle "The Philosopher",[169] the demand for Aristotle's writings grew, and the Greek manuscripts returned to the West, stimulating a revival of Aristotelianism in Europe that continued into the Renaissance.[170] These thinkers blended Aristotelian philosophy with Christianity, bringing the thought of Ancient Greece into the Middle Ages. Scholars such as Boethius, Peter Abelard, and John Buridan worked on Aristotelian logic.[171] The medieval English poet Chaucer describes his student as being happy by having at his beddes heed Twenty bookes, clad in blak or reed, Of aristotle and his philosophie,[172] A cautionary medieval tale held that Aristotle advised his pupil Alexander to avoid the king's seductive mistress, Phyllis, but was himself captivated by her, and allowed her to ride him. Phyllis had secretly told Alexander what to expect, and he witnessed Phyllis proving that a woman's charms could overcome even the greatest philosopher's male intellect. Artists such as Hans Baldung produced a series of illustrations of the popular theme.[173][167] The Italian poet Dante says of Aristotle in The Divine Comedy: Dante L'Inferno, Canto IV. 131–135 Translation Hell vidi 'l maestro di color che sanno seder tra filosofica famiglia. Tutti lo miran, tutti onor li fanno: quivi vid'ïo Socrate e Platone che 'nnanzi a li altri più presso li stanno; I saw the Master there of those who know, Amid the philosophic family, By all admired, and by all reverenced; There Plato too I saw, and Socrates, Who stood beside him closer than the rest. Besides Dante's fellow poets, the classical figure that most influenced the Comedy is Aristotle. Dante built up the philosophy of the Comedy with the works of Aristotle as a foundation, just as the scholastics used Aristotle as the basis for their thinking. Dante knew Aristotle directly from Latin translations of his works and indirectly through quotations in the works of Albert Magnus.[174] Dante even acknowledges Aristotle's influence explicitly in the poem, specifically when Virgil justifies the Inferno's structure by citing the Nicomachean Ethics.[175] On medieval Judaism Moses Maimonides (considered to be the foremost intellectual figure of medieval Judaism)[176] adopted Aristotelianism from the Islamic scholars and based his Guide for the Perplexed on it and that became the basis of Jewish scholastic philosophy. Maimonides also considered Aristotle to be the greatest philosopher that ever lived, and styled him as the "chief of the philosophers".[177][178][179] Also, in his letter to Samuel ibn Tibbon, Maimonides observes that there is no need for Samuel to study the writings of philosophers who preceded Aristotle because the works of the latter are "sufficient by themselves and [superior] to all that were written before them. His intellect, Aristotle's is the extreme limit of human intellect, apart from him upon whom the divine emanation has flowed forth to such an extent that they reach the level of prophecy, there being no level higher".[180] On Early Modern scientists In the Early Modern period, scientists such as William Harvey in England and Galileo Galilei in Italy reacted against the theories of Aristotle and other classical era thinkers like Galen, establishing new theories based to some degree on observation and experiment. Harvey demonstrated the circulation of the blood, establishing that the heart functioned as a pump rather than being the seat of the soul and the controller of the body's heat, as Aristotle thought.[181] Galileo used more doubtful arguments to displace Aristotle's physics, proposing that bodies all fall at the same speed whatever their weight.[182] On 18th/19th-century thinkers The 19th-century German philosopher Friedrich Nietzsche has been said to have taken nearly all of his political philosophy from Aristotle.[183] Aristotle rigidly separated action from production, and argued for the deserved subservience of some people ("natural slaves"),[184] and the natural superiority (virtue, arete) of others. It was Martin Heidegger, not Nietzsche, who elaborated a new interpretation of Aristotle, intended to warrant his deconstruction of scholastic and philosophical tradition.[185] The English mathematician George Boole fully accepted Aristotle's logic, but decided "to go under, over, and beyond" it with his system of algebraic logic in his 1854 book The Laws of Thought. This gives logic a mathematical foundation with equations, enables it to solve equations as well as check validity, and allows it to handle a wider class of problems by expanding propositions of any number of terms, not just two.[186] Charles Darwin regarded Aristotle as the most important contributor to the subject of biology. In an 1882 letter he wrote that "Linnaeus and Cuvier have been my two gods, though in very different ways, but they were mere schoolboys to old Aristotle".[187][188] Also, in later editions of the book "On the Origin of Species', Darwin traced evolutionary ideas as far back as Aristotle;[189] the text he cites is a summary by Aristotle of the ideas of the earlier Greek philosopher Empedocles.[190] James Joyce's favoured philosopher was Aristotle, whom he considered to be "the greatest thinker of all times".[191] Samuel Taylor Coleridge said: Everybody is born either a Platonist or an Aristotelian.[192] Ayn Rand acknowledged Aristotle as her greatest influence[193] and remarked that in the history of philosophy she could only recommend "three A's"—Aristotle, Aquinas, and Ayn Rand.[194] She also regarded Aristotle as the greatest of all philosophers.[195] Karl Marx considered Aristotle to be the "greatest thinker of antiquity", and called him a "giant thinker", a "genius", and "the great scholar".[196][197][198] Modern rejection and rehabilitation During the 20th century, Aristotle's work was widely criticized. The philosopher Bertrand Russell argued that "almost every serious intellectual advance has had to begin with an attack on some Aristotelian doctrine". Russell called Aristotle's ethics "repulsive", and labelled his logic "as definitely antiquated as Ptolemaic astronomy". Russell stated that these errors made it difficult to do historical justice to Aristotle, until one remembered what an advance he made upon all of his predecessors.[12] The Dutch historian of science Eduard Jan Dijksterhuis wrote that Aristotle and his predecessors showed the difficulty of science by "proceed[ing] so readily to frame a theory of such a general character" on limited evidence from their senses.[199] In 1985, the biologist Peter Medawar could still state in "pure seventeenth century"[200] tones that Aristotle had assembled "a strange and generally speaking rather tiresome farrago of hearsay, imperfect observation, wishful thinking and credulity amounting to downright gullibility".[200][201] By the start of the 21st century, however, Aristotle was taken more seriously: Kukkonen noted that "In the best 20th-century scholarship Aristotle comes alive as a thinker wrestling with the full weight of the Greek philosophical tradition."[148] Alasdair MacIntyre has attempted to reform what he calls the Aristotelian tradition in a way that is anti-elitist and capable of disputing the claims of both liberals and Nietzscheans.[202] Kukkonen observed, too, that "that most enduring of romantic images, Aristotle tutoring the future conqueror Alexander" remained current, as in the 2004 film Alexander, while the "firm rules" of Aristotle's theory of drama have ensured a role for the Poetics in Hollywood.[148] Biologists continue to be interested in Aristotle's thinking. Armand Marie Leroi has reconstructed Aristotle's biology,[203] while Niko Tinbergen's four questions, based on Aristotle's four causes, are used to analyse animal behaviour; they examine function, phylogeny, mechanism, and ontogeny.[204][205] Surviving works Corpus Aristotelicum The works of Aristotle that have survived from antiquity through medieval manuscript transmission are collected in the Corpus Aristotelicum. These texts, as opposed to Aristotle's lost works, are technical philosophical treatises from within Aristotle's school.[206] Reference to them is made according to the organization of Immanuel Bekker's Royal Prussian Academy edition (Aristotelis Opera edidit Academia Regia Borussica, Berlin, 1831–1870), which in turn is based on ancient classifications of these works.[207] Loss and preservation Aristotle wrote his works on papyrus scrolls, the common writing medium of that era.[upper-alpha 14] His writings are divisible into two groups: the "exoteric", intended for the public, and the "esoteric", for use within the Lyceum school.[209][upper-alpha 15][210] Aristotle's "lost" works stray considerably in characterization from the surviving Aristotelian corpus. Whereas the lost works appear to have been originally written with a view to subsequent publication, the surviving works mostly resemble lecture notes not intended for publication.[211][209] Cicero's description of Aristotle's literary style as "a river of gold" must have applied to the published works, not the surviving notes.[upper-alpha 16] A major question in the history of Aristotle's works is how the exoteric writings were all lost, and how the ones now possessed came to be found.[213] The consensus is that Andronicus of Rhodes collected the esoteric works of Aristotle's school which existed in the form of smaller, separate works, distinguished them from those of Theophrastus and other Peripatetics, edited them, and finally compiled them into the more cohesive, larger works as they are known today.[214][215] According to Strabo and Plutarch, after Aristotle's death, his library and writings went to Theophrastus (Aristotle's successor as head of the Lycaeum and the Peripatetic school).[216] After the death of Theophrastus, the peripatetic library went to Neleus of Scepsis.[217]: 5 Some time later, the Kingdom of Pergamon began conscripting books for a royal library, and the heirs of Neleus hid their collection in a cellar to prevent it from being seized for that purpose. The library was stored there for about a century and a half, in conditions that were not ideal for document preservation. On the death of Attalus III, which also ended the royal library ambitions, the existence of Aristotelian library was disclosed, and it was purchased by Apellicon and returned to Athens in about 100 BCE.[217]: 5–6 Apellicon sought to recover the texts, many of which were seriously degraded at this point due to the conditions in which they were stored. He had them copied out into new manuscripts, and used his best guesswork to fill in the gaps where the originals were unreadable.[217]: 5–6 When Sulla seized Athens in 86 BCE, he seized the library and transferred it to Rome. There, Andronicus of Rhodes organized the texts into the first complete edition of Aristotle's works (and works attributed to him).[218] The Aristotelian texts we have to day are based on these.[217]: 6–8 Legacy Depictions Paintings Aristotle has been depicted by major artists including Lucas Cranach the Elder,[219] Justus van Gent, Raphael, Paolo Veronese, Jusepe de Ribera,[220] Rembrandt,[221] and Francesco Hayez over the centuries. Among the best-known depictions is Raphael's fresco The School of Athens, in the Vatican's Apostolic Palace, where the figures of Plato and Aristotle are central to the image, at the architectural vanishing point, reflecting their importance.[222] Rembrandt's Aristotle with a Bust of Homer, too, is a celebrated work, showing the knowing philosopher and the blind Homer from an earlier age: as the art critic Jonathan Jones writes, "this painting will remain one of the greatest and most mysterious in the world, ensnaring us in its musty, glowing, pitch-black, terrible knowledge of time."[223][224] • Nuremberg Chronicle anachronistically shows Aristotle in a medieval scholar's clothing. Ink and watercolour on paper, 1493 • Aristotle by Justus van Gent. Oil on panel, c. 1476 • Phyllis and Aristotle by Lucas Cranach the Elder. Oil on panel, 1530 • Aristotle by Paolo Veronese, Biblioteka Marciana. Oil on canvas, 1560s • Aristotle and Campaspe,[upper-alpha 17] Alessandro Turchi (attrib.) Oil on canvas, 1713 • Aristotle by Jusepe de Ribera. Oil on canvas, 1637 • Aristotle with a Bust of Homer by Rembrandt. Oil on canvas, 1653 • Aristotle by Johann Jakob Dorner the Elder. Oil on canvas, by 1813 • Aristotle by Francesco Hayez. Oil on canvas, 1811 Sculptures • Roman copy of 1st or 2nd century from original bronze by Lysippos. Louvre Museum • Roman copy of 117-138 AD of Greek original. Palermo Regional Archeology Museum • Relief of Aristotle and Plato by Luca della Robbia, Florence Cathedral, 1437–1439 • Stone statue in niche, Gladstone's Library, Hawarden, Wales, 1899 • Bronze statue, University of Freiburg, Germany, 1915 Eponyms The Aristotle Mountains in Antarctica are named after Aristotle. He was the first person known to conjecture, in his book Meteorology, the existence of a landmass in the southern high-latitude region and called it Antarctica.[225] Aristoteles is a crater on the Moon bearing the classical form of Aristotle's name.[226] See also • Aristotelian Society • Conimbricenses • Perfectionism References Notes 1. See Shields 2012, pp. 3–16; Düring 1957 covers ancient biographies of Aristotle. 2. That these dates (the first half of the Olympiad year 384/383 BC, and in 322 shortly before the death of Demosthenes) are correct was shown by August Boeckh (Kleine Schriften VI 195); for further discussion, see Felix Jacoby on FGrHist 244 F 38. Ingemar Düring, Aristotle in the Ancient Biographical Tradition, Göteborg, 1957, p. 253 3. This type of syllogism, with all three terms in 'a', is known by the traditional (medieval) mnemonic Barbara.[26] 4. M is the Middle (here, Men), S is the Subject (Greeks), P is the Predicate (mortal).[26] 5. The first equation can be read as 'It is not true that there exists an x such that x is a man and that x is not mortal.'[27] 6. Rhett Allain notes that Newton's First Law is "essentially a direct reply to Aristotle, that the natural state is not to change motion.[42] 7. Leonard Susskind comments that Aristotle had clearly never gone ice skating or he would have seen that it takes force to stop an object.[44] 8. For heavenly bodies like the Sun, Moon, and stars, the observed motions are "to a very good approximation" circular around the Earth's centre, (for example, the apparent rotation of the sky because of the rotation of the Earth, and the rotation of the moon around the Earth) as Aristotle stated.[45] 9. Drabkin quotes numerous passages from Physics and On the Heavens (De Caelo) which state Aristotle's laws of motion.[43] 10. Drabkin agrees that density is treated quantitatively in this passage, but without a sharp definition of density as weight per unit volume.[43] 11. Philoponus and Galileo correctly objected that for the transient phase (still increasing in speed) with heavy objects falling a short distance, the law does not apply: Galileo used balls on a short incline to show this. Rovelli notes that "Two heavy balls with the same shape and different weight do fall at different speeds from an aeroplane, confirming Aristotle's theory, not Galileo's."[45] 12. For a different reading of social and economic processes in the Nicomachean Ethics and Politics see Polanyi, Karl (1957) "Aristotle Discovers the Economy" in Primitive, Archaic and Modern Economies: Essays of Karl Polanyi ed. G. Dalton, Boston 1971, 78–115. 13. "Where, as among the Lacedaemonians, the state of women is bad, almost half of human life is spoilt."[141] 14. "When the Roman dictator Sulla invaded Athens in 86 BC, he brought back to Rome a fantastic prize – Aristotle's library. Books then were papyrus rolls, from 10 to 20 feet long, and since Aristotle's death in 322 BC, worms and damp had done their worst. The rolls needed repairing, and the texts clarifying and copying on to new papyrus (imported from Egypt – Moses' bulrushes). The man in Rome who put Aristotle's library in order was a Greek scholar, Tyrannio."[208] 15. Aristotle: Nicomachean Ethics 1102a26–27. Aristotle himself never uses the term "esoteric" or "acroamatic". For other passages where Aristotle speaks of exōterikoi logoi, see W.D. Ross, Aristotle's Metaphysics (1953), vol. 2 pp= 408–410. Ross defends an interpretation according to which the phrase, at least in Aristotle's own works, usually refers generally to "discussions not peculiar to the Peripatetic school", rather than to specific works of Aristotle's own. 16. "veniet flumen orationis aureum fundens Aristoteles", (Google translation: "Aristotle will come pouring forth a golden stream of eloquence").[212] 17. Compare the medieval tale of Phyllis and Alexander above. Citations 1. Collins English Dictionary. 2. Aristotle (Greek philosopher). 3. McLeisch 1999, p. 5. 4. Aristoteles-Park in Stagira. 5. Borchers, Timothy A.; Hundley, Heather (2018). Rhetorical theory : an introduction (Second ed.). Long Grove, Illinois. ISBN 978-1-4786-3580-2. OCLC 1031145493.{{cite book}}: CS1 maint: location missing publisher (link) 6. Hall 2018, p. 14. 7. Anagnostopoulos 2013, p. 4. 8. Blits 1999, pp. 58–63. 9. Evans 2006. 10. Aristotle 1984, pp. Introduction. 11. Shields 2016. 12. Russell 1972. 13. Green 1991, pp. 58–59. 14. Smith 2007, p. 88. 15. Humphreys 2009. 16. Green 1991, p. 460. 17. Filonik 2013, pp. 72–73. 18. Jones 1980, p. 216. 19. Gigon 2017, p. 41. 20. Düring 1957, p. T44a-e. 21. Britton, Bianca (27 May 2016). "Is this Aristotle's tomb?". CNN. Retrieved 21 January 2023. 22. Haase 1992, p. 3862. 23. Degnan 1994, pp. 81–89. 24. Corcoran 2009, pp. 1–20. 25. Kant 1787, pp. Preface. 26. 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(ed.). Stanford Encyclopedia of Philosophy. • Lahanas, Michael. "Optics and ancient Greeks". Mlahanas.de. Archived from the original on 11 April 2009. Retrieved 26 April 2009. • Lee, Ellen Wardwell; Robinson, Anne (2005). Indianapolis Museum of Art: Highlights of the Collection. Indianapolis Museum of Art. ISBN 978-0-936260-77-8. • Leroi, Armand Marie (Presenter) (3 May 2011). "Aristotle's Lagoon: Embryo Inside a Chicken's Egg". BBC. Retrieved 17 November 2016. • Leroi, Armand Marie (2015). The Lagoon: How Aristotle Invented Science. Bloomsbury. ISBN 978-1-4088-3622-4. • Lindberg, David (1992). The Beginnings of Western Science. University of Chicago Press. ISBN 978-0-226-48205-7. • Lloyd, G. E. R. (1968). The critic of Plato. ISBN 978-0-521-09456-6. {{cite book}}: \|work= ignored (help) • Lloyd, G. E. R. (1996). Causes and correlations. ISBN 978-0-521-55695-8. {{cite book}}: \|work= ignored (help) • Lovejoy, Arthur O. (31 January 1976). The Great Chain of Being: A Study of the History of an Idea. Harvard University Press. ISBN 978-0-674-36153-9. • "Lucas Cranach the Elder\| Phyllis and Aristotle". Sotheby's. 2008. Retrieved 23 March 2018. • Lyell, Charles (1832). Principles of Geology. J. Murray, 1832. OCLC 609586345. • MacDougall-Shackleton, Scott A. (27 July 2011). "The levels of analysis revisited". Philosophical Transactions of the Royal Society B: Biological Sciences. 366 (1574): 2076–2085. doi:10.1098/rstb.2010.0363. PMC 3130367. PMID 21690126. • Machamer, Peter (2017). "Galileo Galilei". In Zalta, Edward N. (ed.). Stanford Encyclopedia of Philosophy. • Magee, Bryan (2010). The Story of Philosophy. Dorling Kindersley. ISBN 978-0-241-24126-4. • Mason, Stephen F. (1979). A History of the Sciences. Collier Books. ISBN 978-0-02-093400-4. OCLC 924760574. • Mayr, Ernst (1982). The Growth of Biological Thought. Belknap Press. ISBN 978-0-674-36446-2. • Mayr, Ernst (1985). The Growth of Biological Thought. Harvard University Press. ISBN 978-0-674-36446-2. • McLeisch, Kenneth Cole (1999). Aristotle: The Great Philosophers. Routledge. ISBN 978-0-415-92392-7. • Medawar, Peter B.; Medawar, J. S. (1984). Aristotle to Zoos: a philosophical dictionary of biology. Oxford University Press. ISBN 978-0-19-283043-2. • Miller, Willard M. (1973). "Aristotle on Necessity, Chance, and Spontaneity". New Scholasticism. 47 (2): 204–213. doi:10.5840/newscholas197347237. • Modrak, Deborah (2009). "Dreams and Method in Aristotle". Skepsis: A Journal for Philosophy and Interdisciplinary Research. 20: 169–181. • Moore, Ruth (1956). The Earth We Live On. Alfred A. Knopf. OCLC 1024467091. • Morsink, Johannes (Spring 1979). "Was Aristotle's Biology Sexist?". Journal of the History of Biology. 12 (1): 83–112. doi:10.1007/bf00128136. JSTOR 4330727. PMID 11615776. S2CID 6090923. • Nasr, Seyyed Hossein (1996). The Islamic Intellectual Tradition in Persia. Curzon Press. ISBN 978-0-7007-0314-2. • Phelan, Joseph (September 2002). "The Philosopher as Hero: Raphael's The School of Athens". ArtCyclopedia. Retrieved 23 March 2018. • "Phyllis and Aristotle". 1 February 2019 – via Musee du Louvre. • Pickover, Clifford A. (2009). The Math Book: From Pythagoras to the 57th Dimension, 250 Milestones in the History of Mathematics. Sterling. ISBN 978-1-4027-5796-9. • "Plutarch – Life of Alexander (Part 1 of 7)". penelope.uchicago.edu. Loeb Classical Library. 1919. Retrieved 31 January 2019. • "Predicate Logic" (PDF). University of Texas. Archived (PDF) from the original on 29 March 2018. Retrieved 29 March 2018. • Rhodes, Frank Harold Trevor (1974). Evolution. Golden Press. ISBN 978-0-307-64360-5. • Robbins, Lionel (2000). Medema, Steven G.; Samuels, Warren J. (eds.). A History of Economic Thought: The LSE Lectures. Princeton University Press. • Rorty, Amélie Oksenberg (1996). "Structuring Rhetoric". In Rorty, Amélie Oksenberg (ed.). Essays on Aristotle's Rhetoric. University of California Press. ISBN 978-0-520-20227-6. • Rovelli, Carlo (2015). "Aristotle's Physics: A Physicist's Look". Journal of the American Philosophical Association. 1 (1): 23–40. arXiv:1312.4057. doi:10.1017/apa.2014.11. S2CID 44193681. • Russell, Bertrand (1972). A history of western philosophy. Simon and Schuster. ISBN 978-0-671-31400-2. • Sedley, David (2007). Creationism and Its Critics in Antiquity. University of California Press. ISBN 978-0-520-25364-3. • Shields, Christopher (2012). The Oxford Handbook of Aristotle. OUP USA. ISBN 978-0-19-518748-9. • Shields, Christopher (2016). "Aristotle's Psychology". In Zalta, Edward N. (ed.). Stanford Encyclopedia of Philosophy (Winter 2016 ed.). • Shute, Clarence (1941). The Psychology of Aristotle: An Analysis of the Living Being. Columbia University Press. OCLC 936606202. • Sikka, Sonya (1997). Forms of Transcendence: Heidegger and Medieval Mystical Theology. SUNY Press. ISBN 978-0-7914-3345-4. • Smith, Robin (2017). "Aristotle's Logic". In Zalta, Edward N. (ed.). Stanford Encyclopedia of Philosophy. • Smith, William George (2007) [1869]. Dictionary of Greek and Roman Biography and Mythology. J. Walton. Retrieved 30 January 2019 – via Internet Archive. • Sorabji, R. (2006). Aristotle on Memory (2nd ed.). Chicago: University of Chicago Press. p. 54. And this is exactly why we hunt for the successor, starting in our thoughts from the present or from something else, and from something similar, or opposite, or neighbouring. By this means recollection occurs... • Sorabji, Richard (1990). Aristotle Transformed. Duckworth. ISBN 978-0-7156-2254-4. • Staley, Kevin (1989). "Al-Kindi on Creation: Aristotle's Challenge to Islam". Journal of the History of Ideas. 50 (3): 355–370. doi:10.2307/2709566. JSTOR 2709566. • Susskind, Leonard (3 October 2011). "Classical Mechanics, Lectures 2, 3". The Theoretical Minimum. Retrieved 11 May 2018. • Taylor, Henry Osborn (1922). "Chapter 3: Aristotle's Biology". Greek Biology and Medicine. Archived from the original on 27 March 2006. Retrieved 3 January 2017. • "The School of Athens by Raphael". Visual Arts Cork. Retrieved 22 March 2018. • Stewart, Jessica (2019). "The Story Behind Raphael's Masterpiece 'The School of Athens'". My Modern Met. Retrieved 29 March 2019. Plato's gesture toward the sky is thought to indicate his Theory of Forms. ... Conversely, Aristotle's hand is a visual representation of his belief that knowledge comes from experience. Empiricism, as it is known, theorizes that humans must have concrete evidence to support their ideas • Tangian, Andranik (2020). Analytical theory of democracy. Vols. 1 and 2. Studies in Choice and Welfare. Cham, Switzerland: Springer. doi:10.1007/978-3-030-39691-6. ISBN 978-3-030-39690-9. S2CID 216190330. • Thompson, D'Arcy (1910). Ross, W. D.; Smith, J. A. (eds.). Historia animalium - The works of Aristotle translated into English. Clarendon Press. OCLC 39273217. Archived from the original on 9 August 2019. Retrieved 19 March 2018. • Warren, Howard C. (1921). A History of the Association of Psychology. C. Scribner's sons. ISBN 978-0-598-91975-5. OCLC 21010604. • Webb, Wilse (1990). Dreamtime and dreamwork: Decoding the language of the night. Jeremy P. Tarcher. ISBN 978-0-87477-594-5. • "When libraries were on a roll". The Telegraph. 19 May 2001. Archived from the original on 10 January 2022. Retrieved 29 June 2017. • Wildberg (2016). "John Philoponus". In Zalta, Edward N. (ed.). Stanford Encyclopedia of Philosophy. • Zalta, Edward N., ed. (2018). "Aristotle's Influence". Stanford Encyclopedia of Philosophy (Spring 2018 ed.). • Darwin, Charles (1872), The Origin of Species by Means of Natural Selection, or the Preservation of Favoured Races in the Struggle for Life (6th ed.), London: John Murray, retrieved 9 January 2009 • Burns, Jennifer (2009). Goddess of the Market: Ayn Rand and the American Right. New York: Oxford University Press. ISBN 978-0-19-532487-7. • Sciabarra, Chris Matthew (1995). Ayn Rand: The Russian Radical. University Park, Pennsylvania: Pennsylvania State University Press. ISBN 978-0-271-01440-1. Further reading The secondary literature on Aristotle is vast. The following is only a small selection. • Ackrill, J. L. (1997). Essays on Plato and Aristotle, Oxford University Press. • Ackrill, J.L. (1981). Aristotle the Philosopher. Oxford University Press. • Adler, Mortimer J. (1978). Aristotle for Everybody. Macmillan. • Ammonius (1991). Cohen, S. Marc; Matthews, Gareth B (eds.). On Aristotle's Categories. Cornell University Press. ISBN 978-0-8014-2688-9. • Aristotle (1908–1952). The Works of Aristotle Translated into English Under the Editorship of W.D. Ross, 12 vols. Clarendon Press. These translations are available in several places online; see External links. • Bakalis, Nikolaos. (2005). Handbook of Greek Philosophy: From Thales to the Stoics Analysis and Fragments, Trafford Publishing, ISBN 978-1-4120-4843-9. • Bocheński, I. M. (1951). Ancient Formal Logic. North-Holland. • Bolotin, David (1998). An Approach to Aristotle's Physics: With Particular Attention to the Role of His Manner of Writing. Albany: SUNY Press. A contribution to our understanding of how to read Aristotle's scientific works. • Burnyeat, Myles F. et al. (1979). Notes on Book Zeta of Aristotle's Metaphysics. Oxford: Sub-faculty of Philosophy. • Cantor, Norman F.; Klein, Peter L., eds. (1969). Ancient Thought: Plato and Aristotle. Monuments of Western Thought. Vol. 1. Blaisdell. • Chappell, V. (1973). "Aristotle's Conception of Matter". Journal of Philosophy. 70 (19): 679–696. doi:10.2307/2025076. JSTOR 2025076. • Code, Alan (1995). Potentiality in Aristotle's Science and Metaphysics, Pacific Philosophical Quarterly 76. • Cohen, S. Marc; Reeve, C. D. C. (21 November 2020). "Aristotle's Metaphysics". Stanford Encyclopedia of Philosophy (Winter 2020 ed.). • Ferguson, John (1972). Aristotle. Twayne Publishers. ISBN 978-0-8057-2064-8. • De Groot, Jean (2014). Aristotle's Empiricism: Experience and Mechanics in the 4th century BC, Parmenides Publishing, ISBN 978-1-930972-83-4. • Frede, Michael (1987). Essays in Ancient Philosophy. Minneapolis: University of Minnesota Press. • Fuller, B.A.G. (1923). Aristotle. History of Greek Philosophy. Vol. 3. Cape. • Gendlin, Eugene T. (2012). Line by Line Commentary on Aristotle's De Anima Archived 27 March 2017 at the Wayback Machine, Volume 1: Books I & II; Volume 2: Book III. The Focusing Institute. • Gill, Mary Louise (1989). Aristotle on Substance: The Paradox of Unity. Princeton University Press. • Guthrie, W.K.C. (1981). A History of Greek Philosophy. Vol. 6. Cambridge University Press. • Halper, Edward C. (2009). One and Many in Aristotle's Metaphysics, Volume 1: Books Alpha – Delta. Parmenides Publishing. ISBN 978-1-930972-21-6. • Halper, Edward C. (2005). One and Many in Aristotle's Metaphysics, Volume 2: The Central Books. Parmenides Publishing. ISBN 978-1-930972-05-6. • Irwin, Terence H. (1988). Aristotle's First Principles (PDF). Oxford: Clarendon Press. ISBN 0-19-824290-5. • Jaeger, Werner (1948). Robinson, Richard (ed.). Aristotle: Fundamentals of the History of His Development (2nd ed.). Clarendon Press. • Jori, Alberto (2003). Aristotele, Bruno Mondadori (Prize 2003 of the "International Academy of the History of Science"), ISBN 978-88-424-9737-0. • Kiernan, Thomas P., ed. (1962). Aristotle Dictionary. Philosophical Library. • Knight, Kelvin (2007). Aristotelian Philosophy: Ethics and Politics from Aristotle to MacIntyre, Polity Press. • Lewis, Frank A. (1991). Substance and Predication in Aristotle. Cambridge University Press. • Lord, Carnes (1984). Introduction to The Politics, by Aristotle. Chicago University Press. • Loux, Michael J. (1991). Primary Ousia: An Essay on Aristotle's Metaphysics Ζ and Η. Ithaca, NY: Cornell University Press. • Maso, Stefano (Ed.), Natali, Carlo (Ed.), Seel, Gerhard (Ed.) (2012) Reading Aristotle: Physics VII. 3: What is Alteration? Proceedings of the International ESAP-HYELE Conference, Parmenides Publishing. ISBN 978-1-930972-73-5. • McKeon, Richard (1973). Introduction to Aristotle (2nd ed.). University of Chicago Press. • Owen, G. E. L. (1965c). "The Platonism of Aristotle". Proceedings of the British Academy. 50: 125–150. [Reprinted in J. Barnes, M. Schofield, and R.R.K. Sorabji, eds.(1975). Articles on Aristotle Vol 1. Science. London: Duckworth 14–34.] • Pangle, Lorraine Smith (2002). Aristotle and the Philosophy of Friendship. doi:10.1017/CBO9780511498282. ISBN 978-0-511-49828-2. • Plato (1979). Allen, Harold Joseph; Wilbur, James B (eds.). The Worlds of Plato and Aristotle. Prometheus Books. • Reeve, C. D. C. (2000). Substantial Knowledge: Aristotle's Metaphysics. Hackett. • Rose, Lynn E. (1968). Aristotle's Syllogistic. Charles C Thomas. • Ross, Sir David (1995). Aristotle (6th ed.). Routledge. • Scaltsas, T. (1994). Substances and Universals in Aristotle's Metaphysics. Cornell University Press. • Strauss, Leo (1964). "On Aristotle's Politics", in The City and Man, Rand McNally. • Swanson, Judith (1992). The Public and the Private in Aristotle's Political Philosophy. Cornell University Press. ISBN 978-0-8014-2319-2. • Veatch, Henry B. (1974). Aristotle: A Contemporary Appreciation. Indiana University Press. • Woods, M. J. (1991b). "Universals and Particular Forms in Aristotle's Metaphysics". Aristotle and the Later Tradition. Oxford Studies in Ancient Philosophy. Vol. Suppl. pp. 41–56. External links Greek Wikisource has original works by or about: Ἀριστοτέλης Library resources about Aristotle • Online books • Resources in your library • Resources in other libraries By Aristotle • Online books • Resources in your library • Resources in other libraries • Aristotle at PhilPapers • 2553 Aristotle at the Indiana Philosophy Ontology Project • At the Internet Encyclopedia of Philosophy: • Aristotle (general article) • Biology • Ethics • Logic • Metaphysics • Motion and its Place in Nature • Poetics • Politics • At the Internet Classics Archive • From the Stanford Encyclopedia of Philosophy: • Aristotle (general article) • Aristotle in the Renaissance • Biology • Causality • Commentators on Aristotle • Ethics • Logic • Mathematics • Metaphysics • Natural philosophy • Non-contradiction • Political theory • Psychology • Rhetoric • Turner, William (1907). "Aristotle" . Catholic Encyclopedia. Vol. 1. • Laërtius, Diogenes (1925). "The Peripatetics: Aristotle" . Lives of the Eminent Philosophers. Vol. 1:5. Translated by Hicks, Robert Drew (Two volume ed.). Loeb Classical Library. Collections of works • Works by Aristotle in eBook form at Standard Ebooks • At Massachusetts Institute of Technology • Works by Aristotle at Project Gutenberg • Works by or about Aristotle at Internet Archive • Works by Aristotle at LibriVox (public domain audiobooks) • Works by Aristotle at Open Library • (in English and Greek) Perseus Project at Tufts University • At the University of Adelaide Archived 15 February 2017 at the Wayback Machine • (in Greek and French) P. Remacle • The 11-volume 1837 Bekker edition of Aristotle's Works in Greek (PDF · DJVU) Aristotelianism Overview • Aristotle • Physics • Biology • Ethics • Logic Ideas and interests • Active intellect • Antiperistasis • Arete • Category of being • Catharsis • Correspondence theory of truth • Essence–accident • Eudaimonia • Four causes • Future contingents • Genus–differentia • Hexis • Hylomorphism • Lexis • Magnanimity • Mimesis • Minima naturalia • Moderate realism • Mythos • Philia • Horror vacui (physics) • Rational animal • Phronesis • Potentiality and actuality • Substance theory (hypokeimenon, ousia) • Syllogism • Telos • Temporal finitism • Quiddity • Haecceity • Unmoved mover • Virtue ethics • mathematical realism Works • Organon • Categories • On Interpretation • Prior Analytics • Posterior Analytics • Topics • Sophistical Refutations • Physics • On the Heavens • On Generation and Corruption • Meteorology • On the Universe • On the Soul • Parva Naturalia • Sense and Sensibilia • On Memory • On Sleep • On Dreams • On Divination in Sleep • On Length and Shortness of Life • On Youth, Old Age, Life and Death, and Respiration • On Breath • On Animals • History • Parts • Movement • Progression • Generation • On Colors • On Things Heard • Physiognomonics • On Plants • On Marvellous Things Heard • Mechanics • Problems • On Indivisible Lines • The Situations and Names of Winds • On Melissus, Xenophanes, and Gorgias • Metaphysics • Nicomachean Ethics • Magna Moralia • Eudemian Ethics • On Virtues and Vices • Politics • Economics • Constitution of the Athenians • Rhetoric • Rhetoric to Alexander • Poetics • Constitution of the Athenians (Aristotle) • Lost • Protrepticus Followers Peripatetic school • Aristoxenus • Clearchus of Soli • Dicaearchus • Eudemus of Rhodes • Theophrastus • Strato of Lampsacus • Lyco of Troas • Aristo of Ceos • Critolaus • Diodorus of Tyre • Erymneus • Andronicus of Rhodes Islamic Golden Age • Al-Kindi • Al-Farabi • Avicenna • Avicennism • Averroes • Averroism Jewish • Maimonides Scholasticism • Peter Lombard • Albertus Magnus • Thomas Aquinas • Thomism • Duns Scotus • Scotism • Peter of Spain • Jacopo Zabarella • Pietro Pomponazzi • Cesar Cremonini Modern • Newman • Trendelenburg • Brentano • Adler • Foot • MacIntyre • Smith • Hursthouse • Nussbaum • Rand Related topics • Plato • Platonism • Lyceum • Commentaries on Aristotle • Pseudo-Aristotle • Neoplatonism • Transmission of the Greek Classics • Aristotle's views on women • Aristotle's wheel paradox • Metabasis paradox • Category • Philosophy portal Links to related articles Peripatetic philosophers Greek era • Aristotle • Eudemus • Theophrastus • Aristoxenus • Chamaeleon • Phaenias • Praxiphanes • Dicaearchus • Nicomachus • Demetrius of Phalerum • Strato of Lampsacus • Clearchus • Hieronymus of Rhodes • Lyco of Troas • Aristo of Ceos • Satyrus • Critolaus • Diodorus of Tyre Roman era • Cratippus • Andronicus of Rhodes • Boethus of Sidon • Aristocles of Messene • Aspasius • Adrastus • Alexander of Aphrodisias • Themistius • Olympiodorus the Elder Metaphysics Theories • Abstract object theory • Action theory • Anti-realism • Determinism • Dualism • Enactivism • Essentialism • Existentialism • Free will • Idealism • Libertarianism • Liberty • Materialism • Meaning of life • Monism • Naturalism • Nihilism • Phenomenalism • Realism • Physicalism • Relativism • Scientific realism • Solipsism • Subjectivism • Substance theory • Theory of forms • Truthmaker theory • Type theory Concepts • Abstract object • Anima mundi • Category of being • Causality • Causal closure • Cogito, ergo sum • Concept • Embodied cognition • Essence • Existence • Experience • Hypostatic abstraction • Idea • Identity • Information • Insight • Intelligence • Intention • Linguistic modality • Matter • Meaning • Mental representation • Mind • Motion • Nature • Necessity • Object • Pattern • Perception • Physical object • Principle • Property • Qualia • Quality • Reality • Relation • Soul • Subject • Substantial form • Thought • Time • Truth • Type–token distinction • Universal • Unobservable • Value • more ... 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The Strange Logic of Random Graphs The Strange Logic of Random Graphs is a book on zero-one laws for random graphs. It was written by Joel Spencer and published in 2001 by Springer-Verlag as volume 22 of their book series Algorithms and Combinatorics. Topics The random graphs of the book are generated from the Erdős–Rényi–Gilbert model $G(n,p)$ in which $n$ vertices are given and a random choice is made whether to connect each pair of vertices by an edge, independently for each pair, with probability $p$ of making a connection. A zero-one law is a theorem stating that, for certain properties of graphs, and for certain choices of $p$, the probability of generating a graph with the property tends to zero or one in the limit as $n$ goes to infinity.[1] A fundamental result in this area, proved independently by Glebskiĭ et al. and by Ronald Fagin, is that there is a zero-one law for $G(n,1/2)$ for every property that can be described in the first-order logic of graphs.[2] Moreover, the limiting probability is one if and only if the infinite Rado graph has the property. For instance, a random graph in this model contains a triangle with probability tending to one; it contains a universal vertex with probability tending to zero. For other choices of $p$, other outcomes can occur. For instance, the limiting probability of containing a triangle is between 0 and 1 when $p=c/n$ for a constant $c$; it tends to 0 for smaller choices of $p$ and to 1 for larger choices. The function $1/n$ is said to be a threshold for the property of containing a triangle, meaning that it separates the values of $p$ with limiting probability 0 from the values with limiting probability 1.[1] The main result of the book (proved by Spencer with Saharon Shelah) is that irrational powers of $n$ are never threshold functions. That is, whenever $a>0$ is an irrational number, there is a zero-one law for the first-order properties of the random graphs $G(n,n^{-a})$.[1] A key tool in the proof is the Ehrenfeucht–Fraïssé game.[3] Audience and reception Although it is essentially the proof of a single theorem, aimed at specialists in the area, the book is written in a readable style that introduces the reader to many important topics in finite model theory and the theory of random graphs. Reviewer Valentin Kolchin, himself the author of another book on random graphs, writes that the book is "self-contained, easily read, and is distinguished by elegant writing", recommending it to probability theorists and logicians.[2] Reviewer Alessandro Berarducci calls the book "beautifully written" and its subject "fascinating".[1] References 1. Berarducci, Alessandro (2003), "Review of The Strange Logic of Random Graphs", Mathematical Reviews, MR 1847951 2. Kolchin, V. F. (January 2007), translated by Kolchin, A. V., "Review of The Strange Logic of Random Graphs", Theory of Probability and Its Applications, 51 (3): 554–555, doi:10.1137/s0040585x97982608 3. Frank, Ove, "Review of The Strange Logic of Random Graphs", zbMATH, Zbl 0976.05001	Wikipedia
The Symmetries of Things The Symmetries of Things is a book on mathematical symmetry and the symmetries of geometric objects, aimed at audiences of multiple levels. It was written over the course of many years by John Horton Conway, Heidi Burgiel, and Chaim Goodman-Strauss,[1] and published in 2008 by A K Peters. Its critical reception was mixed, with some reviewers praising it for its accessible and thorough approach to its material and for its many inspiring illustrations,[2][3][4][5] and others complaining about its inconsistent level of difficulty,[6] overuse of neologisms, failure to adequately cite prior work, and technical errors.[7] Topics The Symmetries of Things has three major sections, subdivided into 26 chapters.[8] The first of the sections discusses the symmetries of geometric objects. It includes both the symmetries of finite objects in two and three dimensions, and two-dimensional infinite structures such as frieze patterns and tessellations,[2] and develops a new notation for these symmetries based on work of Alexander Murray MacBeath that, as proven by the authors using a simplified form of the Riemann–Hurwitz formula, covers all possibilities.[9] Other topics include Euler's polyhedral formula and the classification of two-dimensional surfaces.[8] It is heavily illustrated with both artworks and objects depicting these symmetries, such as the art of M. C. Escher[2] and Bathsheba Grossman,[3][4] as well as new illustrations created by the authors using custom software.[2] The second section of the book considers symmetries more abstractly and combinatorially, considering both the color-preserving symmetries of colored objects, the symmetries of topological spaces described in terms of orbifolds, and abstract forms of symmetry described by group theory and presentations of groups. This section culminates with a classification of all of the finite groups with up to 2009 elements.[2] The third section of the book provides a classification of the three-dimensional space groups[2] and examples of honeycombs such as the Weaire–Phelan structure.[3] It also considers the symmetries of less familiar geometries: higher dimensional spaces, non-Euclidean spaces,[2] and three-dimensional flat manifolds.[9] Hyperbolic groups are used to provide a new explanation of the problem of hearing the shape of a drum.[8][9] It includes the first published classification of four-dimensional convex uniform polytopes announced by Conway and Richard K. Guy in 1965, and a discussion of William Thurston's geometrization conjecture, proved by Grigori Perelman shortly before the publication of the book, according to which all three-dimensional manifolds can be realized as symmetric spaces.[2] One omission lamented by Jaron Lanier is the set of regular projective polytopes such as the 11-cell.[4] Audience and reception Reviewer Darren Glass writes that different parts of the book are aimed at different audiences, resulting in "a wonderful book which can be appreciated on many levels"[2] and providing an unusual level of depth for a popular mathematics book.[5] Its first section, on symmetries of low-dimensional Euclidean spaces, is suitable for a general audience. The second part involves some understanding of group theory, as would be expected of undergraduate mathematics students, and some additional familiarity with abstract algebra towards its end. And the third part, more technical, is primarily aimed at researchers in these topics,[2] although much still remains accessible at the undergraduate level.[9] It also has exercises making it useful as a textbook, and its heavy use of color illustration would make it suitable as a coffee table book.[2] However, reviewer Robert Moyer finds fault with its choice to include material at significantly different levels of difficulty, writing that for most of its audience, too much of the book will be unreadable.[6] Much of the material in the book is either new, or previously known only through technical publications aimed at specialists,[1][6][8] and much of the previously-known material that it presents is described in new notation and nomenclature.[1][8] Although there are many other books on symmetry,[2] reviewer N. G. Macleod writes that this one "may well become the definitive guide in this area for many years".[3] Jaron Lanier calls it "a plaything, an inexhaustible exercise in brain expansion for the reader, a work of art and a bold statement of what the culture of math can be like", and "a masterpiece".[4] Despite these positive reviews, Branko Grünbaum, himself an authority on geometric symmetry, is much less enthusiastic, writing that the book has "some serious shortcomings". These include the unnecessary use of "cute" neologisms for concepts that already have well-established terminology, an inadequate treatment of MacBeath's and Andreas Dress's contributions to the book's notation, sloppy reasoning in some arguments, inaccurate claims of novelty and failure to credit previous work in the classification of colored plane patterns, missing cases in this classification, likely errors in other of the more technical parts, poor copyediting, and a lack of clear definitions that ends up leaving out such central notions as the symmetries of a circle without providing any explanation of why they were omitted.[7] References 1. Wilson, Phil (December 2008), "Review of The Symmetries of Things" (PDF), Plus Magazine, Millennium Mathematics Project, University of Cambridge 2. Glass, Darren (July 2008), "Review of The Symmetries of Things", MAA Reviews 3. Macleod, N. G. (March 2010), "Review of The Symmetries of Things", Mathematics in School, 39 (2): 43–44, JSTOR 20696997 4. Lanier, Jaron (January–February 2009), "From planar patterns to polytopes (review of The Symmetries of Things)", American Scientist, 97 (1): 73–75, JSTOR 27859276 5. Eaton, Charles (March 2012), The Mathematical Gazette, 96 (535): 188–190, doi:10.1017/S002555720000437X, JSTOR 23249557{{citation}}: CS1 maint: untitled periodical (link) 6. Moyer, Robert (May 2009), "Review of The Symmetries of Things", The Mathematics Teacher, 102 (9): 716–717, JSTOR 20876494 7. Grünbaum, Branko (June–July 2009), "Review of The Symmetries of Things", The American Mathematical Monthly, 116 (6): 555–562, doi:10.4169/193009709X470470, JSTOR 40391162 8. Chirteş, Florentina, "Review of The Symmetries of Things", zbMATH, Zbl 1173.00001 9. Conder, Marston (2009), "Review of The Symmetries of Things", MathSciNet, MR 2410150	Wikipedia
The Taylor Prize in Mathematics The Taylor Prize in Mathematics is a cash prize awarded annually to an outstanding graduate student of mathematics, displaying excellence in graduate research and overall accomplishments, at The George Washington University in Washington, DC. The prize is named after James Henry Taylor, a professor of mathematics at GW from 1929 to 1958.[1] History James Henry Taylor was a mathematics professor at GW from 1929 to 1958 and then professor emeritus until his death in 1972. Several years after his death the president of the university, Lloyd H. Elliott, along with several of the professors in the Mathematics Department, including Taylor's good friend Fritz Joachim Weyl, decided to create an award in his memory.[2] Money was deposited in an account from which the prize funds would be drawn annually. The interest that builds throughout the year on the account makes up the majority of the annual prize. For example, in 1983 first year graduate student Karma Dajani won the prize, which at the time was $500 but when she won the prize again in 1986 it was worth only $300 because interest had not accumulated for as many years.[3] Thomas J. Carter became the first recipient of the prize in 1977. About the recipients The GWU bulletin simply describes the criteria for recipients of the prize as, "awarded to an outstanding mathematics graduate student."[3] Many of the thirty-four winners of the Taylor Prize have gone on to become professors of mathematics at various universities. Most have published essays and books or given lectures on their specific subjects. Complete list of recipients[4] Year Name Current occupation 2022 Gabriel Montoya Vega and Guanning Zhang NSF Postdoc at CUNY, PhD candidate at the George Washington University 2021 Dionne Ibarra Research Fellow at Monash University[5] 2020 Rhea Palak Bakshi, Pavel Avdeyev Junior Fellow at the Institute for Theoretical Studies at ETH Zurich,[6] PhD candidate at the George Washington University 2019 Iva Bilanovic PhD student at the George Washington University 2018 Sujoy Mukherjee, Kai Yang Ross Assistant Professor at Ohio State University, postdoctoral associate at Florida International University 2017 Xiao Wang Jilin University 2016 Chong Wang postdoctoral fellow at McMaster University 2015 Seung Yeop Yang visiting assistant professor at the University of Denver 2014 Leah Marshall works at the U.S. Census Bureau,[7] also a Professorial Lecturer at The George Washington University[8] 2013 Carl Hammarsten, Jing Wang (Hammarsten) visiting assistant professor at Lafayette University[9] (Wang) assistant professor at Christian Brothers University [10] 2012 Kai Maeda, Tyler White 2010 Forest Fisher, Michael Coleman (Fisher) assistant professor of mathematics at Guttman Community College[11] (Coleman) works at U.S. Naval Research Laboratory,[12] also a professorial lecturer at the George Washington University [13] 2009 Radmila Sazdanovic assistant professor at North Carolina State University 2008 Jennifer Chubb and Hillary Einziger (Chubb) assistant professor at the University of San Francisco[14] (Einziger) educator assistant professor at the University of Cincinnati [15] 2007 Kerry Luse, Maciej Niebrzydowski (Niebrzydowski) assistant professor at University of Gdansk 2006 Malgorzata Dabkowska senior lecturer of mathematics at the University of Texas at Dallas[16] 2005 Eric Ufferman visiting assistant professor at GWU[17] 2004 Laure Helme-Guizon co-wrote "Torsion in Graph Technology" with Józef H. Przytycki[18] 2003 Amir Togha associate professor of mathematics at Bronx Community College[19] 2002 Mietek Dabkowski associate professor at University of Texas at Dallas 2001 Rumen Dimitrov associate professor of mathematics at Western Illinois University[20] 2000 Maxim Sokolov works at a Chicago investment company[21] 1999 Hongxun Qin works at Mitre Corporation [22] 1998 William Collier – 1997 Jun Zhang – 1996 Adam Sikora assistant professor of mathematics at SUNY at Buffalo[23] 1995 William Miller 1994 Qing Shen – 1992 Sita Ramamurti associate professor of mathematics at Trinity Washington University[24] 1991 Gary Schwartz – 1990 Uma Shivapuram – 1988 Claire Hackett – 1987 Hassan Sedaghat professor of mathematics at Virginia Commonwealth University[25] 1986 Karma Dajani senior lecturer and researcher at the University of Utrecht (Netherlands)[26] 1983 Karma Dajani senior lecturer and researcher at the University of Utrecht (Netherlands)[26] 1978 John Petro – 1977 Thomas J Carter professor and chair of computer science/cognitive studies at CSU Stanislaus[27] See also • List of mathematics awards References 1. "The George Washington University Bulletin < the George Washington University". 2. Fritz Joachim Weyl 3. "Prize winners". Archived from the original on 2010-06-02. Retrieved 2009-12-10. 4. "Undergraduate Student Opportunities \| the Department of Mathematics \| the George Washington University".{{cite web}}: CS1 maint: url-status (link) 5. "Dr Dionne Ibarra". Science. Retrieved 2022-11-04. 6. "People". 7. "Archived copy" (PDF). Archived from the original (PDF) on 2018-04-26. Retrieved 2018-04-25.{{cite web}}: CS1 maint: archived copy as title (link) 8. "Leah Marshall \| the Department of Mathematics \| the George Washington University". 9. "Carl Hammarsten · Math · Lafayette College". Archived from the original on 2018-04-26. Retrieved 2018-04-25. 10. https://www.cbu.edu/spotlight-jing-wang 11. "– New Faculty Members Join Guttman this FallGuttman Community College". 12. "Archived copy" (PDF). Archived from the original (PDF) on 2018-04-26. Retrieved 2018-04-25.{{cite web}}: CS1 maint: archived copy as title (link) 13. "Michael Coleman \| the Department of Mathematics \| the George Washington University". 14. "Usf \| math \| jennifer chubb". 15. http://www.artsci.uc.edu/faculty-staff/listing/by_dept/math.html?eid=einzighy&thecomp=uceprof 16. "Dabkowska, Malgorzata - Mathematical Sciences - the University of Texas at Dallas". 17. http://home.gwu.edu/~ufferman/%5B%5D 18. "Helme-Guizon". Archived from the original on 2012-07-16. Retrieved 2009-12-10. 19. "List of Faculty". Archived from the original on 2009-12-14. Retrieved 2009-12-10. 20. "Student and Employee Directory - Higher Values in Higher Education - Western Illinois University". 21. "Maxim Sokolov's Home Page: Mathematics". 22. "Hongxun Qin - Email, Phone - Multi-discipline Systems Engineer, Lead, MITRE". 23. "Adam Sikora, Associate Professor of Mathematics at University at Buffalo, SUNY". 24. http://www.trinitydc.edu/directory/directory.php?detail=32&type=people%5B%5D 25. "Prof. H. Sedaghat". 26. "Homepage for Karma Dajani". 27. http://csustan.csustan.edu/~tom/vita/TJC-CV-2006.pdf The George Washington University Colleges and schools • Columbian College of Arts and Sciences • Corcoran School of the Arts and Design • Elliott School of International Affairs • Graduate School of Political Management • Law School • School of Business • School of Engineering and Applied Science • School of Media and Public Affairs • Trachtenberg School of Public Policy and Public Administration Publications • Anthropological Quarterly • Law Review • Federal Circuit Bar Journal • Public Contract Law Journal • International Law Review • AIPLA Quarterly Journal • The Federal Communications Law Journal • The Washington Quarterly • Women's Health Issues • Planet Forward Centers and institutes • List of centers and research institutes at George Washington University • National Security Archive • Textile Museum (Washington, D.C.) • Institute for International Economic Policy • Munich Intellectual Property Law Center Athletics • George Washington Revolutionaries • Men's basketball • Women's basketball • Men's baseball • Softball • Men's soccer • Charles E. Smith Center • Tucker Field • "Hail to the Buff and Blue" • Football (defunct) Campuses • Campuses • Foggy Bottom (main campus) • Virginia Science & Technology Campus • Mount Vernon Campus Buildings and places • 2000 Pennsylvania Avenue • Anniversary Park • University Art Galleries • Corcoran Gallery of Art • Corcoran Hall • John J. Earley Office and Studio • Engine Company 23 • F Street House • Foggy Bottom–GWU station • Fulbright Hall • Fairbanks' George Washington • Jacqueline Bouvier Kennedy Onassis Hall • Lisner Auditorium • Madison Hall • Munson Hall • President's Office • Rawlins Park • Residence halls • River Horse • School Without Walls (Washington, D.C.) • Charles E. Smith Center • Snows Court (Washington, D.C.) • Stockton Hall • Hattie M. Strong Residence Hall • Oscar W. Underwood House • Washington Circle • Margaret Wetzel House • Maxwell Woodhull House Student life • GW-TV • Student Association • The GW Hatchet • Enosinian Society • The Taylor Prize in Mathematics • ΔΦΕ Libraries • Gelman Library • Jacob Burns Law Library • Himmelfarb Health Sciences Library People • President of the University (Thomas LeBlanc) • Notable Alumni & Notable Faculty (Law School Alumni · Elliott School Alumni & Faculty · Columbian College Alumni & Faculty · GW Business School Alumni & Faculty) Medicine and health • Medical Faculty Associates • George Washington University Hospital • School of Medicine and Health Sciences • School of Nursing • Milken Institute School of Public Health • Dr. Cyrus and Myrtle Katzen Cancer Research Center See also • Benjamin Franklin University • Mount Vernon College for Women • National University School of Law	Wikipedia
Theory of relativity The theory of relativity usually encompasses two interrelated physics theories by Albert Einstein: special relativity and general relativity, proposed and published in 1905 and 1915, respectively.[1] Special relativity applies to all physical phenomena in the absence of gravity. General relativity explains the law of gravitation and its relation to the forces of nature.[2] It applies to the cosmological and astrophysical realm, including astronomy.[3] The theory transformed theoretical physics and astronomy during the 20th century, superseding a 200-year-old theory of mechanics created primarily by Isaac Newton.[3][4][5] It introduced concepts including 4-dimensional spacetime as a unified entity of space and time, relativity of simultaneity, kinematic and gravitational time dilation, and length contraction. In the field of physics, relativity improved the science of elementary particles and their fundamental interactions, along with ushering in the nuclear age. With relativity, cosmology and astrophysics predicted extraordinary astronomical phenomena such as neutron stars, black holes, and gravitational waves.[3][4][5] Development and acceptance General relativity $G_{\mu \nu }+\Lambda g_{\mu \nu }={\kappa }T_{\mu \nu }$ • Introduction • History • Timeline • Tests • Mathematical formulation Fundamental concepts • Equivalence principle • Special relativity • World line • Pseudo-Riemannian manifold Phenomena • Kepler problem • Gravitational lensing • Gravitational waves • Frame-dragging • Geodetic effect • Event horizon • Singularity • Black hole Spacetime • Spacetime diagrams • Minkowski spacetime • Einstein–Rosen bridge • Equations • Formalisms Equations • Linearized gravity • Einstein field equations • Friedmann • Geodesics • Mathisson–Papapetrou–Dixon • Hamilton–Jacobi–Einstein Formalisms • ADM • BSSN • Post-Newtonian Advanced theory • Kaluza–Klein theory • Quantum gravity Solutions • Schwarzschild (interior) • Reissner–Nordström • Gödel • Kerr • Kerr–Newman • Kasner • Lemaître–Tolman • Taub–NUT • Milne • Robertson–Walker • Oppenheimer-Snyder • pp-wave • van Stockum dust • Weyl−Lewis−Papapetrou Scientists • Einstein • Lorentz • Hilbert • Poincaré • Schwarzschild • de Sitter • Reissner • Nordström • Weyl • Eddington • Friedman • Milne • Zwicky • Lemaître • Oppenheimer • Gödel • Wheeler • Robertson • Bardeen • Walker • Kerr • Chandrasekhar • Ehlers • Penrose • Hawking • Raychaudhuri • Taylor • Hulse • van Stockum • Taub • Newman • Yau • Thorne • others • Physics portal • Category Albert Einstein published the theory of special relativity in 1905, building on many theoretical results and empirical findings obtained by Albert A. Michelson, Hendrik Lorentz, Henri Poincaré and others. Max Planck, Hermann Minkowski and others did subsequent work. Einstein developed general relativity between 1907 and 1915, with contributions by many others after 1915. The final form of general relativity was published in 1916.[3] The term "theory of relativity" was based on the expression "relative theory" (German: Relativtheorie) used in 1906 by Planck, who emphasized how the theory uses the principle of relativity. In the discussion section of the same paper, Alfred Bucherer used for the first time the expression "theory of relativity" (German: Relativitätstheorie).[6][7] By the 1920s, the physics community understood and accepted special relativity.[8] It rapidly became a significant and necessary tool for theorists and experimentalists in the new fields of atomic physics, nuclear physics, and quantum mechanics. By comparison, general relativity did not appear to be as useful, beyond making minor corrections to predictions of Newtonian gravitation theory.[3] It seemed to offer little potential for experimental test, as most of its assertions were on an astronomical scale. Its mathematics seemed difficult and fully understandable only by a small number of people. Around 1960, general relativity became central to physics and astronomy. New mathematical techniques to apply to general relativity streamlined calculations and made its concepts more easily visualized. As astronomical phenomena were discovered, such as quasars (1963), the 3-kelvin microwave background radiation (1965), pulsars (1967), and the first black hole candidates (1981),[3] the theory explained their attributes, and measurement of them further confirmed the theory. Special relativity Main article: Special relativity Special relativity is a theory of the structure of spacetime. It was introduced in Einstein's 1905 paper "On the Electrodynamics of Moving Bodies" (for the contributions of many other physicists and mathematicians, see History of special relativity). Special relativity is based on two postulates which are contradictory in classical mechanics: 1. The laws of physics are the same for all observers in any inertial frame of reference relative to one another (principle of relativity). 2. The speed of light in a vacuum is the same for all observers, regardless of their relative motion or of the motion of the light source. The resultant theory copes with experiment better than classical mechanics. For instance, postulate 2 explains the results of the Michelson–Morley experiment. Moreover, the theory has many surprising and counterintuitive consequences. Some of these are: • Relativity of simultaneity: Two events, simultaneous for one observer, may not be simultaneous for another observer if the observers are in relative motion. • Time dilation: Moving clocks are measured to tick more slowly than an observer's "stationary" clock. • Length contraction: Objects are measured to be shortened in the direction that they are moving with respect to the observer. • Maximum speed is finite: No physical object, message or field line can travel faster than the speed of light in a vacuum. • The effect of gravity can only travel through space at the speed of light, not faster or instantaneously. • Mass–energy equivalence: E = mc2, energy and mass are equivalent and transmutable. • Relativistic mass, idea used by some researchers.[9] The defining feature of special relativity is the replacement of the Galilean transformations of classical mechanics by the Lorentz transformations. (See Maxwell's equations of electromagnetism.) General relativity Main articles: General relativity and Introduction to general relativity General relativity is a theory of gravitation developed by Einstein in the years 1907–1915. The development of general relativity began with the equivalence principle, under which the states of accelerated motion and being at rest in a gravitational field (for example, when standing on the surface of the Earth) are physically identical. The upshot of this is that free fall is inertial motion: an object in free fall is falling because that is how objects move when there is no force being exerted on them, instead of this being due to the force of gravity as is the case in classical mechanics. This is incompatible with classical mechanics and special relativity because in those theories inertially moving objects cannot accelerate with respect to each other, but objects in free fall do so. To resolve this difficulty Einstein first proposed that spacetime is curved. Einstein discussed his idea with mathematician Marcel Grossmann and they concluded that general relativity could be formulated in the context of Riemannian geometry which had been developed in the 1800s.[10] In 1915, he devised the Einstein field equations which relate the curvature of spacetime with the mass, energy, and any momentum within it. Some of the consequences of general relativity are: • Gravitational time dilation: Clocks run slower in deeper gravitational wells.[11] • Precession: Orbits precess in a way unexpected in Newton's theory of gravity. (This has been observed in the orbit of Mercury and in binary pulsars). • Light deflection: Rays of light bend in the presence of a gravitational field. • Frame-dragging: Rotating masses "drag along" the spacetime around them. • Expansion of the universe: The universe is expanding, and certain components within the universe can accelerate the expansion. Technically, general relativity is a theory of gravitation whose defining feature is its use of the Einstein field equations. The solutions of the field equations are metric tensors which define the topology of the spacetime and how objects move inertially. Experimental evidence Einstein stated that the theory of relativity belongs to a class of "principle-theories". As such, it employs an analytic method, which means that the elements of this theory are not based on hypothesis but on empirical discovery. By observing natural processes, we understand their general characteristics, devise mathematical models to describe what we observed, and by analytical means we deduce the necessary conditions that have to be satisfied. Measurement of separate events must satisfy these conditions and match the theory's conclusions.[2] Tests of special relativity Relativity is a falsifiable theory: It makes predictions that can be tested by experiment. In the case of special relativity, these include the principle of relativity, the constancy of the speed of light, and time dilation.[12] The predictions of special relativity have been confirmed in numerous tests since Einstein published his paper in 1905, but three experiments conducted between 1881 and 1938 were critical to its validation. These are the Michelson–Morley experiment, the Kennedy–Thorndike experiment, and the Ives–Stilwell experiment. Einstein derived the Lorentz transformations from first principles in 1905, but these three experiments allow the transformations to be induced from experimental evidence. Maxwell's equations—the foundation of classical electromagnetism—describe light as a wave that moves with a characteristic velocity. The modern view is that light needs no medium of transmission, but Maxwell and his contemporaries were convinced that light waves were propagated in a medium, analogous to sound propagating in air, and ripples propagating on the surface of a pond. This hypothetical medium was called the luminiferous aether, at rest relative to the "fixed stars" and through which the Earth moves. Fresnel's partial ether dragging hypothesis ruled out the measurement of first-order (v/c) effects, and although observations of second-order effects (v2/c2) were possible in principle, Maxwell thought they were too small to be detected with then-current technology.[13][14] The Michelson–Morley experiment was designed to detect second-order effects of the "aether wind"—the motion of the aether relative to the earth. Michelson designed an instrument called the Michelson interferometer to accomplish this. The apparatus was sufficiently accurate to detect the expected effects, but he obtained a null result when the first experiment was conducted in 1881,[15] and again in 1887.[16] Although the failure to detect an aether wind was a disappointment, the results were accepted by the scientific community.[14] In an attempt to salvage the aether paradigm, FitzGerald and Lorentz independently created an ad hoc hypothesis in which the length of material bodies changes according to their motion through the aether.[17] This was the origin of FitzGerald–Lorentz contraction, and their hypothesis had no theoretical basis. The interpretation of the null result of the Michelson–Morley experiment is that the round-trip travel time for light is isotropic (independent of direction), but the result alone is not enough to discount the theory of the aether or validate the predictions of special relativity.[18][19] While the Michelson–Morley experiment showed that the velocity of light is isotropic, it said nothing about how the magnitude of the velocity changed (if at all) in different inertial frames. The Kennedy–Thorndike experiment was designed to do that, and was first performed in 1932 by Roy Kennedy and Edward Thorndike.[20] They obtained a null result, and concluded that "there is no effect ... unless the velocity of the solar system in space is no more than about half that of the earth in its orbit".[19][21] That possibility was thought to be too coincidental to provide an acceptable explanation, so from the null result of their experiment it was concluded that the round-trip time for light is the same in all inertial reference frames.[18][19] The Ives–Stilwell experiment was carried out by Herbert Ives and G.R. Stilwell first in 1938[22] and with better accuracy in 1941.[23] It was designed to test the transverse Doppler effect – the redshift of light from a moving source in a direction perpendicular to its velocity—which had been predicted by Einstein in 1905. The strategy was to compare observed Doppler shifts with what was predicted by classical theory, and look for a Lorentz factor correction. Such a correction was observed, from which was concluded that the frequency of a moving atomic clock is altered according to special relativity.[18][19] Those classic experiments have been repeated many times with increased precision. Other experiments include, for instance, relativistic energy and momentum increase at high velocities, experimental testing of time dilation, and modern searches for Lorentz violations. Tests of general relativity General relativity has also been confirmed many times, the classic experiments being the perihelion precession of Mercury's orbit, the deflection of light by the Sun, and the gravitational redshift of light. Other tests confirmed the equivalence principle and frame dragging. Modern applications Far from being simply of theoretical interest, relativistic effects are important practical engineering concerns. Satellite-based measurement needs to take into account relativistic effects, as each satellite is in motion relative to an Earth-bound user and is thus in a different frame of reference under the theory of relativity. Global positioning systems such as GPS, GLONASS, and Galileo, must account for all of the relativistic effects, such as the consequences of Earth's gravitational field, in order to work with precision.[24] This is also the case in the high-precision measurement of time.[25] Instruments ranging from electron microscopes to particle accelerators would not work if relativistic considerations were omitted.[26] See also • Doubly special relativity • Galilean invariance • General relativity references • Special relativity references References 1. Einstein A. (1916), Relativity: The Special and General Theory (Translation 1920), New York: H. Holt and Company 2. Einstein, Albert (28 November 1919). "Time, Space, and Gravitation" . The Times. 3. Will, Clifford M (2010). "Relativity". Grolier Multimedia Encyclopedia. Archived from the original on 21 May 2020. Retrieved 1 August 2010. 4. Will, Clifford M (2010). "Space-Time Continuum". Grolier Multimedia Encyclopedia. Retrieved 1 August 2010. 5. Will, Clifford M (2010). "Fitzgerald–Lorentz contraction". Grolier Multimedia Encyclopedia. Archived from the original on 25 January 2013. Retrieved 1 August 2010. 6. Planck, Max (1906), "Die Kaufmannschen Messungen der Ablenkbarkeit der β-Strahlen in ihrer Bedeutung für die Dynamik der Elektronen (The Measurements of Kaufmann on the Deflectability of β-Rays in their Importance for the Dynamics of the Electrons)" , Physikalische Zeitschrift, 7: 753–761 7. Miller, Arthur I. (1981), Albert Einstein's special theory of relativity. Emergence (1905) and early interpretation (1905–1911), Reading: Addison–Wesley, ISBN 978-0-201-04679-3 8. Hey, Anthony J.G.; Walters, Patrick (2003). The New Quantum Universe (illustrated, revised ed.). Cambridge University Press. p. 227. Bibcode:2003nqu..book.....H. ISBN 978-0-521-56457-1. 9. Greene, Brian. "The Theory of Relativity, Then and Now". Retrieved 26 September 2015. 10. Einstein, A.; Grossmann, M. (1913). "Entwurf einer verallgemeinerten Relativitätstheorie und einer Theorie der Gravitation" [Outline of a Generalized Theory of Relativity and of a Theory of Gravitation]. Zeitschrift für Mathematik und Physik. 62: 225–261. 11. Feynman, Richard Phillips; Morínigo, Fernando B.; Wagner, William; Pines, David; Hatfield, Brian (2002). Feynman Lectures on Gravitation. West view Press. p. 68. ISBN 978-0-8133-4038-8., Lecture 5 12. Roberts, T; Schleif, S; Dlugosz, JM, eds. (2007). "What is the experimental basis of Special Relativity?". Usenet Physics FAQ. University of California, Riverside. Retrieved 31 October 2010. 13. Maxwell, James Clerk (1880), "On a Possible Mode of Detecting a Motion of the Solar System through the Luminiferous Ether" , Nature, 21 (535): 314–315, Bibcode:1880Natur..21S.314., doi:10.1038/021314c0 14. Pais, Abraham (1982). "Subtle is the Lord ...": The Science and the Life of Albert Einstein (1st ed.). Oxford: Oxford Univ. Press. pp. 111–113. ISBN 978-0-19-280672-7. 15. Michelson, Albert A. (1881). "The Relative Motion of the Earth and the Luminiferous Ether" . American Journal of Science. 22 (128): 120–129. Bibcode:1881AmJS...22..120M. doi:10.2475/ajs.s3-22.128.120. S2CID 130423116. 16. Michelson, Albert A. & Morley, Edward W. (1887). "On the Relative Motion of the Earth and the Luminiferous Ether" . American Journal of Science. 34 (203): 333–345. Bibcode:1887AmJS...34..333M. doi:10.2475/ajs.s3-34.203.333. S2CID 124333204.{{cite journal}}: CS1 maint: multiple names: authors list (link) 17. Pais, Abraham (1982). "Subtle is the Lord ...": The Science and the Life of Albert Einstein (1st ed.). Oxford: Oxford Univ. Press. p. 122. ISBN 978-0-19-280672-7. 18. Robertson, H.P. (July 1949). "Postulate versus Observation in the Special Theory of Relativity" (PDF). Reviews of Modern Physics. 21 (3): 378–382. Bibcode:1949RvMP...21..378R. doi:10.1103/RevModPhys.21.378. 19. Taylor, Edwin F.; John Archibald Wheeler (1992). Spacetime physics: Introduction to Special Relativity (2nd ed.). New York: W.H. Freeman. pp. 84–88. ISBN 978-0-7167-2327-1. 20. Kennedy, R.J.; Thorndike, E.M. (1932). "Experimental Establishment of the Relativity of Time" (PDF). Physical Review. 42 (3): 400–418. Bibcode:1932PhRv...42..400K. doi:10.1103/PhysRev.42.400. S2CID 121519138. Archived from the original (PDF) on 6 July 2020. 21. Robertson, H.P. (July 1949). "Postulate versus Observation in the Special Theory of Relativity" (PDF). Reviews of Modern Physics. 21 (3): 381. Bibcode:1949RvMP...21..378R. doi:10.1103/revmodphys.21.378. 22. Ives, H.E.; Stilwell, G.R. (1938). "An experimental study of the rate of a moving atomic clock". Journal of the Optical Society of America. 28 (7): 215. Bibcode:1938JOSA...28..215I. doi:10.1364/JOSA.28.000215. 23. Ives, H.E.; Stilwell, G.R. (1941). "An experimental study of the rate of a moving atomic clock. II". Journal of the Optical Society of America. 31 (5): 369. Bibcode:1941JOSA...31..369I. doi:10.1364/JOSA.31.000369. 24. Ashby, N. Relativity in the Global Positioning System. Living Rev. Relativ. 6, 1 (2003). doi:10.12942/lrr-2003-1"Archived copy" (PDF). Archived from the original (PDF) on 5 November 2015. Retrieved 9 December 2015.{{cite web}}: CS1 maint: archived copy as title (link) 25. Francis, S.; B. Ramsey; S. Stein; Leitner, J.; Moreau, J.M.; Burns, R.; Nelson, R.A.; Bartholomew, T.R.; Gifford, A. (2002). "Timekeeping and Time Dissemination in a Distributed Space-Based Clock Ensemble" (PDF). Proceedings 34th Annual Precise Time and Time Interval (PTTI) Systems and Applications Meeting: 201–214. Archived from the original (PDF) on 17 February 2013. Retrieved 14 April 2013. 26. Hey, Tony; Hey, Anthony J. G.; Walters, Patrick (1997). Einstein's Mirror (illustrated ed.). Cambridge University Press. p. x (preface). ISBN 978-0-521-43532-1. Further reading • Einstein, Albert (2005). Relativity: The Special and General Theory. Translated by Robert W. Lawson (The masterpiece science ed.). New York: Pi Press. ISBN 978-0-13-186261-6. • Einstein, Albert (1920). Relativity: The Special and General Theory (PDF). Henry Holt and Company. • Einstein, Albert; trans. Schilpp; Paul Arthur (1979). Albert Einstein, Autobiographical Notes (A Centennial ed.). La Salle, Illinois: Open Court Publishing Co. ISBN 978-0-87548-352-8. • Einstein, Albert (2009). Einstein's Essays in Science. Translated by Alan Harris (Dover ed.). Mineola, New York: Dover Publications. ISBN 978-0-486-47011-5. • Einstein, Albert (1956) [1922]. The Meaning of Relativity (5 ed.). Princeton University Press. • The Meaning of Relativity Albert Einstein: Four lectures delivered at Princeton University, May 1921 • How I created the theory of relativity Albert Einstein, December 14, 1922; Physics Today August 1982 • Relativity Sidney Perkowitz Encyclopædia Britannica External links Wikiquote has quotations related to Theory of relativity. 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Twin paradox • Terrell rotation Spacetime • Light cone • World line • Minkowski diagram • Biquaternions • Minkowski space General relativity Background • Introduction • Mathematical formulation Fundamental concepts • Equivalence principle • Riemannian geometry • Penrose diagram • Geodesics • Mach's principle Formulation • ADM formalism • BSSN formalism • Einstein field equations • Linearized gravity • Post-Newtonian formalism • Raychaudhuri equation • Hamilton–Jacobi–Einstein equation • Ernst equation Phenomena • Black hole • Event horizon • Singularity • Two-body problem • Gravitational waves: astronomy • detectors (LIGO and collaboration • Virgo • LISA Pathfinder • GEO) • Hulse–Taylor binary • Other tests: precession of Mercury • lensing (together with Einstein cross and Einstein rings) • redshift • Shapiro delay • frame-dragging / geodetic effect (Lense–Thirring precession) • pulsar timing arrays Advanced theories • Brans–Dicke theory • Kaluza–Klein • Quantum gravity Solutions • 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The Tower of Hanoi – Myths and Maths The Tower of Hanoi – Myths and Maths is a book in recreational mathematics, on the tower of Hanoi, baguenaudier, and related puzzles. It was written by Andreas M. Hinz, Sandi Klavžar, Uroš Milutinović, and Ciril Petr, and published in 2013 by Birkhäuser,[1][2][3][4][5][6][7][8] with an expanded second edition in 2018.[9] The Basic Library List Committee of the Mathematical Association of America has suggested its inclusion in undergraduate mathematics libraries.[2] The Tower of Hanoi – Myths and Maths First edition Author • Andreas M. Hinz • Sandi Klavžar • Uroš Milutinović • Ciril Petr SubjectTower of Hanoi and related puzzles PublisherBirkhäuser Publication date 2013 Topics Although this book is in recreational mathematics, it takes its subject seriously,[8] and brings in material from automata theory, computational complexity, the design and analysis of algorithms, graph theory, and group theory,[3] topology, fractal geometry, chemical graph theory, and even psychology[1] (where related puzzles have applications in psychological testing).[8] The 1st edition of the book had 10 chapters, and the 2nd edition has 11. In both cases they begin with chapter zero, on the background and history of the Tower of Hanoi puzzle, covering its real-world invention by Édouard Lucas and in the mythical backstory he invented for it. Chapter one considers the Baguenaudier puzzle (or, as it is often called, the Chinese rings), related to the tower of Hanoi both in the structure of its state space and in the fact that it takes an exponential number of moves to solve, and likely the inspiration for Lucas. Chapter two introduces the main topic of the book, the tower of Hanoi, in its classical form in which one must move disks one-by-one between three towers, always keeping the disks on each tower sorted by size. It provides several different algorithms for solving the classical puzzle (in which the disks begin and end all on a single tower) in as few moves as possible, and for collecting all disks on a single tower when they begin in other configurations, again as quickly as possible. It introduces the Hanoi graphs describing the state space of the puzzle, and relates numbers of puzzle steps to distances within this graph. After a chapter on "irregular" puzzles in which the initial placement of disks on their towers is not sorted, chapter four discusses the "Sierpiński graphs" derived from the Sierpiński triangle; these are closely related to the three-tower Hanoi graphs but diverge from them for higher numbers of towers of Hanoi or higher-dimensional Sierpinski fractals.[4][7][9] The next four chapters concern additional variants of the tower of Hanoi, in which more than three towers are used, the disks are only allowed to move between some of the towers or in restricted directions between the towers, or the rules for which disks can be placed on which are modified or relaxed.[4][9] A particularly important case is the Reve's puzzle, in which the rules are unchanged except that there are four towers instead of three. An old conjecture concerning the minimum possible number of moves between two states with all disks on a single tower was finally proven in 2014, after the publication of the first edition of the book, and the second edition includes this material.[7][10] Some of the definitions and proofs are extended into the book's many exercises.[7] A new chapter in the second edition provides hints and partial solutions,[9] and the final chapter collects open problems and (in the second edition) provides updates to previously-listed problems.[4][9] Many color illustrations and photographs are included throughout the book.[8] Audience The book can be read both by mathematicians working on topics related to the tower of Hanoi puzzle, and by a general audience interested in recreational mathematics. Reviewer László Kozma describes the book as essential reading for the first type of audience and (despite occasional heavy notation and encyclopedic detail) accessible and interesting to the second type, even for readers with only a high school level background in mathematics.[4] On the other hand, reviewer Cory Palmer cautions that "this book is not for a casual reader", adding that a good understanding of combinatorics is necessary to read it,[6] and reviewer Charles Ashbacher suggests that it has enough depth of content to be the topic of an advanced undergraduate elective course.[2] Although generally positive, reviewer S. V. Nagaraj complains about a "significant number of errors" in the book.[5] Reviewer Andrew Percy calls it "an enjoyable adventure", "humorous, and very thorough".[7] Reviewer Martin Klazar calls the book "wonderful", recommending it to anyone interested in recreational mathematics or mathematics more generally.[9] References 1. Allouche, Jean-Paul (2014), "Review of The Tower of Hanoi - Myths and Maths (1st ed.)" (PDF), European Mathematical Society Newsletter, 93: 56 2. Ashbacher, Charles (May 2013), "Review of The Tower of Hanoi - Myths and Maths (1st ed.)", MAA Reviews, Mathematical Association of America 3. Bultheel, Adhemar (February 2013), "Review of The Tower of Hanoi - Myths and Maths (1st ed.)", EMS Reviews, European Mathematical Society 4. Kozma, László (September 2014), "Review of The Tower of Hanoi - Myths and Maths (1st ed.)" (PDF), SIGACT News, 45 (3): 29–31, doi:10.1145/2670418.2670430 5. Nagaraj, S. V. (December 2013), "Review of The Tower of Hanoi - Myths and Maths (1st ed.)", ACM Computing Reviews 6. Palmer, Cory (December 2014), "Review of The Tower of Hanoi - Myths and Maths (1st ed.)", The Mathematics Enthusiast, 11 (3): 753–754 7. Percy, Andrew, "Review of The Tower of Hanoi - Myths and Maths (1st ed.)", zbMATH, Zbl 1285.00003 8. Sangwin, Chris (August 2015), "Review of The Tower of Hanoi - Myths and Maths (1st ed.)", The Mathematical Intelligencer, 37 (4): 87–88, doi:10.1007/s00283-015-9552-y 9. Klazar, Martin, "Review of The Tower of Hanoi - Myths and Maths (2nd ed.)", Mathematical Reviews, MR 3791459 10. From the publisher's description of the second edition, as quoted by Zbl 1387.00002 External links • Home page	Wikipedia
Triangle inequality In mathematics, the triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side.[1][2] This statement permits the inclusion of degenerate triangles, but some authors, especially those writing about elementary geometry, will exclude this possibility, thus leaving out the possibility of equality.[3] If x, y, and z are the lengths of the sides of the triangle, with no side being greater than z, then the triangle inequality states that $z\leq x+y,$ This article is about the basic inequality $z\leq x+y$. For other inequalities associated with triangles, see List of triangle inequalities. with equality only in the degenerate case of a triangle with zero area. In Euclidean geometry and some other geometries, the triangle inequality is a theorem about distances, and it is written using vectors and vector lengths (norms): $\\|\mathbf {x} +\mathbf {y} \\|\leq \\|\mathbf {x} \\|+\\|\mathbf {y} \\|,$ where the length z of the third side has been replaced by the vector sum x + y. When x and y are real numbers, they can be viewed as vectors in R1, and the triangle inequality expresses a relationship between absolute values. In Euclidean geometry, for right triangles the triangle inequality is a consequence of the Pythagorean theorem, and for general triangles, a consequence of the law of cosines, although it may be proved without these theorems. The inequality can be viewed intuitively in either R2 or R3. The figure at the right shows three examples beginning with clear inequality (top) and approaching equality (bottom). In the Euclidean case, equality occurs only if the triangle has a 180° angle and two 0° angles, making the three vertices collinear, as shown in the bottom example. Thus, in Euclidean geometry, the shortest distance between two points is a straight line. In spherical geometry, the shortest distance between two points is an arc of a great circle, but the triangle inequality holds provided the restriction is made that the distance between two points on a sphere is the length of a minor spherical line segment (that is, one with central angle in [0, π]) with those endpoints.[4][5] The triangle inequality is a defining property of norms and measures of distance. This property must be established as a theorem for any function proposed for such purposes for each particular space: for example, spaces such as the real numbers, Euclidean spaces, the Lp spaces (p ≥ 1), and inner product spaces. Euclidean geometry Euclid proved the triangle inequality for distances in plane geometry using the construction in the figure.[6] Beginning with triangle ABC, an isosceles triangle is constructed with one side taken as BC and the other equal leg BD along the extension of side AB. It then is argued that angle β has larger measure than angle α, so side AD is longer than side AC. But AD = AB + BD = AB + BC, so the sum of the lengths of sides AB and BC is larger than the length of AC. This proof appears in Euclid's Elements, Book 1, Proposition 20.[7] Mathematical expression of the constraint on the sides of a triangle For a proper triangle, the triangle inequality, as stated in words, literally translates into three inequalities (given that a proper triangle has side lengths a, b, c that are all positive and excludes the degenerate case of zero area): $a+b>c,\quad b+c>a,\quad c+a>b.$ A more succinct form of this inequality system can be shown to be $\|a-b\|<c<a+b.$ Another way to state it is $\max(a,b,c)<a+b+c-\max(a,b,c)$ implying $2\max(a,b,c)<a+b+c$ and thus that the longest side length is less than the semiperimeter. A mathematically equivalent formulation is that the area of a triangle with sides a, b, c must be a real number greater than zero. Heron's formula for the area is ${\begin{aligned}4\cdot {\text{area}}&={\sqrt {(a+b+c)(-a+b+c)(a-b+c)(a+b-c)}}\\&={\sqrt {-a^{4}-b^{4}-c^{4}+2a^{2}b^{2}+2a^{2}c^{2}+2b^{2}c^{2}}}.\end{aligned}}$ In terms of either area expression, the triangle inequality imposed on all sides is equivalent to the condition that the expression under the square root sign be real and greater than zero (so the area expression is real and greater than zero). The triangle inequality provides two more interesting constraints for triangles whose sides are a, b, c, where a ≥ b ≥ c and $\phi $ is the golden ratio, as $1<{\frac {a+c}{b}}<3$ $1\leq \min \left({\frac {a}{b}},{\frac {b}{c}}\right)<\phi .$[8] Right triangle In the case of right triangles, the triangle inequality specializes to the statement that the hypotenuse is greater than either of the two sides and less than their sum.[9] The second part of this theorem is already established above for any side of any triangle. The first part is established using the lower figure. In the figure, consider the right triangle ADC. An isosceles triangle ABC is constructed with equal sides AB = AC. From the triangle postulate, the angles in the right triangle ADC satisfy: $\alpha +\gamma =\pi /2\ .$ Likewise, in the isosceles triangle ABC, the angles satisfy: $2\beta +\gamma =\pi \ .$ Therefore, $\alpha =\pi /2-\gamma ,\ \mathrm {while} \ \beta =\pi /2-\gamma /2\ ,$ and so, in particular, $\alpha <\beta \ .$ That means side AD opposite angle α is shorter than side AB opposite the larger angle β. But AB = AC. Hence: ${\overline {\mathrm {AC} }}>{\overline {\mathrm {AD} }}\ .$ A similar construction shows AC > DC, establishing the theorem. An alternative proof (also based upon the triangle postulate) proceeds by considering three positions for point B:[10] (i) as depicted (which is to be proved), or (ii) B coincident with D (which would mean the isosceles triangle had two right angles as base angles plus the vertex angle γ, which would violate the triangle postulate), or lastly, (iii) B interior to the right triangle between points A and D (in which case angle ABC is an exterior angle of a right triangle BDC and therefore larger than π/2, meaning the other base angle of the isosceles triangle also is greater than π/2 and their sum exceeds π in violation of the triangle postulate). This theorem establishing inequalities is sharpened by Pythagoras' theorem to the equality that the square of the length of the hypotenuse equals the sum of the squares of the other two sides. Examples of use Consider a triangle whose sides are in an arithmetic progression and let the sides be a, a + d, a + 2d. Then the triangle inequality requires that $0<a<2a+3d$ $0<a+d<2a+2d$ $0<a+2d<2a+d.$ To satisfy all these inequalities requires $a>0{\text{ and }}-{\frac {a}{3}}<d<a.$[11] When d is chosen such that d = a/3, it generates a right triangle that is always similar to the Pythagorean triple with sides 3, 4, 5. Now consider a triangle whose sides are in a geometric progression and let the sides be a, ar, ar2. Then the triangle inequality requires that $0<a<ar+ar^{2}$ $0<ar<a+ar^{2}$ $0<ar^{2}<a+ar.$ The first inequality requires a > 0; consequently it can be divided through and eliminated. With a > 0, the middle inequality only requires r > 0. This now leaves the first and third inequalities needing to satisfy ${\begin{aligned}r^{2}+r-1&{}>0\\r^{2}-r-1&{}<0.\end{aligned}}$ The first of these quadratic inequalities requires r to range in the region beyond the value of the positive root of the quadratic equation r2 + r − 1 = 0, i.e. r > φ − 1 where φ is the golden ratio. The second quadratic inequality requires r to range between 0 and the positive root of the quadratic equation r2 − r − 1 = 0, i.e. 0 < r < φ. The combined requirements result in r being confined to the range $\varphi -1<r<\varphi \,{\text{ and }}a>0.$[12] When r the common ratio is chosen such that r = √φ it generates a right triangle that is always similar to the Kepler triangle. Generalization to any polygon The triangle inequality can be extended by mathematical induction to arbitrary polygonal paths, showing that the total length of such a path is no less than the length of the straight line between its endpoints. Consequently, the length of any polygon side is always less than the sum of the other polygon side lengths. Example of the generalized polygon inequality for a quadrilateral Consider a quadrilateral whose sides are in a geometric progression and let the sides be a, ar, ar2, ar3. Then the generalized polygon inequality requires that $0<a<ar+ar^{2}+ar^{3}$ $0<ar<a+ar^{2}+ar^{3}$ $0<ar^{2}<a+ar+ar^{3}$ $0<ar^{3}<a+ar+ar^{2}.$ These inequalities for a > 0 reduce to the following $r^{3}+r^{2}+r-1>0$ $r^{3}-r^{2}-r-1<0.$[13] The left-hand side polynomials of these two inequalities have roots that are the tribonacci constant and its reciprocal. Consequently, r is limited to the range 1/t < r < t where t is the tribonacci constant. Relationship with shortest paths This generalization can be used to prove that the shortest curve between two points in Euclidean geometry is a straight line. No polygonal path between two points is shorter than the line between them. This implies that no curve can have an arc length less than the distance between its endpoints. By definition, the arc length of a curve is the least upper bound of the lengths of all polygonal approximations of the curve. The result for polygonal paths shows that the straight line between the endpoints is the shortest of all the polygonal approximations. Because the arc length of the curve is greater than or equal to the length of every polygonal approximation, the curve itself cannot be shorter than the straight line path.[14] Converse The converse of the triangle inequality theorem is also true: if three real numbers are such that each is less than the sum of the others, then there exists a triangle with these numbers as its side lengths and with positive area; and if one number equals the sum of the other two, there exists a degenerate triangle (that is, with zero area) with these numbers as its side lengths. In either case, if the side lengths are a, b, c we can attempt to place a triangle in the Euclidean plane as shown in the diagram. We need to prove that there exists a real number h consistent with the values a, b, and c, in which case this triangle exists. By the Pythagorean theorem we have b2 = h2 + d2 and a2 = h2 + (c − d)2 according to the figure at the right. Subtracting these yields a2 − b2 = c2 − 2cd. This equation allows us to express d in terms of the sides of the triangle: $d={\frac {-a^{2}+b^{2}+c^{2}}{2c}}.$ For the height of the triangle we have that h2 = b2 − d2. By replacing d with the formula given above, we have $h^{2}=b^{2}-\left({\frac {-a^{2}+b^{2}+c^{2}}{2c}}\right)^{2}.$ For a real number h to satisfy this, $h^{2}$ must be non-negative: $b^{2}-\left({\frac {-a^{2}+b^{2}+c^{2}}{2c}}\right)^{2}\geq 0,$ $\left(b-{\frac {-a^{2}+b^{2}+c^{2}}{2c}}\right)\left(b+{\frac {-a^{2}+b^{2}+c^{2}}{2c}}\right)\geq 0,$ $\left(a^{2}-(b-c)^{2})((b+c)^{2}-a^{2}\right)\geq 0,$ $(a+b-c)(a-b+c)(b+c+a)(b+c-a)\geq 0,$ $(a+b-c)(a+c-b)(b+c-a)\geq 0,$ which holds if the triangle inequality is satisfied for all sides. Therefore, there does exist a real number h consistent with the sides a, b, c, and the triangle exists. If each triangle inequality holds strictly, h > 0 and the triangle is non-degenerate (has positive area); but if one of the inequalities holds with equality, so h = 0, the triangle is degenerate. Generalization to higher dimensions See also: Distance geometry § Cayley–Menger determinants The area of a triangular face of a tetrahedron is less than or equal to the sum of the areas of the other three triangular faces. More generally, in Euclidean space the hypervolume of an (n − 1)-facet of an n-simplex is less than or equal to the sum of the hypervolumes of the other n facets. Much as the triangle inequality generalizes to a polygon inequality, the inequality for a simplex of any dimension generalizes to a polytope of any dimension: the hypervolume of any facet of a polytope is less than or equal to the sum of the hypervolumes of the remaining facets. In some cases the tetrahedral inequality is stronger than several applications of the triangle inequality. For example, the triangle inequality appears to allow the possibility of four points A, B, C, and Z in Euclidean space such that distances AB = BC = CA = 26 and AZ = BZ = CZ = 14. However, points with such distances cannot exist: the area of the 26–26–26 equilateral triangle ABC is 169√3, which is larger than three times 39√3, the area of a 26–14–14 isosceles triangle (all by Heron's formula), and so the arrangement is forbidden by the tetrahedral inequality. Normed vector space In a normed vector space V, one of the defining properties of the norm is the triangle inequality: $\\|x+y\\|\leq \\|x\\|+\\|y\\|\quad \forall \,x,y\in V$ that is, the norm of the sum of two vectors is at most as large as the sum of the norms of the two vectors. This is also referred to as subadditivity. For any proposed function to behave as a norm, it must satisfy this requirement.[15] If the normed space is Euclidean, or, more generally, strictly convex, then $\\|x+y\\|=\\|x\\|+\\|y\\|$ if and only if the triangle formed by x, y, and x + y, is degenerate, that is, x and y are on the same ray, i.e., x = 0 or y = 0, or x = α y for some α > 0. This property characterizes strictly convex normed spaces such as the ℓp spaces with 1 < p < ∞. However, there are normed spaces in which this is not true. For instance, consider the plane with the ℓ1 norm (the Manhattan distance) and denote x = (1, 0) and y = (0, 1). Then the triangle formed by x, y, and x + y, is non-degenerate but $\\|x+y\\|=\\|(1,1)\\|=\|1\|+\|1\|=2=\\|x\\|+\\|y\\|.$ Example norms • Absolute value as norm for the real line. To be a norm, the triangle inequality requires that the absolute value satisfy for any real numbers x and y: $\|x+y\|\leq \|x\|+\|y\|,$ which it does. Proof:[16] $-\left\vert x\right\vert \leq x\leq \left\vert x\right\vert $ $-\left\vert y\right\vert \leq y\leq \left\vert y\right\vert $ After adding, $-(\left\vert x\right\vert +\left\vert y\right\vert )\leq x+y\leq \left\vert x\right\vert +\left\vert y\right\vert $ Use the fact that $\left\vert b\right\vert \leq a\Leftrightarrow -a\leq b\leq a$ (with b replaced by x+y and a by $\left\vert x\right\vert +\left\vert y\right\vert $), we have $\|x+y\|\leq \|x\|+\|y\|$ The triangle inequality is useful in mathematical analysis for determining the best upper estimate on the size of the sum of two numbers, in terms of the sizes of the individual numbers. There is also a lower estimate, which can be found using the reverse triangle inequality which states that for any real numbers x and y: $\|x-y\|\geq {\biggl \|}\|x\|-\|y\|{\biggr \|}.$ • Inner product as norm in an inner product space. If the norm arises from an inner product (as is the case for Euclidean spaces), then the triangle inequality follows from the Cauchy–Schwarz inequality as follows: Given vectors $x$ and $y$, and denoting the inner product as $\langle x,y\rangle $:[17] $\\|x+y\\|^{2}$$=\langle x+y,x+y\rangle $ $=\\|x\\|^{2}+\langle x,y\rangle +\langle y,x\rangle +\\|y\\|^{2}$ $\leq \\|x\\|^{2}+2\|\langle x,y\rangle \|+\\|y\\|^{2}$ $\leq \\|x\\|^{2}+2\\|x\\|\\|y\\|+\\|y\\|^{2}$ (by the Cauchy–Schwarz inequality) $=\left(\\|x\\|+\\|y\\|\right)^{2}$. The Cauchy–Schwarz inequality turns into an equality if and only if x and y are linearly dependent. The inequality $\langle x,y\rangle +\langle y,x\rangle \leq 2\left\|\left\langle x,y\right\rangle \right\|$ turns into an equality for linearly dependent $x$ and $y$ if and only if one of the vectors x or y is a nonnegative scalar of the other. Taking the square root of the final result gives the triangle inequality. • p-norm: a commonly used norm is the p-norm: $\\|x\\|_{p}=\left(\sum _{i=1}^{n}\|x_{i}\|^{p}\right)^{1/p}\ ,$ where the xi are the components of vector x. For p = 2 the p-norm becomes the Euclidean norm: $\\|x\\|_{2}=\left(\sum _{i=1}^{n}\|x_{i}\|^{2}\right)^{1/2}=\left(\sum _{i=1}^{n}x_{i}^{2}\right)^{1/2}\ ,$ which is Pythagoras' theorem in n-dimensions, a very special case corresponding to an inner product norm. Except for the case p = 2, the p-norm is not an inner product norm, because it does not satisfy the parallelogram law. The triangle inequality for general values of p is called Minkowski's inequality.[18] It takes the form: $\\|x+y\\|_{p}\leq \\|x\\|_{p}+\\|y\\|_{p}\ .$ Metric space In a metric space M with metric d, the triangle inequality is a requirement upon distance: $d(x,\ z)\leq d(x,\ y)+d(y,\ z)\ ,$ for all x, y, z in M. That is, the distance from x to z is at most as large as the sum of the distance from x to y and the distance from y to z. The triangle inequality is responsible for most of the interesting structure on a metric space, namely, convergence. This is because the remaining requirements for a metric are rather simplistic in comparison. For example, the fact that any convergent sequence in a metric space is a Cauchy sequence is a direct consequence of the triangle inequality, because if we choose any xn and xm such that d(xn, x) < ε/2 and d(xm, x) < ε/2, where ε > 0 is given and arbitrary (as in the definition of a limit in a metric space), then by the triangle inequality, d(xn, xm) ≤ d(xn, x) + d(xm, x) < ε/2 + ε/2 = ε, so that the sequence {xn} is a Cauchy sequence, by definition. This version of the triangle inequality reduces to the one stated above in case of normed vector spaces where a metric is induced via d(x, y) ≔ ‖x − y‖, with x − y being the vector pointing from point y to x. Reverse triangle inequality The reverse triangle inequality is an elementary consequence of the triangle inequality that gives lower bounds instead of upper bounds. For plane geometry, the statement is:[19] Any side of a triangle is greater than or equal to the difference between the other two sides. In the case of a normed vector space, the statement is: ${\bigg \|}\\|x\\|-\\|y\\|{\bigg \|}\leq \\|x-y\\|,$ or for metric spaces, \|d(y, x) − d(x, z)\| ≤ d(y, z). This implies that the norm $\\|\cdot \\|$ as well as the distance function $d(x,\cdot )$ are Lipschitz continuous with Lipschitz constant 1, and therefore are in particular uniformly continuous. The proof for the reverse triangle uses the regular triangle inequality, and $\\|y-x\\|=\\|{-}1(x-y)\\|=\|{-}1\|\cdot \\|x-y\\|=\\|x-y\\|$: $\\|x\\|=\\|(x-y)+y\\|\leq \\|x-y\\|+\\|y\\|\Rightarrow \\|x\\|-\\|y\\|\leq \\|x-y\\|,$ $\\|y\\|=\\|(y-x)+x\\|\leq \\|y-x\\|+\\|x\\|\Rightarrow \\|x\\|-\\|y\\|\geq -\\|x-y\\|,$ Combining these two statements gives: $-\\|x-y\\|\leq \\|x\\|-\\|y\\|\leq \\|x-y\\|\Rightarrow {\bigg \|}\\|x\\|-\\|y\\|{\bigg \|}\leq \\|x-y\\|.$ Triangle inequality for cosine similarity By applying the cosine function to the triangle inequality and reverse triangle inequality for arc lengths and employing the angle addition and subtraction formulas for cosines, it follows immediately that $\operatorname {sim} (x,z)\geq \operatorname {sim} (x,y)\cdot \operatorname {sim} (y,z)-{\sqrt {\left(1-\operatorname {sim} (x,y)^{2}\right)\cdot \left(1-\operatorname {sim} (y,z)^{2}\right)}}$ and $\operatorname {sim} (x,z)\leq \operatorname {sim} (x,y)\cdot \operatorname {sim} (y,z)+{\sqrt {\left(1-\operatorname {sim} (x,y)^{2}\right)\cdot \left(1-\operatorname {sim} (y,z)^{2}\right)}}\,.$ With these formulas, one needs to compute a square root for each triple of vectors {x, y, z} that is examined rather than arccos(sim(x,y)) for each pair of vectors {x, y} examined, and could be a performance improvement when the number of triples examined is less than the number of pairs examined. Reversal in Minkowski space The Minkowski space metric $\eta _{\mu \nu }$ is not positive-definite, which means that $\\|x\\|^{2}=\eta _{\mu \nu }x^{\mu }x^{\nu }$ can have either sign or vanish, even if the vector x is non-zero. Moreover, if x and y are both timelike vectors lying in the future light cone, the triangle inequality is reversed: $\\|x+y\\|\geq \\|x\\|+\\|y\\|.$ A physical example of this inequality is the twin paradox in special relativity. The same reversed form of the inequality holds if both vectors lie in the past light cone, and if one or both are null vectors. The result holds in n + 1 dimensions for any n ≥ 1. If the plane defined by x and y is spacelike (and therefore a Euclidean subspace) then the usual triangle inequality holds. See also • Subadditivity • Minkowski inequality • Ptolemy's inequality Notes 1. Wolfram MathWorld – http://mathworld.wolfram.com/TriangleInequality.html 2. Mohamed A. Khamsi; William A. Kirk (2001). "§1.4 The triangle inequality in Rn". An introduction to metric spaces and fixed point theory. Wiley-IEEE. ISBN 0-471-41825-0. 3. for instance, Jacobs, Harold R. (1974), Geometry, W. H. Freeman & Co., p. 246, ISBN 0-7167-0456-0 4. Oliver Brock; Jeff Trinkle; Fabio Ramos (2009). Robotics: Science and Systems IV. MIT Press. p. 195. ISBN 978-0-262-51309-8. 5. Arlan Ramsay; Robert D. Richtmyer (1995). Introduction to hyperbolic geometry. Springer. p. 17. ISBN 0-387-94339-0. 6. Harold R. Jacobs (2003). Geometry: seeing, doing, understanding (3rd ed.). Macmillan. p. 201. ISBN 0-7167-4361-2. 7. David E. Joyce (1997). "Euclid's elements, Book 1, Proposition 20". Euclid's elements. Dept. Math and Computer Science, Clark University. Retrieved 2010-06-25. 8. American Mathematical Monthly, pp. 49-50, 1954. 9. Claude Irwin Palmer (1919). Practical mathematics for home study: being the essentials of arithmetic, geometry, algebra and trigonometry. McGraw-Hill. p. 422. 10. Alexander Zawaira; Gavin Hitchcock (2009). "Lemma 1: In a right-angled triangle the hypotenuse is greater than either of the other two sides". A primer for mathematics competitions. Oxford University Press. ISBN 978-0-19-953988-8. 11. Wolfram\|Alpha. "input: solve 0<a<2a+3d, 0<a+d<2a+2d, 0<a+2d<2a+d,". Wolfram Research. Retrieved 2010-09-07. 12. Wolfram\|Alpha. "input: solve 0<a<ar+ar2, 0<ar<a+ar2, 0<ar2<a+ar". Wolfram Research. Retrieved 2010-09-07. 13. Wolfram\|Alpha. "input: solve 0<a<ar+ar2+ar3, 0<ar3<a+ar+ar2". Wolfram Research. Retrieved 2012-07-29. 14. John Stillwell (1997). Numbers and Geometry. Springer. ISBN 978-0-387-98289-2. p. 95. 15. Rainer Kress (1988). "§3.1: Normed spaces". Numerical analysis. Springer. p. 26. ISBN 0-387-98408-9. 16. James Stewart (2008). Essential Calculus. Thomson Brooks/Cole. p. A10. ISBN 978-0-495-10860-3. 17. John Stillwell (2005). The four pillars of geometry. Springer. p. 80. ISBN 0-387-25530-3. 18. Karen Saxe (2002). Beginning functional analysis. Springer. p. 61. ISBN 0-387-95224-1. 19. Anonymous (1854). "Exercise I. to proposition XIX". The popular educator; fourth volume. Ludgate Hill, London: John Cassell. p. 196. References • Pedoe, Daniel (1988). Geometry: A comprehensive course. Dover. ISBN 0-486-65812-0.. • Rudin, Walter (1976). Principles of Mathematical Analysis. New York: McGraw-Hill. ISBN 0-07-054235-X.. External links • Triangle inequality at ProofWiki	Wikipedia
The Unimaginable Mathematics of Borges' Library of Babel The Unimaginable Mathematics of Borges' Library of Babel is a popular mathematics book on Jorge Luis Borges and mathematics. It describes several mathematical concepts related to the short story "The Library of Babel", by Jorge Luis Borges. Written by mathematics professor William Goldbloom Bloch, and published in 2008 by the Oxford University Press, it received an honorable mention in the 2008 PROSE Awards.[1][2] Topics "The Library of Babel" was originally written by Borges in 1941,[3] based on an earlier essay he had published in 1939 while working as a librarian.[4] It concerns a fictional library containing every possible book of a certain fixed length, over a 25-symbol alphabet (which, including spacing and punctuation, is sufficient for the Spanish language).[5] These assumptions, based on the dimensions of his own library[4] and spelled out in more detail in the story, imply that the total number of books in the library is 251312000, an enormous number.[5][6] The story also describes, with an attitude of some horror,[4][2][7] the physical layout of the library that holds all of these books, and some of the behavior of its inhabitants.[5] After a copy of "The Library of Babel" itself, as translated into English by Andrew Hurley,[3][5] The Unimaginable Mathematics of Borges' Library of Babel has seven chapters on its mathematics. The first chapter, on combinatorics, repeats the calculation above, of the number of books in the library, putting it in context with the size of the known universe and with other huge numbers, and uses this material as an excuse to branch off into a discussion of logarithms and their use in estimation. The second chapter concerns a line in the story about the existence of a library catalog for the library, using information theory to prove that such a catalog would necessarily equal in size the library itself, and touching on topics including the prime number theorem. The third chapter considers the mathematics of the infinite, and the possibility of books with infinitely many, infinitely thin pages, connecting these topics both to a footnote in "The Library of Babel" and to another Borges story, "The Book of Sand", about such an infinite book.[6][8] Chapters four and five concern the architecture of the library, described as a set of interconnected hexagonal rooms, exploring the possibilities for their connections in terms of geometry, topology, and graph theory.[6][8] They also use mathematics to deduce unexpected conclusions about the library's structure: it must have at least one room whose shelves are not full (because the number of books per room does not divide the total number of books evenly), and the rooms on each floor of the library must either be connected into a single Hamiltonian cycle, or possibly be disconnected into subsets that cannot reach each other.[7] Chapter six considers the ways the books might be distributed through these rooms, and chapter seven views the library and its interactions with its inhabitants as analogous to Turing machines. A concluding chapter provides references to the literature on the story, critiques the scholarship on this story from the point of view of its mathematics, and discusses how much of this mathematics might have been familiar to Borges.[6][8] Author William Bloch, a mathematics professor at Wheaton College (Massachusetts), says that his book was originally intended as a short paper, based on his research from a sabbatical visit to Borges's home city Buenos Aires, but that it "grew and grew and grew".[9] The endpapers of the book are decorated with reproductions of Borges's original manuscript for his story.[4] Audience and reception Reading The Unimaginable Mathematics of Borges' Library of Babel requires only high-school mathematics,[3][5] and its chapters are independent of each other and can be read in any order.[3] Although written for a popular audience, it has enough depth of content to interest professional mathematicians as well.[6] The book's reviewers point to some minor issues with the book, including a too-facile derivation of the (correct) conclusion that an index for the library would be as large as the library itself,[2] a miscalculation of the number of permutations of books that are possible,[7] a missed easy explanation of logarithms as approximating the number of digits in a number,[5] an incorrect statement that a book with infinitely many infinitely thin pages would necessarily itself be infinitely thin,[2] the choice for an example of a letter that does not appear in Borges's descriptions,[6] and a failure to address the Spanish-language literature on Borges's work.[2] Nevertheless, reviewer James V. Rauff calls it "a treat for anyone with a passion for infinity, logic, language, and the philosophy of mathematics".[6] And reviewer Dan King, who himself has taught the mathematics of Borges's writing, writes that the book is "as eloquent and provocative as Borges’ story itself", and a must-read for all fans of Borges.[5] References 1. "2008 Winners", PROSE Awards, retrieved 2020-05-27 2. Tuckey, Curtis (2010), "The imaginative mathematics of Bloch's "Unimaginable Mathematics of Borges's Library of Babel"" (PDF), Variaciones Borges, 30: 217–231, JSTOR 24881592 3. Boslaugh, Sarah (August 2008), "Review of The Unimaginable Mathematics of Borges' Library of Babel", MAA Reviews, Mathematical Association of America 4. Manguel, Alberto (September 24, 2008), "A Universe of Books: Borges's 'Library of Babel' (review of The Unimaginable Mathematics of Borges' Library of Babel)", The New York Sun 5. King, Dan (November 2010), "Review of The Unimaginable Mathematics of Borges' Library of Babel", The College Mathematics Journal, 41 (5): 416–418, doi:10.4169/074683410x522053, JSTOR 10.4169/074683410x522053 6. Rauff, James V. (Spring 2010), "Review of The Unimaginable Mathematics of Borges' Library of Babel", Mathematics and Computer Education, 44 (2): 182–183, ProQuest 503549780 7. Hayes, Brian (January–February 2009), "Books-a-million (review of The Unimaginable Mathematics of Borges' Library of Babel)", American Scientist, 97 (1): 78–79, doi:10.1511/2009.76.78, JSTOR 27859279 8. Pambuccian, Victor V., "Review of The Unimaginable Mathematics of Borges' Library of Babel", zbMATH, Zbl 1152.01013 9. "Bloch explores intersection of math and literature", Wheaton News, Wheaton College (Massachusetts), October 31, 2008, retrieved 2020-05-27	Wikipedia
The Vectors of Mind The Vectors of Mind[1] is a book published by American psychologist Louis Leon Thurstone in 1935 that summarized Thurstone's methodology for multiple factor analysis.[2] The Vectors of Mind; Multiple-Factor Analysis for the Isolation of Primary Traits AuthorL. L. Thurstone CountryUnited States LanguageEnglish SubjectsFactor analysis, psychometrics PublisherUniversity of Chicago Press Publication date August 1935 Media typePrint Pages266 Overview The Vectors of Mind presents Thurstone's methods for conducting a factor analysis on a set of variables that allow for more than one factor, an important extension of Spearman's unifactor method. Having multiple factors adds significant complications and much of the book is focussed on the problem of rotation. It attempts to solve this problem by providing an objective basis for the rotation factors, called simple structure, and advocates the use of oblique (correlated) factors to achieve a simple structure. The book utilizes his centroid method of factor extraction, which made it feasible to complete the arduous calculations necessary for a factor analysis at a time when fast electronic computers had not even been imagined. This is a predominantly technical book that relies heavily upon mathematical presentations and provides multiple numerical examples. However, the early chapters delve into philosophical questions of the nature of science and present Thurstone's understanding of measurement theory. Synopsis Preface. This book extends and presents more formally the findings from the author's Multiple Factor Analysis paper of 1931. The author notes that he only recently learned matrix theory and presumes that other psychologists have had similar limitations in their training. He finds existing textbooks on the topic inadequate and the book begins with a presentation of matrix theory, written for those with undergraduate instruction in analytic geometry and real number calculus. The author expresses indebted to various professors in the mathematics department of the University of Chicago for helping him to develop his ideas. He also expresses appreciation to his computer (a person, Leone Chesire), who also wrote the appendix on the calculations used in the centroid method. He foresees a bright future for the use of factor analysis and expects to see the simplification of the computational methods. He expects factor analysis to become an important technique it the early stages of science. For example, the laws of classical mechanics could have been revealed by a factor analysis, by analyzing a great many attributes of objects that are dropped or thrown from an elevated point, with the time of fall factor uncorrelated with the weight factor. Work by Sewell Wright on path coefficients and Truman L. Kelley on multiple factors differs from factor analysis, which Thurstone sees as an extension of professor Spearman's work. Mathematical Introduction. A brief presentation of matrices, determinants, matrix multiplication, diagonal matrices, the inverse, the characteristic equation, summation notation, linear dependence, geometric interpretations, orthogonal transformations, and oblique transformations. Chapter I. The Factor Problem. Natural phenomena are only comprehensible through constructs that are man-made inventions. A scientific law is not part of nature; it is but man's way of understanding nature. Examples are provided of such man-made constructs from physics. He responds to skepticism from the practitioners of "rigorous science" that human behavior can ever be brought into the fold of such science by pointing out that there is considerable individuality in physical events even though described by rigorous scientific laws, such as the fact that every explosion is unique. Human abilities are the cause of individual differences in the "completion of a task". The science of psychology will reduce a large number of psychological abilities down to primary reference traits. Formal definitions are provided for the concepts of trait, ability, test, score, linear independence, statistical independence, experimental independence, reference abilities, primary abilities, and unitary ability. These conceptions constitute a theory of measurement that defines factors common to all tests in a battery-the communality of the test battery-, a specific factor that is unique to one test-the specificity of the test-, and the error variance. Factor analysis can determine the communality of a test, but cannot separate the uniqueness into the specific factor and the error factors. The reliability coefficient is the sum of communality and the specificity of a test. Chapter II. The Fundamental Factor Theorem. The factor matrix post-multiplied by its transpose gives the reduced correlation matrix: this is the fundamental factor theorem. The task of factor analysis is to find a factor matrix of the lowest possible rank (the least number of factors) that can reproduce the off-diagonal members of the observed correlation matrix as close as can be expected, allowing for sample variation. The bulk of the chapter considers mathematical issues, including the rank of a matrix and methods for estimating the commonalities of the correlation matrix (the diagonal elements). Chapter III: The Centroid Method. A computation method is developed for factoring a correlation matrix, which is a symmetric matrix of real elements. After a conceptual presentation of the method, some worked examples are provided, including one with eight variables, and another with fifteen variables that are factored into four factors. The mechanics of the calculations are given in Appendix I, which provides the specific steps in making the calculation (the algorithm). Chapter IV: The Principal Axes. A method is presented for determining a desirable rotation of the orthogonal factors called the principal axes. The mathematical foundations are provided, as well as worked examples. This approach is distinguished from Hotelling's method, which the author feels has limited usefulness to factor analysis. The unrotated solution for 15 psychological tests given in chapter III are rotated to their principal axes. Chapter V: The Special Case of Rank One. Spearman presented factor analysis with a single factor (a matrix with rank one) thirty years, but recent advances have made it possible to extend factor analysis to multiple factors. The shortcomings of Spearman's method of tetrad differences are detailed and the current approach found to be more accurate. A numerical example is given. Chapter VI: Primary Traits. Rotation does not affect the results of the fundamental factor theorem. All rotations result in the same reduced correlation matrix so other criteria must be used to ascertain the best rotation. Criteria refer to "simple structure": the book presents very detailed criteria for simple structure, but more generally it consists of minimizing the number of loading for each variable and wide variance for loadings of each factor. Realizing simple structure may require uses of oblique (correlated) factors. Three additional criteria are given that define when the simple structure is unique. Graphical-mathematical methods are developed for understanding and defining the structure that reveals primary traits–the scientific goal of factor analysis. The prior worked example of fifteen psychological traits is rotated to an oblique simple structure to reveal three intercorrelated primary traits. Chapters VII - X: The remaining chapters explore more specific details and problems that can arise. Chapter VII considers several methods for isolating primary traits, with numerical examples given. Chapter VIII addresses the methodological problems that can arise when the correlation matrix has negative correlations. Though most scientific investigations of primary abilities will entail oblique factors, there are situations where the factors are likely to be orthogonal. Chapter IX looks at techniques for achieving orthogonal rotations. The results of a factor analysis can be used to estimate each individual's score on the primary abilities based upon the individual's scores on the tests. Chapter X presents a method for obtaining the regression weights for estimating primary abilities from subject scores, and well as for estimating subjects scores from the primary traits (for estimating the components of variance of the subject scores). Appendices. I: Outline of Calculations for the Centroid Method with Unknown Diagonals. II: A Method of Finding the Roots of a Polynomial. III: A Method of Determining the Square Root on the Calculating Machine. Historical context In 1904 Charles Spearman published a paper that largely founded the field of psychometrics and included a crude form of factor analysis that attempted to determine if a single factor model was appropriate.[3] There was limited subsequent work on factor analysis until Thurstone published a paper in 1931 called Multiple Factor Analysis,[4] which expanded Spearman's single-factor analysis to include more than one factor. In 1932, Hotelling presented a more accurate method of extracting factors, which he called principal components analysis.[5] Thurstone rejected Hotelling's approach because it set the commonalities to 1.0, and Thurstone realized that will introduce distortions to the factor loadings when variables include unique components. Hotelling's method was also limited by the fact that it required too much calculation to be usable with more than about ten variables.[6] A year after Hotelling's paper, Thurstone presented a more efficient way of extracting factors, called the centroid method,[7] which allowed the factor analysis of a far larger number of variables. Later that year he gave his presidential address to the American Psychological Association wherein he presented the results of several factor analyses, including a factor analysis of 60 adjectives describing personality traits, showing how they could be reduced to five personality traits. He also presented analyses of 37 mental health symptoms, of attitudes towards 12 controversial social issues, and of 9 IQ tests.[8] In those analyses, Thurstone had made use of tetrachoric correlation coefficients, a method for estimating continuous variable correlations from dichotomous variables. Tetrachorics require extensive calculations but in early 1933, he and two colleagues at the University of Chicago published a set of computing diagrams that greatly reduce the calculations needed for those coefficients,[9] another aspect of making his method of factor analysis practical with more than just a few variables. His 1933 presidential address was published in early 1934 with the title Vectors of the Mind. It lacked methodological and mathematical details of his technique, which is then the subject of this book. A 2004 conference called Factor Analysis at 100 produced a book with two chapters that document the historical importance Thurstone's contributions to factor analysis.[10][11] Thurstone's approach to factor analysis remains an important method in psychological research and it has since been used in numerous other fields of study.[12] It is now considered part of a family of methods for analyzing the covariance structure of variables, which includes principal components analysis, exploratory factor analysis, confirmatory factor analysis, and structural equation modeling.[13] External links • Online Archive.org • The vectors of mind: Multiple-factor analysis for the isolation of primary traits. APA PsychNET References 1. Thurstone, L. L. (1935). The Vectors of Mind. Chicago, Illinois: The University of Chicago Press. 2. Wilks, S. S. Review: L. L. Thurstone, The Vectors of Mind . Bull. Amer. Math. Soc. 42 (1936), no. 11, 790--791. http://projecteuclid.org/euclid.bams/1183499382. 3. Spearson, Charles (1904). "General intelligence objectively determined and measured". American Journal of Psychology. 15 (2): 201–293. doi:10.2307/1412107. JSTOR 1412107. 4. Thurstone, Louis (1931). "Multiple factor analysis". Psychological Review. 38 (5): 406–427. doi:10.1037/h0069792. 5. Hotelling, H. (1933). "Analysis of a complex of statistical variables into principal components". Journal of Educational Psychology. 24 (6): 417–441, 498–520. doi:10.1037/h0071325. hdl:2027/wu.89097139406. 6. Harman, Harry (1976). Modern Factor Analysis. Third Edition Revised. Chicago, Illinois: The University of Chicago Press. p. 5. ISBN 0-226-31652-1. 7. Mulaik, Stanley (2010). Foundations of Factor Analysis. Second Edition. Boca Raton, Florida: CRC Press. pp. 147–151. ISBN 978-1-4200-9961-4. 8. Thurstone, Louis (1934). "The Vectors of Mind". The Psychological Review. 41: 1–32. doi:10.1037/h0075959. 9. Chesire, Leone; Saffir, Milton; Thurstone, L.L. (1933). Computing Diagrams for the Tetrachoric Correlation Coefficient. Chicago, Illinois: The University of Chicago Bookstore. 10. Bock, Darrell (2007). "Rethinking Thurstone". In Cudeck, Robert; MacCallum, Robert C. (eds.). Factor Analysis at 100. Historical Developments and Future Directions. Mahwah, New Jersey: Lawrence Erlbaum Associates. ISBN 978-0-8058-5347-6. 11. Bock, Darrell (2007). "Rethinking Thurstone". In Cudeck, Robert; MacCallum, Robert C. (eds.). Factor Analysis at 100. Historical Developments and Future Directions. Mahwah, New Jersey: Lawrence Erlbaum Associates. ISBN 978-0-8058-5347-6. 12. Harman, Harry H. (1976). Modern Factor Analysis. Third Edition Revised. Chicago, Illinois: University of Chicago Press. pp. 6–8. ISBN 0-226-31652-1. 13. Mulaik, Stanley A. (2010). Foundations of Factor Analysis. Second Edition. Boca Raton, Florida: CRC Press. pp. 1–3. ISBN 978-1-4200-9961-4.	Wikipedia
TWIRL In cryptography and number theory, TWIRL (The Weizmann Institute Relation Locator) is a hypothetical hardware device designed to speed up the sieving step of the general number field sieve integer factorization algorithm.[1] During the sieving step, the algorithm searches for numbers with a certain mathematical relationship. In distributed factoring projects, this is the step that is parallelized to a large number of processors. TWIRL is still a hypothetical device — no implementation has been publicly reported. However, its designers, Adi Shamir and Eran Tromer, estimate that if TWIRL were built, it would be able to factor 1024-bit numbers in one year at the cost of "a few dozen million US dollars". TWIRL could therefore have enormous repercussions in cryptography and computer security — many high-security systems still use 1024-bit RSA keys, which TWIRL would be able to break in a reasonable amount of time and for reasonable costs. The security of some important cryptographic algorithms, notably RSA and the Blum Blum Shub pseudorandom number generator, rests in the difficulty of factorizing large integers. If factorizing large integers becomes easier, users of these algorithms will have to resort to using larger keys (computationally more expensive) or to using different algorithms, whose security rests on some other computationally hard problem (like the discrete logarithm problem). See also • Custom hardware attack • TWINKLE References 1. Shamir, Adi; Tromer, Eran (2003), "Factoring Large Numbers with the TWIRL Device", Advances in Cryptology – CRYPTO 2003, Springer Berlin Heidelberg, pp. 1–26, doi:10.1007/978-3-540-45146-4_1, ISBN 9783540406747 External links • "The TWIRL integer factorization device" - homepage	Wikipedia
The Whetstone of Witte The Whetstone of Witte is the shortened title of Robert Recorde's mathematics book published in 1557, the full title being The whetstone of witte, whiche is the seconde parte of Arithmetike: containyng thextraction of Rootes: The Coßike practise, with the rule of Equation: and the woorkes of Surde Nombers. The book covers topics including whole numbers, the extraction of roots and irrational numbers.[2] The work is notable for containing the first recorded use of the equals sign[3] and also for being the first book in English to use the plus and minus signs.[4] Recordian notation for exponentiation, however, differed from the later Cartesian notation $p^{q}=p\times p\times p\cdots \times p$. Recorde expressed indices and surds larger than 3 in a systematic form based on the prime factorization of the exponent: a factor of two he termed a zenzic, and a factor of three, a cubic. Recorde termed the larger prime numbers appearing in this factorization sursolids, distinguishing between them by use of ordinal numbers: that is, he defined 5 as the first sursolid, written as ʃz and 7 as the second sursolid, written as Bʃz.[5] He also devised symbols for these factors: a zenzic was denoted by z, and a cubic by &. For instance, he referred to p8=p2×2×2 as zzz (the zenzizenzizenzic), and q12=q2×2×3 as zz& (the zenzizenzicubic).[6] Later in the book he includes a chart of exponents all the way up to p80=p2×2×2×2×5 written as zzzzʃz. There is an error in the chart, however, writing p69 as Sʃz, despite it not being a prime. It should be p3×23 or &Gʃz.[7] References 1. Robert Recorde (1557). The whetstone of witte, whiche is the seconde parte of Arithmetike: containyng thextraction of Rootes: The Coßike practise, with the rule of Equation: and the woorkes of Surde Nombers (PDF). London: Jhon Kyngstone.. Page 238 in the pdf file. 2. Williams, Jack (2011), "The Whetstone of Witte", Robert Recorde: Tudor Polymath, Expositor and Practitioner of Computation, History of Computing, Springer, pp. 173–196, doi:10.1007/978-0-85729-862-1_10, ISBN 9780857298621. 3. Atkins, Peter (2004), Galileo's Finger:The Ten Great Ideas of Science, Oxford University Press, p. 484, ISBN 9780191622502. 4. Cajori, Florian (2007), A History of Mathematical Notations, Cosimo, p. 164, ISBN 9781602066847. 5. Williams (2011), p. 147. 6. Williams (2011), p. 154. 7. Williams (2011), p. 163. External links • The Whetstone of Witte at The Internet Archive	Wikipedia
Association of Mathematics Teachers of India The Association of Mathematics Teachers of India or AMTI is an academically oriented body of professionals and students interested in the fields of mathematics and mathematics education. The AMTI's main base is Tamil Nadu, but it has recently been spreading its network in other parts of India, particularly in South India. Examinations and Olympiads National Mathematics Talent Contest Further information: National Mathematics Talent Contest AMTI conducts a National Mathematics Talent Contest or NMTC at Primary(Gauss Contest) (Standards 4 to 6), Sub-junior (Kaprekar Contest) (Standards 7 and 8), Junior (Bhaskara Contest) (Standards 9 and 10), Inter(Ramanujan Contest) (Standards 11 and 12) and Senior (Aryabhata Contest) (B.Sc.) levels. For students at the Junior and Inter levels from Tamil Nadu, the NMTC also plays the role of Regional Mathematical Olympiad. Although the question papers are different for Junior and Inter levels, students from both levels may be chosen to appear at INMO based on their performance. The NMTC is usually held around the last week of October. A preliminary examination is conducted earlier (in September) for all levels except B.Sc. students. Students (Junior and Inter) qualifying the preliminary examination are invited for an Orientation Camp one week before the NMTC, where Olympiad problems and theories are taught. This is also useful for those students qualifying further for INMO. Grand Achievement Test This test is for students studying in 12th standard under the Tamil Nadu State Board. It is intended to give a perfectly simulated atmosphere of the board's examination. Training Activities Ten-week training session In 2005, AMTI started a ten-week training programme for students for Olympiad-related problems. The training batches were split into: • Primary level: Standards 4 to 6 • Sub-junior level: Standards 7 and 8 • Junior level: Standards 9 and 10 • Inter level: Standards 11 and 12 Around 85 students attended the ten-week training session. AMTI conducted the programme again in 2006, and received a much better response. Workshops and conferences The AMTI has been organizing conferences in different parts of the country to meet and deliberate issues of mathematics education, particularly at the school level. Notable office bearers P. K. Srinivasan, a famous teacher of mathematics, was the first Editor of the magazine Junior Mathematician (1990 to 1994) and the Academic Secretary of AMTI from 1981 to 1994. External links • AMTI official page	Wikipedia
Cohomology operation In mathematics, the cohomology operation concept became central to algebraic topology, particularly homotopy theory, from the 1950s onwards, in the shape of the simple definition that if F is a functor defining a cohomology theory, then a cohomology operation should be a natural transformation from F to itself. Throughout there have been two basic points: 1. the operations can be studied by combinatorial means; and 2. the effect of the operations is to yield an interesting bicommutant theory. The origin of these studies was the work of Pontryagin, Postnikov, and Norman Steenrod, who first defined the Pontryagin square, Postnikov square, and Steenrod square operations for singular cohomology, in the case of mod 2 coefficients. The combinatorial aspect there arises as a formulation of the failure of a natural diagonal map, at cochain level. The general theory of the Steenrod algebra of operations has been brought into close relation with that of the symmetric group. In the Adams spectral sequence the bicommutant aspect is implicit in the use of Ext functors, the derived functors of Hom-functors; if there is a bicommutant aspect, taken over the Steenrod algebra acting, it is only at a derived level. The convergence is to groups in stable homotopy theory, about which information is hard to come by. This connection established the deep interest of the cohomology operations for homotopy theory, and has been a research topic ever since. An extraordinary cohomology theory has its own cohomology operations, and these may exhibit a richer set on constraints. Formal definition A cohomology operation $\theta $ of type $(n,q,\pi ,G)\,$ is a natural transformation of functors $\theta :H^{n}(-,\pi )\to H^{q}(-,G)\,$ defined on CW complexes. Relation to Eilenberg–MacLane spaces Cohomology of CW complexes is representable by an Eilenberg–MacLane space, so by the Yoneda lemma a cohomology operation of type $(n,q,\pi ,G)$ is given by a homotopy class of maps $K(\pi ,n)\to K(G,q)$. Using representability once again, the cohomology operation is given by an element of $H^{q}(K(\pi ,n),G)$. Symbolically, letting $[A,B]$ denote the set of homotopy classes of maps from $A$ to $B$, ${\begin{aligned}\displaystyle \mathrm {Nat} (H^{n}(-,\pi ),H^{q}(-,G))&=\mathrm {Nat} ([-,K(\pi ,n)],[-,K(G,q)])\\&=[K(\pi ,n),K(G,q)]\\&=H^{q}(K(\pi ,n);G).\end{aligned}}$ See also • Secondary cohomology operation References • Mosher, Robert E.; Tangora, Martin C. (2008) [1968], Cohomology operations and applications in homotopy theory, New York: Dover Publications, ISBN 978-0-486-46664-4, MR 0226634 • Steenrod, N. E. (1962), Epstein, D. B. A. (ed.), Cohomology operations, Annals of Mathematics Studies, vol. 50, Princeton University Press, ISBN 978-0-691-07924-0, MR 0145525	Wikipedia
100 prisoners problem The 100 prisoners problem is a mathematical problem in probability theory and combinatorics. In this problem, 100 numbered prisoners must find their own numbers in one of 100 drawers in order to survive. The rules state that each prisoner may open only 50 drawers and cannot communicate with other prisoners. At first glance, the situation appears hopeless, but a clever strategy offers the prisoners a realistic chance of survival. Danish computer scientist Peter Bro Miltersen first proposed the problem in 2003. Problem The 100 prisoners problem has different renditions in the literature. The following version is by Philippe Flajolet and Robert Sedgewick:[1] The director of a prison offers 100 death row prisoners, who are numbered from 1 to 100, a last chance. A room contains a cupboard with 100 drawers. The director randomly puts one prisoner's number in each closed drawer. The prisoners enter the room, one after another. Each prisoner may open and look into 50 drawers in any order. The drawers are closed again afterwards. If, during this search, every prisoner finds their number in one of the drawers, all prisoners are pardoned. If even one prisoner does not find their number, all prisoners die. Before the first prisoner enters the room, the prisoners may discuss strategy — but may not communicate once the first prisoner enters to look in the drawers. What is the prisoners' best strategy? If every prisoner selects 50 drawers at random, the probability that a single prisoner finds their number is 50%. Therefore, the probability that all prisoners find their numbers is the product of the single probabilities, which is (1/2)100 ≈ 0.0000000000000000000000000000008, a vanishingly small number. The situation appears hopeless. Solution Strategy Surprisingly, there is a strategy that provides a survival probability of more than 30%. The key to success is that the prisoners do not have to decide beforehand which drawers to open. Each prisoner can use the information gained from the contents of every drawer they already opened to decide which one to open next. Another important observation is that this way the success of one prisoner is not independent of the success of the other prisoners, because they all depend on the way the numbers are distributed.[2] To describe the strategy, not only the prisoners, but also the drawers, are numbered from 1 to 100; for example, row by row starting with the top left drawer. The strategy is now as follows:[3] 1. Each prisoner first opens the drawer labeled with their own number. 2. If this drawer contains their number, they are done and were successful. 3. Otherwise, the drawer contains the number of another prisoner, and they next open the drawer labeled with this number. 4. The prisoner repeats steps 2 and 3 until they find their own number, or fail because the number is not found in the first fifty opened drawers. If the prisoner could continue indefinitely this way, they would inevitably loop back to the drawer they started with, forming a permutation cycle (see below). By starting with their own number, the prisoner guarantees they are on the specific cycle of drawers containing their number. The only question is whether any cycle is longer than fifty drawers - and only one cycle can possibly be too long, since at most one can comprise more than half of the total drawers. Examples The reason this is a promising strategy is illustrated with the following example using 8 prisoners and drawers, whereby each prisoner may open 4 drawers. The prison director has distributed the prisoners' numbers into the drawers in the following fashion: number of drawer 1 2 3 4 5 6 7 8 number of prisoner 74681352 The prisoners now act as follows: • Prisoner 1 first opens drawer 1 and finds number 7. Then they open drawer 7 and find number 5. Then they open drawer 5, where they find their own number and are successful. • Prisoner 2 opens drawers 2, 4, and 8 in this order. In the last drawer they find their own number, 2. • Prisoner 3 opens drawers 3 and 6, where they find their own number. • Prisoner 4 opens drawers 4, 8, and 2, where they find their own number. This is the same cycle that was encountered by prisoner 2 and will be encountered by prisoner 8. Each of these prisoners will find their own number in the third opened drawer. • Prisoners 5 to 8 will also each find their numbers in a similar fashion. In this case, all prisoners find their numbers. This is, however, not always the case. For example, the small change to the numbers of swapping drawers 5 and 8 would cause prisoner 1 to fail after opening 1, 7, 5, and 2 (and not finding their own number): number of drawer 1 2 3 4 5 6 7 8 number of prisoner 74682351 And in the following arrangement, prisoner 1 opens drawers 1, 3, 7, and 4, at which point they have to stop unsuccessfully: number of drawer 1 2 3 4 5 6 7 8 number of prisoner 31758642 Indeed, all prisoners except 6 (who succeeds directly) fail. Permutation representation Graph representations of the permutations (1 7 5)(2 4 8)(3 6) and (1 3 7 4 5 8 2)(6) The prison director's assignment of prisoner numbers to drawers can mathematically be described as a permutation of the numbers 1 to 100. Such a permutation is a one-to-one mapping of the set of natural numbers from 1 to 100 to itself. A sequence of numbers which after repeated application of the permutation returns to the first number is called a cycle of the permutation. Every permutation can be decomposed into disjoint cycles, that is, cycles which have no common elements. The permutation of the first example above can be written in cycle notation as $(1~7~5)(2~4~8)(3~6)$ and thus consists of two cycles of length 3 and one cycle of length 2. The permutation of the third example is accordingly $(1~3~7~4~5~8~2)(6)$ and consists of a cycle of length 7 and a cycle of length 1. The cycle notation is not unique since a cycle of length $l$ can be written in $l$ different ways depending on the starting number of the cycle. During the opening the drawers in the above strategy, each prisoner follows a single cycle which always ends with their own number. In the case of eight prisoners, this cycle-following strategy is successful if and only if the length of the longest cycle of the permutation is at most 4. If a permutation contains a cycle of length 5 or more, all prisoners whose numbers lie in such a cycle do not reach their own number after four steps. Probability of success In the initial problem, the 100 prisoners are successful if the longest cycle of the permutation has a length of at most 50. Their survival probability is therefore equal to the probability that a random permutation of the numbers 1 to 100 contains no cycle of length greater than 50. This probability is determined in the following. A permutation of the numbers 1 to 100 can contain at most one cycle of length $l>50$. There are exactly ${\tbinom {100}{l}}$ ways to select the numbers of such a cycle (see combination). Within this cycle, these numbers can be arranged in $(l-1)!$ ways since there are $l$ permutations to represent distinct cycles of length $l$ because of cyclic symmetry. The remaining numbers can be arranged in $(100-l)!$ ways. Therefore, the number of permutations of the numbers 1 to 100 with a cycle of length $l>50$ is equal to ${\binom {100}{l}}\cdot (l-1)!\cdot (100-l)!={\frac {100!}{l}}.$ The probability, that a (uniformly distributed) random permutation contains no cycle of length greater than 50 is calculated with the formula for single events and the formula for complementary events thus given by $1-{\frac {1}{100!}}\left({\frac {100!}{51}}+\ldots +{\frac {100!}{100}}\right)=1-\left({\frac {1}{51}}+\ldots +{\frac {1}{100}}\right)=1-(H_{100}-H_{50})\approx 0.31183,$ where $H_{n}$ is the $n$-th harmonic number. Therefore, using the cycle-following strategy the prisoners survive in a surprising 31% of cases.[3] Asymptotics If $2n$ instead of 100 prisoners are considered, where $n$ an arbitrary natural number, the prisoners' survival probability with the cycle-following strategy is given by $1-(H_{2n}-H_{n})=1-(H_{2n}-\ln 2n)+(H_{n}-\ln n)-\ln 2.$ With the Euler–Mascheroni constant $\gamma $, for $n\to \infty $ $\lim _{n\to \infty }(H_{n}-\ln n)=\gamma $ holds, which results in an asymptotic survival probability of $\lim _{n\to \infty }(1-H_{2n}+H_{n})=1-\gamma +\gamma -\ln 2=1-\ln 2\approx 0.30685.$ Since the sequence of probabilities is monotonically decreasing, the prisoners survive with the cycle-following strategy in more than 30% of cases independently of the number of prisoners.[3] Optimality In 2006, Eugene Curtin and Max Warshauer gave a proof for the optimality of the cycle-following strategy. The proof is based on an equivalence to a related problem in which all prisoners are allowed to be present in the room and observe the opening of the drawers. Mathematically, this equivalence is based on Foata's transition lemma, a one-to-one correspondence of the (canonical) cycle notation and the one-line notation of permutations. In the second problem, the survival probability is independent of the chosen strategy and equal to the survival probability in the original problem with the cycle-following strategy. Since an arbitrary strategy for the original problem can also be applied to the second problem, but cannot attain a higher survival probability there, the cycle-following strategy has to be optimal.[2] History The 100 prisoners problem was first considered in 2003 by Danish computer scientist Peter Bro Miltersen who published it with Anna Gál in the proceedings of the 30. International Colloquium on Automata, Languages and Programming (ICALP).[4] In their version, player A (the prison director) randomly colors strips of paper with the names of the players of team B (the prisoners) in red or blue and puts each strip into a different box. Some of the boxes may be empty (see below). Every player of team B must guess their color correctly after opening half of the boxes for their team to win.[4] Initially, Miltersen assumed that the winning probability quickly tends to zero with increasing number of players. However, Sven Skyum, a colleague of Miltersen at Aarhus University, brought his attention to the cycle-following strategy for the case of this problem where there are no empty boxes. To find this strategy was left open as an exercise in the publication. The paper was honored with the best paper award.[2] In spring 2004, the problem appeared in Joe Buhler and Elwyn Berlekamp's puzzle column of the quarterly The Emissary of the Mathematical Sciences Research Institute. Thereby, the authors replaced boxes by ROMs and colored strips of paper by signed numbers. The authors noted that the winning probability can be increased also in the case where the team members don't find their own numbers. If the given answer is the product of all the signs found and if the length of the longest cycle is half the (even) number of players plus one, then the team members in this cycle either all guess wrong or all guess right. Even if this extension of the strategy offers a visible improvement for a small number of players, it becomes negligible when the number of players becomes large.[5] In the following years, the problem entered the mathematical literature, where it was shaped in further different ways, for example with cards on a table[6] or wallets in lockers (locker puzzle).[2] In the form of a prisoner problem it was posed in 2006 by Christoph Pöppe in the journal Spektrum der Wissenschaft and by Peter Winkler in the College Mathematics Journal.[7][8] With slight alterations this form was adopted by Philippe Flajolet, Robert Sedgewick and Richard P. Stanley in their textbooks on combinatorics.[1][3] Variants Empty boxes At first, Gál and Miltersen considered in their paper the case that the number of boxes is twice the number of team members while half of the boxes are empty. This is a more difficult problem since empty boxes lead nowhere and thus the cycle-following strategy cannot be applied. It is an open problem whether in this case the winning probability tends to zero with growing number of team members.[4] In 2005, Navin Goyal and Michael Saks developed a strategy for team B based on the cycle-following strategy for a more general problem in which the fraction of empty boxes as well as the fraction of boxes each team member is allowed to open are variable. The winning probability still tends to zero in this case, but slower than suggested by Gál and Miltersen. If the number of team members and the fraction of boxes which are opened is fixed, the winning probability stays strictly larger than zero when more empty boxes are added.[9] David Avis and Anne Broadbent considered in 2009 a quantum theoretical variant in which team B wins with certainty.[10] The malicious director In case the prison director does not have to distribute the numbers into the drawers randomly, and realizes that the prisoners may apply the above-mentioned strategy and guesses the box numbering the prisoners will apply (such as numbers indicated on the boxes), they can foil the strategy. To this end, they just have to ensure that their assignment of prisoners' numbers to drawers constitutes a permutation with a cycle of length larger than 50. The prisoners in turn can counter this by agreeing among themselves on a specific random numbering of the drawers, provided that the director does not overhear this or does not bother to respond by replacing numbers in the boxes before the prisoners are let in.[11] One prisoner may make one change In the case that one prisoner may enter the room first, inspect all boxes, and then switch the contents of two boxes, all prisoners will survive. This is so since any cycle of length larger than 50 can be broken, so that it can be guaranteed that all cycles are of length at most 50. Any prisoner who finds their number is free In the variant where any prisoner who finds their number is free, the expected probability of an individual's survival given a random permutation is as follows: Without strategy: ${\frac {1}{2}}$ With the strategy for the original problem: $(1-\ln(2))\cdot 1+\sum _{k=\lfloor n/2\rfloor +1}^{N}{\frac {1}{k}}\left(1-{\frac {k}{n}}\right)=1-\ln(2)+\sum _{\lfloor n/2\rfloor +1}^{N}{\frac {1}{k}}-\sum _{k=\lfloor n/2\rfloor +1}^{N}{\frac {1}{n}}=1-\ln(2)+\ln(2)-{\frac {1}{2}}={\frac {1}{2}}$ It is noteworthy that although we receive the same expected values, they are from very different distributions. With the second strategy, some prisoners are simply destined to die or live given a particular permutation, and with the first strategy (i.e., no strategy), there is "truly" a 1/2 chance for every permutation. Monty Hall problem In 2009, Adam S. Landsberg proposed the following simpler variant of the 100 prisoners problem which is based on the well-known Monty Hall problem:[12] Behind three closed doors a car, the car keys and a goat are randomly distributed. There are two players: the first player has to find the car, the second player the keys to the car. Only if both players are successful they may drive the car home. The first player enters the room and may consecutively open two of the three doors. If they are successful, the doors are closed again and the second player enters the room. The second player may also open two of the three doors, but they cannot communicate with the first player in any form. What is the winning probability if both players act optimally? If the players select their doors randomly, the winning probability is only 4/9 (about 44%). The optimal strategy is, however, as follows: • Player 1 first opens door 1. If the car is behind the door, the player is successful. If the keys were behind the door, the player next opens door 2; if instead the goat was behind the door, the player next opens door 3. • Player 2 first opens door 2. If the keys are behind the door, the player is successful. If the goat was behind the door, the player next opens door 3; whereas if the car was behind the door, the player next opens door 1. In the six possible distributions of car, keys and goat behind the three doors, the players open the following doors (in the green cases, the player was successful): Car − Keys − GoatCar − Goat − KeysKeys − Car − GoatKeys − Goat − CarGoat − Car − KeysGoat − Keys − Car Player 1 Door 1: Car Door 1: Car Door 1: Keys Door 2: Car Door 1: Keys Door 2: Goat Door 1: Goat Door 3: Keys Door 1: Goat Door 3: Car Player 2 Door 2: Keys Door 2: Goat Door 3: Keys Door 2: Car Door 1: Keys (Door 2: Goat) (Door 3: Car) (Door 2: Car) (Door 1: Goat) Door 2: Keys The success of the strategy is based on building a correlation between the successes and failures of the two players. Here, the winning probability is 2/3, which is optimal since the first player cannot have a higher winning probability than that.[12] In a further variant, three prizes are hidden behind the three doors and three players have to independently find their assigned prizes with two tries. In this case the winning probability is also 2/3 when the optimal strategy is employed.[13] See also • Prisoner's dilemma • Three prisoners problem • Unexpected hanging paradox • Random permutation statistics • Golomb–Dickman constant References 1. Philippe Flajolet, Robert Sedgewick (2009), Analytic Combinatorics, Cambridge University Press, p. 124 2. Eugene Curtin, Max Warshauer (2006), "The locker puzzle", Mathematical Intelligencer, 28: 28–31, doi:10.1007/BF02986999, S2CID 123089718 3. Richard P. Stanley (2013), Algebraic Combinatorics: Walks, Trees, Tableaux, and More, Springer, pp. 187–189 4. Anna Gál, Peter Bro Miltersen (2003), "The cell probe complexity of succinct data structures", Proceedings 30th International Colloquium on Automata, Languages and Programming (ICALP), pp. 332–344 5. Joe Buhler, Elwyn Berlekamp (2004), "Puzzles Column", The Emissary, Spring 2004: 3 6. Richard E. Blahut (2014), Cryptography and Secure Communication, Cambridge University Press, pp. 29–30 7. Christoph Pöppe (2006), "Mathematische Unterhaltungen: Freiheit für die Kombinatoriker", Spektrum der Wissenschaft (in German), 6/2006: 106–108 8. Peter Winkler (2006), "Names in Boxes Puzzle", College Mathematics Journal, 37 (4): 260, 285, 289 9. Navin Goyal, Michael Saks (2005), "A parallel search game", Random Structures & Algorithms, 27 (2): 227–234, doi:10.1002/rsa.20068, S2CID 90893 10. David Avis, Anne Broadbent (2009), "The quantum locker puzzle", Third International Conference on Quantum, Nano and Micro Technologies ICQNM '09, pp. 63–66 11. Philippe Flajolet, Robert Sedgewick (2009), Analytic Combinatorics, Cambridge University Press, p. 177 12. Adam S. Landsberg (2009), "The Return of Monty Hall", Mathematical Intelligencer, 31 (2): 1, doi:10.1007/s00283-008-9016-8 13. Eric Grundwald (2010), "Re: The Locker Puzzle", Mathematical Intelligencer, 32 (2): 1, doi:10.1007/s00283-009-9107-1 Literature • Philippe Flajolet, Robert Sedgewick (2009), Analytic Combinatorics, Cambridge University Press, ISBN 978-1-139-47716-1 • Richard P. Stanley (2013), Algebraic Combinatorics: Walks, Trees, Tableaux, and More, Undergraduate Texts in Mathematics, Springer, ISBN 978-1-461-46998-8 • Peter Winkler (2007), Mathematical Mind-Benders, Taylor and Francis, ISBN 978-1-568-81336-3 External links • Rob Heaton: Mathematicians hate civil liberties - 100 prisoners and 100 boxes, 13 January 2014 • Oliver Nash: Pity the prisoners Archived 2014-07-14 at the Wayback Machine, 12 December 2009 • Jamie Mulholland: Prisoners in Boxes, Spring 2011 (PDF) • MinutePhysics: An Impossible Bet on YouTube and Solution to The Impossible Bet on YouTube, 8 December 2014 • Robert Feldt: Stochastic simulation in Julia to check the optimal strategy, 6 July 2022	Wikipedia
Direct method in the calculus of variations In mathematics, the direct method in the calculus of variations is a general method for constructing a proof of the existence of a minimizer for a given functional,[1] introduced by Stanisław Zaremba and David Hilbert around 1900. The method relies on methods of functional analysis and topology. As well as being used to prove the existence of a solution, direct methods may be used to compute the solution to desired accuracy.[2] Part of a series of articles about Calculus • Fundamental theorem • Limits • Continuity • Rolle's theorem • Mean value theorem • Inverse function theorem Differential Definitions • Derivative (generalizations) • Differential • infinitesimal • of a function • total Concepts • Differentiation notation • Second derivative • Implicit differentiation • Logarithmic differentiation • Related rates • Taylor's theorem Rules and identities • Sum • Product • Chain • Power • Quotient • L'Hôpital's rule • Inverse • General Leibniz • Faà di Bruno's formula • Reynolds Integral • Lists of integrals • Integral transform • Leibniz integral rule Definitions • Antiderivative • Integral (improper) • Riemann integral • Lebesgue integration • Contour integration • Integral of inverse functions Integration by • Parts • Discs • Cylindrical shells • Substitution (trigonometric, tangent half-angle, Euler) • Euler's formula • Partial fractions • Changing order • Reduction formulae • Differentiating under the integral sign • Risch algorithm Series • Geometric (arithmetico-geometric) • Harmonic • Alternating • Power • Binomial • Taylor Convergence tests • Summand limit (term test) • Ratio • Root • Integral • Direct comparison • Limit comparison • Alternating series • Cauchy condensation • Dirichlet • Abel Vector • Gradient • Divergence • Curl • Laplacian • Directional derivative • Identities Theorems • Gradient • Green's • Stokes' • Divergence • generalized Stokes Multivariable Formalisms • Matrix • Tensor • Exterior • Geometric Definitions • Partial derivative • Multiple integral • Line integral • Surface integral • Volume integral • Jacobian • Hessian Advanced • Calculus on Euclidean space • Generalized functions • Limit of distributions Specialized • Fractional • Malliavin • Stochastic • Variations Miscellaneous • Precalculus • History • Glossary • List of topics • Integration Bee • Mathematical analysis • Nonstandard analysis The method The calculus of variations deals with functionals $J:V\to {\bar {\mathbb {R} }}$, where $V$ is some function space and ${\bar {\mathbb {R} }}=\mathbb {R} \cup \{\infty \}$. The main interest of the subject is to find minimizers for such functionals, that is, functions $v\in V$ such that:$J(v)\leq J(u)\forall u\in V.$ The standard tool for obtaining necessary conditions for a function to be a minimizer is the Euler–Lagrange equation. But seeking a minimizer amongst functions satisfying these may lead to false conclusions if the existence of a minimizer is not established beforehand. The functional $J$ must be bounded from below to have a minimizer. This means $\inf\{J(u)\|u\in V\}>-\infty .\,$ This condition is not enough to know that a minimizer exists, but it shows the existence of a minimizing sequence, that is, a sequence $(u_{n})$ in $V$ such that $J(u_{n})\to \inf\{J(u)\|u\in V\}.$ The direct method may be broken into the following steps 1. Take a minimizing sequence $(u_{n})$ for $J$. 2. Show that $(u_{n})$ admits some subsequence $(u_{n_{k}})$, that converges to a $u_{0}\in V$ with respect to a topology $\tau $ on $V$. 3. Show that $J$ is sequentially lower semi-continuous with respect to the topology $\tau $. To see that this shows the existence of a minimizer, consider the following characterization of sequentially lower-semicontinuous functions. The function $J$ is sequentially lower-semicontinuous if $\liminf _{n\to \infty }J(u_{n})\geq J(u_{0})$ for any convergent sequence $u_{n}\to u_{0}$ in $V$. The conclusions follows from $\inf\{J(u)\|u\in V\}=\lim _{n\to \infty }J(u_{n})=\lim _{k\to \infty }J(u_{n_{k}})\geq J(u_{0})\geq \inf\{J(u)\|u\in V\}$, in other words $J(u_{0})=\inf\{J(u)\|u\in V\}$. Details Banach spaces The direct method may often be applied with success when the space $V$ is a subset of a separable reflexive Banach space $W$. In this case the sequential Banach–Alaoglu theorem implies that any bounded sequence $(u_{n})$ in $V$ has a subsequence that converges to some $u_{0}$ in $W$ with respect to the weak topology. If $V$ is sequentially closed in $W$, so that $u_{0}$ is in $V$, the direct method may be applied to a functional $J:V\to {\bar {\mathbb {R} }}$ by showing 1. $J$ is bounded from below, 2. any minimizing sequence for $J$ is bounded, and 3. $J$ is weakly sequentially lower semi-continuous, i.e., for any weakly convergent sequence $u_{n}\to u_{0}$ it holds that $\liminf _{n\to \infty }J(u_{n})\geq J(u_{0})$. The second part is usually accomplished by showing that $J$ admits some growth condition. An example is $J(x)\geq \alpha \lVert x\rVert ^{q}-\beta $ for some $\alpha >0$, $q\geq 1$ and $\beta \geq 0$. A functional with this property is sometimes called coercive. Showing sequential lower semi-continuity is usually the most difficult part when applying the direct method. See below for some theorems for a general class of functionals. Sobolev spaces The typical functional in the calculus of variations is an integral of the form $J(u)=\int _{\Omega }F(x,u(x),\nabla u(x))dx$ where $\Omega $ is a subset of $\mathbb {R} ^{n}$ and $F$ is a real-valued function on $\Omega \times \mathbb {R} ^{m}\times \mathbb {R} ^{mn}$. The argument of $J$ is a differentiable function $u:\Omega \to \mathbb {R} ^{m}$, and its Jacobian $\nabla u(x)$ is identified with a $mn$-vector. When deriving the Euler–Lagrange equation, the common approach is to assume $\Omega $ has a $C^{2}$ boundary and let the domain of definition for $J$ be $C^{2}(\Omega ,\mathbb {R} ^{m})$. This space is a Banach space when endowed with the supremum norm, but it is not reflexive. When applying the direct method, the functional is usually defined on a Sobolev space $W^{1,p}(\Omega ,\mathbb {R} ^{m})$ with $p>1$, which is a reflexive Banach space. The derivatives of $u$ in the formula for $J$ must then be taken as weak derivatives. Another common function space is $W_{g}^{1,p}(\Omega ,\mathbb {R} ^{m})$ which is the affine sub space of $W^{1,p}(\Omega ,\mathbb {R} ^{m})$ of functions whose trace is some fixed function $g$ in the image of the trace operator. This restriction allows finding minimizers of the functional $J$ that satisfy some desired boundary conditions. This is similar to solving the Euler–Lagrange equation with Dirichlet boundary conditions. Additionally there are settings in which there are minimizers in $W_{g}^{1,p}(\Omega ,\mathbb {R} ^{m})$ but not in $W^{1,p}(\Omega ,\mathbb {R} ^{m})$. The idea of solving minimization problems while restricting the values on the boundary can be further generalized by looking on function spaces where the trace is fixed only on a part of the boundary, and can be arbitrary on the rest. The next section presents theorems regarding weak sequential lower semi-continuity of functionals of the above type. Sequential lower semi-continuity of integrals As many functionals in the calculus of variations are of the form $J(u)=\int _{\Omega }F(x,u(x),\nabla u(x))dx$, where $\Omega \subseteq \mathbb {R} ^{n}$ is open, theorems characterizing functions $F$ for which $J$ is weakly sequentially lower-semicontinuous in $W^{1,p}(\Omega ,\mathbb {R} ^{m})$ with $p\geq 1$ is of great importance. In general one has the following:[3] Assume that $F$ is a function that has the following properties: 1. The function $F$ is a Carathéodory function. 2. There exist $a\in L^{q}(\Omega ,\mathbb {R} ^{mn})$ with Hölder conjugate $q={\tfrac {p}{p-1}}$ and $b\in L^{1}(\Omega )$ such that the following inequality holds true for almost every $x\in \Omega $ and every $(y,A)\in \mathbb {R} ^{m}\times \mathbb {R} ^{mn}$: $F(x,y,A)\geq \langle a(x),A\rangle +b(x)$. Here, $\langle a(x),A\rangle $ denotes the Frobenius inner product of $a(x)$ and $A$ in $\mathbb {R} ^{mn}$). If the function $A\mapsto F(x,y,A)$ is convex for almost every $x\in \Omega $ and every $y\in \mathbb {R} ^{m}$, then $J$ is sequentially weakly lower semi-continuous. When $n=1$ or $m=1$ the following converse-like theorem holds[4] Assume that $F$ is continuous and satisfies $\|F(x,y,A)\|\leq a(x,\|y\|,\|A\|)$ for every $(x,y,A)$, and a fixed function $a(x,\|y\|,\|A\|)$ increasing in $\|y\|$ and $\|A\|$, and locally integrable in $x$. If $J$ is sequentially weakly lower semi-continuous, then for any given $(x,y)\in \Omega \times \mathbb {R} ^{m}$ the function $A\mapsto F(x,y,A)$ is convex. In conclusion, when $m=1$ or $n=1$, the functional $J$, assuming reasonable growth and boundedness on $F$, is weakly sequentially lower semi-continuous if, and only if the function $A\mapsto F(x,y,A)$ is convex. However, there are many interesting cases where one cannot assume that $F$ is convex. The following theorem[5] proves sequential lower semi-continuity using a weaker notion of convexity: Assume that $F:\Omega \times \mathbb {R} ^{m}\times \mathbb {R} ^{mn}\to [0,\infty )$ is a function that has the following properties: 1. The function $F$ is a Carathéodory function. 2. The function $F$ has $p$-growth for some $p>1$: There exists a constant $C$ such that for every $y\in \mathbb {R} ^{m}$ and for almost every $x\in \Omega $ $\|F(x,y,A)\|\leq C(1+\|y\|^{p}+\|A\|^{p})$. 3. For every $y\in \mathbb {R} ^{m}$ and for almost every $x\in \Omega $, the function $A\mapsto F(x,y,A)$ is quasiconvex: there exists a cube $D\subseteq \mathbb {R} ^{n}$ such that for every $A\in \mathbb {R} ^{mn},\varphi \in W_{0}^{1,\infty }(\Omega ,\mathbb {R} ^{m})$ it holds: $F(x,y,A)\leq \|D\|^{-1}\int _{D}F(x,y,A+\nabla \varphi (z))dz$ where $\|D\|$ is the volume of $D$. Then $J$ is sequentially weakly lower semi-continuous in $W^{1,p}(\Omega ,\mathbb {R} ^{m})$. A converse like theorem in this case is the following: [6] Assume that $F$ is continuous and satisfies $\|F(x,y,A)\|\leq a(x,\|y\|,\|A\|)$ for every $(x,y,A)$, and a fixed function $a(x,\|y\|,\|A\|)$ increasing in $\|y\|$ and $\|A\|$, and locally integrable in $x$. If $J$ is sequentially weakly lower semi-continuous, then for any given $(x,y)\in \Omega \times \mathbb {R} ^{m}$ the function $A\mapsto F(x,y,A)$ is quasiconvex. The claim is true even when both $m,n$ are bigger than $1$ and coincides with the previous claim when $m=1$ or $n=1$, since then quasiconvexity is equivalent to convexity. Notes 1. Dacorogna, pp. 1–43. 2. I. M. Gelfand; S. V. Fomin (1991). Calculus of Variations. Dover Publications. ISBN 978-0-486-41448-5. 3. Dacorogna, pp. 74–79. 4. Dacorogna, pp. 66–74. 5. Acerbi-Fusco 6. Dacorogna, pp. 156. References and further reading • Dacorogna, Bernard (1989). Direct Methods in the Calculus of Variations. Springer-Verlag. ISBN 0-387-50491-5. • Fonseca, Irene; Giovanni Leoni (2007). Modern Methods in the Calculus of Variations: $L^{p}$ Spaces. Springer. ISBN 978-0-387-35784-3. • Morrey, C. B., Jr.: Multiple Integrals in the Calculus of Variations. Springer, 1966 (reprinted 2008), Berlin ISBN 978-3-540-69915-6. • Jindřich Nečas: Direct Methods in the Theory of Elliptic Equations. (Transl. from French original 1967 by A.Kufner and G.Tronel), Springer, 2012, ISBN 978-3-642-10455-8. • T. Roubíček (2000). "Direct method for parabolic problems". Adv. Math. Sci. Appl. Vol. 10. pp. 57–65. MR 1769181. • Acerbi Emilio, Fusco Nicola. "Semicontinuity problems in the calculus of variations." Archive for Rational Mechanics and Analysis 86.2 (1984): 125-145	Wikipedia
Edge-of-the-wedge theorem In mathematics, Bogoliubov's edge-of-the-wedge theorem implies that holomorphic functions on two "wedges" with an "edge" in common are analytic continuations of each other provided they both give the same continuous function on the edge. It is used in quantum field theory to construct the analytic continuation of Wightman functions. The formulation and the first proof of the theorem were presented[1][2] by Nikolay Bogoliubov at the International Conference on Theoretical Physics, Seattle, USA (September, 1956) and also published in the book Problems in the Theory of Dispersion Relations.[3] Further proofs and generalizations of the theorem were given by R. Jost and H. Lehmann (1957),[4] F. Dyson (1958), H. Epstein (1960), and by other researchers. The one-dimensional case Continuous boundary values In one dimension, a simple case of the edge-of-the-wedge theorem can be stated as follows. • Suppose that f is a continuous complex-valued function on the complex plane that is holomorphic on the upper half-plane, and on the lower half-plane. Then it is holomorphic everywhere. In this example, the two wedges are the upper half-plane and the lower half plane, and their common edge is the real axis. This result can be proved from Morera's theorem. Indeed, a function is holomorphic provided its integral round any contour vanishes; a contour which crosses the real axis can be broken up into contours in the upper and lower half-planes and the integral round these vanishes by hypothesis.[5][6] Distributional boundary values on a circle The more general case is phrased in terms of distributions.[7][8] This is technically simplest in the case where the common boundary is the unit circle $\|z\|=1$ in the complex plane. In that case holomorphic functions f, g in the regions $r<\|z\|<1$ and $1<\|z\|<R$ have Laurent expansions $f(z)=\sum _{-\infty }^{\infty }a_{n}z^{n},\,\,\,\,g(z)=\sum _{-\infty }^{\infty }b_{n}z^{n}$ absolutely convergent in the same regions and have distributional boundary values given by the formal Fourier series $f(\theta )=\sum _{-\infty }^{\infty }a_{n}e^{in\theta },\,\,\,\,g(\theta )=\sum _{-\infty }^{\infty }b_{n}e^{in\theta }.$ Their distributional boundary values are equal if $a_{n}=b_{n}$ for all n. It is then elementary that the common Laurent series converges absolutely in the whole region $r<\|z\|<R$. Distributional boundary values on an interval In general given an open interval $I=(a,b)$ on the real axis and holomorphic functions $f_{+},\,\,\ f_{-}$ defined in $(a,b)\times (0,R)$ and $(a,b)\times (-R,0)$ satisfying $\|f_{\pm }(x+iy)\|<C\|y\|^{-N}$ for some non-negative integer N, the boundary values $T_{\pm }$ of $f_{\pm }$ can be defined as distributions on the real axis by the formulas[9][8] $\langle T_{\pm },\varphi \rangle =\lim _{\varepsilon \downarrow 0}\int f(x\pm i\varepsilon )\varphi (x)\,dx.$ Existence can be proved by noting that, under the hypothesis, $f_{\pm }(z)$ is the $(N+1)$-th complex derivative of a holomorphic function which extends to a continuous function on the boundary. If f is defined as $f_{\pm }$ above and below the real axis and F is the distribution defined on the rectangle $(a,b)\times (-R,R)$ by the formula $\langle F,\varphi \rangle =\iint f(x+iy)\varphi (x,y)\,dx\,dy,$ then F equals $f_{\pm }$ off the real axis and the distribution $F_{\overline {z}}$ is induced by the distribution $ {1 \over 2}(T_{+}-T_{-})$ on the real axis. In particular if the hypotheses of the edge-of-the-wedge theorem apply, i.e. $T_{+}=T_{-}$, then $F_{\overline {z}}=0.$ By elliptic regularity it then follows that the function F is holomorphic in $(a,b)\times (-R,R)$. In this case elliptic regularity can be deduced directly from the fact that $(\pi z)^{-1}$ is known to provide a fundamental solution for the Cauchy–Riemann operator $\partial /\partial {\overline {z}}$.[10] Using the Cayley transform between the circle and the real line, this argument can be rephrased in a standard way in terms of Fourier series and Sobolev spaces on the circle. Indeed, let $f$ and $g$ be holomorphic functions defined exterior and interior to some arc on the unit circle such that locally they have radial limits in some Sobolev space, Then, letting $D=z{\partial \over \partial z},$ the equations $D^{k}F=f,\,\,\,D^{k}G=g$ can be solved locally in such a way that the radial limits of G and F tend locally to the same function in a higher Sobolev space. For k large enough, this convergence is uniform by the Sobolev embedding theorem. By the argument for continuous functions, F and G therefore patch to give a holomorphic function near the arc and hence so do f and g. The general case A wedge is a product of a cone with some set. Let $C$ be an open cone in the real vector space $\mathbb {R} ^{n}$, with vertex at the origin. Let E be an open subset of $\mathbb {R} ^{n}$, called the edge. Write W for the wedge $E\times iC$ in the complex vector space $\mathbb {C} ^{n}$, and write W' for the opposite wedge $E\times -iC$. Then the two wedges W and W' meet at the edge E, where we identify E with the product of E with the tip of the cone. • Suppose that f is a continuous function on the union $W\cup E\cup W'$ that is holomorphic on both the wedges W and W' . Then the edge-of-the-wedge theorem says that f is also holomorphic on E (or more precisely, it can be extended to a holomorphic function on a neighborhood of E). The conditions for the theorem to be true can be weakened. It is not necessary to assume that f is defined on the whole of the wedges: it is enough to assume that it is defined near the edge. It is also not necessary to assume that f is defined or continuous on the edge: it is sufficient to assume that the functions defined on either of the wedges have the same distributional boundary values on the edge. Application to quantum field theory In quantum field theory the Wightman distributions are boundary values of Wightman functions W(z1, ..., zn) depending on variables zi in the complexification of Minkowski spacetime. They are defined and holomorphic in the wedge where the imaginary part of each zi−zi−1 lies in the open positive timelike cone. By permuting the variables we get n! different Wightman functions defined in n! different wedges. By applying the edge-of-the-wedge theorem (with the edge given by the set of totally spacelike points) one can deduce that the Wightman functions are all analytic continuations of the same holomorphic function, defined on a connected region containing all n! wedges. (The equality of the boundary values on the edge that we need to apply the edge-of-the-wedge theorem follows from the locality axiom of quantum field theory.) Connection with hyperfunctions The edge-of-the-wedge theorem has a natural interpretation in the language of hyperfunctions. A hyperfunction is roughly a sum of boundary values of holomorphic functions, and can also be thought of as something like a "distribution of infinite order". The analytic wave front set of a hyperfunction at each point is a cone in the cotangent space of that point, and can be thought of as describing the directions in which the singularity at that point is moving. In the edge-of-the-wedge theorem, we have a distribution (or hyperfunction) f on the edge, given as the boundary values of two holomorphic functions on the two wedges. If a hyperfunction is the boundary value of a holomorphic function on a wedge, then its analytic wave front set lies in the dual of the corresponding cone. So the analytic wave front set of f lies in the duals of two opposite cones. But the intersection of these duals is empty, so the analytic wave front set of f is empty, which implies that f is analytic. This is the edge-of-the-wedge theorem. In the theory of hyperfunctions there is an extension of the edge-of-the-wedge theorem to the case when there are several wedges instead of two, called Martineau's edge-of-the-wedge theorem. See the book by Hörmander for details. Notes 1. Vladimirov, V. S. (1966), Methods of the Theory of Functions of Many Complex Variables, Cambridge, Mass.: M.I.T. Press 2. V. S. Vladimirov, V. V. Zharinov, A. G. Sergeev (1994). "Bogolyubov's “edge of the wedge” theorem, its development and applications", Russian Math. Surveys, 49(5): 51—65. 3. Bogoliubov, N. N.; Medvedev, B. V.; Polivanov, M. K. (1958), Problems in the Theory of Dispersion Relations, Princeton: Institute for Advanced Study Press 4. Jost, R.; Lehmann, H. (1957). "Integral-Darstellung kausaler Kommutatoren". Nuovo Cimento. 5 (6): 1598–1610. Bibcode:1957NCim....5.1598J. doi:10.1007/BF02856049. 5. Rudin 1971 6. Streater & Wightman 2000 7. Hörmander 1990, pp. 63–65, 343–344 8. Berenstein & Gay 1991, pp. 256–265 9. Hörmander 1990, pp. 63–66 10. Hörmander 1990, p. 63,81,110 References • Berenstein, Carlos A.; Gay, Roger (1991), Complex variables: an introduction, Graduate texts in mathematics, vol. 125 (2nd ed.), Springer, ISBN 978-0-387-97349-4 Further reading • Bogoliubov, N.N.; Logunov, A.A.; Todorov, I.T. (1975), Introduction to Axiomatic Quantum Field Theory, Mathematical Physics Monograph Series, vol. 18, Reading, Massachusetts: W.A. Benjamin, ISBN 978-0-8053-0982-9, Zbl 1114.81300. • Bogoliubov, N.N.; Logunov, A.A.; Oksak, A.I.; I.T., Todorov (1990), General Principles of Quantum Field Theory, Mathematical Physics and Applied Mathematics, vol. 10, Dordrecht-Boston-London: Kluwer Academic Publishers, ISBN 978-0-7923-0540-8, Zbl 0732.46040 The connection with hyperfunctions is described in: • Hörmander, Lars (1990), The analysis of linear partial differential operators I, Grundlehren der Mathematischen Wissenschaft, vol. 256 (2 ed.), Berlin-Heidelberg-New York: Springer-Verlag, ISBN 978-0-387-52343-9, Zbl 0712.35001. • Rudin, Walter (1971), Lectures on the edge-of-the-wedge theorem, CMBS Regional Conference Series in Mathematics, vol. 6, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-1655-4, MR 0310288, Zbl 0214.09001 For the application of the edge-of-the-wedge theorem to quantum field theory see: • Streater, R.F.; Wightman, A.S. (2000), PCT, Spin and Statistics, and All That, Princeton Landmarks in Mathematics and Physics (1978 ed.), Princeton, NJ: Princeton University Press, ISBN 978-0-691-07062-9, Zbl 1026.81027 • Vladimirov, V.S. (2001) [1994], "Bogolyubov's theorem", Encyclopedia of Mathematics, EMS Press	Wikipedia
Group of rational points on the unit circle In mathematics, the rational points on the unit circle are those points (x, y) such that both x and y are rational numbers ("fractions") and satisfy x2 + y2 = 1. The set of such points turns out to be closely related to primitive Pythagorean triples. Consider a primitive right triangle, that is, with integer side lengths a, b, c, with c the hypotenuse, such that the sides have no common factor larger than 1. Then on the unit circle there exists the rational point (a/c, b/c), which, in the complex plane, is just a/c + ib/c, where i is the imaginary unit. Conversely, if (x, y) is a rational point on the unit circle in the 1st quadrant of the coordinate system (i.e. x > 0, y > 0), then there exists a primitive right triangle with sides xc, yc, c, with c being the least common multiple of the denominators of x and y. There is a correspondence between points (a, b) in the x-y plane and points a + ib in the complex plane which is used below. Group operation The set of rational points on the unit circle, shortened G in this article, forms an infinite abelian group under rotations. The identity element is the point (1, 0) = 1 + i0 = 1. The group operation, or "product" is (x, y) * (t, u) = (xt − uy, xu + yt). This product is angle addition since x = cos(A) and y = sin(A), where A is the angle that the vector (x, y) makes with the vector (1,0), measured counter-clockwise. So with (x, y) and (t, u) forming angles A and B with (1, 0) respectively, their product (xt − uy, xu + yt) is just the rational point on the unit circle forming the angle A + B with (1, 0). The group operation is expressed more easily with complex numbers: identifying the points (x, y) and (t, u) with x + iy and t + iu respectively, the group product above is just the ordinary complex number multiplication (x + iy)(t + iu) = xt − yu + i(xu + yt), which corresponds to the point (xt − uy, xu + yt) as above. Example 3/5 + 4/5i and 5/13 + 12/13i (which correspond to the two most famous Pythagorean triples (3,4,5) and (5,12,13)) are rational points on the unit circle in the complex plane, and thus are elements of G. Their group product is −33/65 + 56/65i, which corresponds to the Pythagorean triple (33,56,65). The sum of the squares of the numerators 33 and 56 is 1089 + 3136 = 4225, which is the square of the denominator 65. Other ways to describe the group $G\cong \mathrm {SO} (2,\mathbb {Q} ).$ The set of all 2×2 rotation matrices with rational entries coincides with G. This follows from the fact that the circle group $S^{1}$ is isomorphic to $\mathrm {SO} (2,\mathbb {R} )$, and the fact that their rational points coincide. Group structure The structure of G is an infinite sum of cyclic groups. Let G2 denote the subgroup of G generated by the point 0 + 1i. G2 is a cyclic subgroup of order 4. For a prime p of form 4k + 1, let Gp denote the subgroup of elements with denominator pn where n is a non-negative integer. Gp is an infinite cyclic group, and the point (a2 − b2)/p + (2ab/p)i is a generator of Gp. Furthermore, by factoring the denominators of an element of G, it can be shown that G is a direct sum of G2 and the Gp. That is: $G\cong G_{2}\oplus \bigoplus _{p\,\equiv \,1\,({\text{mod }}4)}G_{p}.$ Since it is a direct sum rather than direct product, only finitely many of the values in the Gps are non-zero. Example Viewing G as an infinite direct sum, consider the element ({0}; 2, 0, 1, 0, 0, ..., 0, ...) where the first coordinate 0 is in C4 and the other coordinates give the powers of (a2 − b2)/p(r) + i2ab/p(r), where p(r) is the rth prime number of form 4k + 1. Then this corresponds to, in G, the rational point (3/5 + i4/5)2 · (8/17 + i15/17)1 = −416/425 + i87/425. The denominator 425 is the product of the denominator 5 twice, and the denominator 17 once, and as in the previous example, the square of the numerator −416 plus the square of the numerator 87 is equal to the square of the denominator 425. It should also be noted, as a connection to help retain understanding, that the denominator 5 = p(1) is the 1st prime of form 4k + 1, and the denominator 17 = p(3) is the 3rd prime of form 4k + 1. The unit hyperbola's group of rational points There is a close connection between this group on the unit hyperbola and the group discussed above. If ${\frac {a+ib}{c}}$ is a rational point on the unit circle, where a/c and b/c are reduced fractions, then (c/a, b/a) is a rational point on the unit hyperbola, since $(c/a)^{2}-(b/a)^{2}=1,$ satisfying the equation for the unit hyperbola. The group operation here is $(x,y)\times (u,v)=(xu+yv,xv+yu),$and the group identity is the same point (1, 0) as above. In this group there is a close connection with the hyperbolic cosine and hyperbolic sine, which parallels the connection with cosine and sine in the unit circle group above. Copies inside a larger group There are isomorphic copies of both groups, as subgroups (and as geometric objects) of the group of the rational points on the abelian variety in four-dimensional space given by the equation $w^{2}+x^{2}-y^{2}+z^{2}=0.$ Note that this variety is the set of points with Minkowski metric relative to the origin equal to 0. The identity in this larger group is (1, 0, 1, 0), and the group operation is $(a,b,c,d)\times (w,x,y,z)=(aw-bx,ax+bw,cy+dz,cz+dy).$ For the group on the unit circle, the appropriate subgroup is the subgroup of points of the form (w, x, 1, 0), with $w^{2}+x^{2}=1,$ and its identity element is (1, 0, 1, 0). The unit hyperbola group corresponds to points of form (1, 0, y, z), with $y^{2}-z^{2}=1,$ and the identity is again (1, 0, 1, 0). (Of course, since they are subgroups of the larger group, they both must have the same identity element.) See also • Circle group References • The Group of Rational Points on the Unit Circle, Lin Tan, Mathematics Magazine Vol. 69, No. 3 (June, 1996), pp. 163–171 • The Group of Primitive Pythagorean Triangles, Ernest J. Eckert, Mathematics Magazine Vol 57 No. 1 (January, 1984), pp 22–26 • ’’Rational Points on Elliptic Curves’’ Joseph Silverman	Wikipedia
History of mathematical notation The history of mathematical notation[1] includes the commencement, progress, and cultural diffusion of mathematical symbols and the conflict of the methods of notation confronted in a notation's move to popularity or inconspicuousness. Mathematical notation[2] comprises the symbols used to write mathematical equations and formulas. Notation generally implies a set of well-defined representations of quantities and symbols operators.[3] The history includes Hindu–Arabic numerals, letters from the Roman, Greek, Hebrew, and German alphabets, and a host of symbols invented by mathematicians over the past several centuries. See also: Timeline of mathematics and Foundations of mathematics The development of mathematical notation can be divided in stages:[4][5] • The "rhetorical" stage is where calculations are performed by words and no symbols are used.[6] • The "syncopated" stage is where frequently used operations and quantities are represented by symbolic syntactical abbreviations. From ancient times through the post-classical age,[note 1] bursts of mathematical creativity were often followed by centuries of stagnation. As the early modern age opened and the worldwide spread of knowledge began, written examples of mathematical developments came to light. • The "symbolic" stage is where comprehensive systems of notation supersede rhetoric. Beginning in Italy in the 16th century, new mathematical developments, interacting with new scientific discoveries were made at an increasing pace that continues through the present day. This symbolic system was in use by medieval Indian mathematicians and in Europe since the middle of the 17th century,[7] and has continued to develop in the contemporary era. The area of study known as the history of mathematics is primarily an investigation into the origin of discoveries in mathematics and the focus here, the investigation into the mathematical methods and notation of the past. Rhetorical stage See also: Measurement Although the history commences with that of the Ionian schools, there is no doubt that those Ancient Greeks who paid attention to it were largely indebted to the previous investigations of the Ancient Egyptians and Ancient Phoenicians. Numerical notation's distinctive feature, i.e. symbols having local as well as intrinsic values (arithmetic), implies a state of civilization at the period of its invention. Our knowledge of the mathematical attainments of these early peoples, to which this section is devoted, is imperfect and the following brief notes be regarded as a summary of the conclusions which seem most probable, and the history of mathematics begins with the symbolic sections. Many areas of mathematics began with the study of real world problems, before the underlying rules and concepts were identified and defined as abstract structures. For example, geometry has its origins in the calculation of distances and areas in the real world; algebra started with methods of solving problems in arithmetic. There can be no doubt that most early peoples which have left records knew something of numeration and mechanics and that a few were also acquainted with the elements of land-surveying. In particular, the Egyptians paid attention to geometry and numbers, and the Phoenicians to practical arithmetic, book-keeping, navigation, and land-surveying. The results attained by these people seem to have been accessible, under certain conditions, to travelers. It is probable that the knowledge of the Egyptians and Phoenicians was largely the result of observation and measurement, and represented the accumulated experience of many ages. Beginning of notation See also: Ancient history, History of writing ancient numbers, and History of science in early cultures Written mathematics began with numbers expressed as tally marks, with each tally representing a single unit. The numerical symbols consisted probably of strokes or notches cut in wood or stone, and intelligible alike to all nations.[note 2] For example, one notch in a bone represented one animal, or person, or anything else. The peoples with whom the Greeks of Asia Minor (amongst whom notation in western history begins) were likely to have come into frequent contact were those inhabiting the eastern littoral of the Mediterranean: and Greek tradition uniformly assigned the special development of geometry to the Egyptians, and that of the science of numbers[note 3] either to the Egyptians or to the Phoenicians. The Ancient Egyptians had a symbolic notation which was the numeration by Hieroglyphics.[8][9] The Egyptian mathematics had a symbol for one, ten, one hundred, one thousand, ten thousand, one hundred thousand, and one million. Smaller digits were placed on the left of the number, as they are in Hindu–Arabic numerals. Later, the Egyptians used hieratic instead of hieroglyphic script to show numbers. Hieratic was more like cursive and replaced several groups of symbols with individual ones. For example, the four vertical lines used to represent four were replaced by a single horizontal line. This is found in the Rhind Mathematical Papyrus (c. 2000–1800 BC) and the Moscow Mathematical Papyrus (c. 1890 BC). The system the Egyptians used was discovered and modified by many other civilizations in the Mediterranean. The Egyptians also had symbols for basic operations: legs going forward represented addition, and legs walking backward to represent subtraction. The Mesopotamians had symbols for each power of ten.[10] Later, they wrote their numbers in almost exactly the same way done in modern times. Instead of having symbols for each power of ten, they would just put the coefficient of that number. Each digit was separated by only a space, but by the time of Alexander the Great, they had created a symbol that represented zero and was a placeholder. The Mesopotamians also used a sexagesimal system, that is base sixty. It is this system that is used in modern times when measuring time and angles. Babylonian mathematics is derived from more than 400 clay tablets unearthed since the 1850s.[11] Written in Cuneiform script, tablets were inscribed whilst the clay was moist, and baked hard in an oven or by the heat of the sun. Some of these appear to be graded homework. The earliest evidence of written mathematics dates back to the ancient Sumerians and the system of metrology from 3000 BC. From around 2500 BC onwards, the Sumerians wrote multiplication tables on clay tablets and dealt with geometrical exercises and division problems. The earliest traces of the Babylonian numerals also date back to this period.[12] The majority of Mesopotamian clay tablets date from 1800 to 1600 BC, and cover topics which include fractions, algebra, quadratic and cubic equations, and the calculation of regular, reciprocal and pairs.[13] The tablets also include multiplication tables and methods for solving linear and quadratic equations. The Babylonian tablet YBC 7289 gives an approximation of √2 accurate to five decimal places. Babylonian mathematics were written using a sexagesimal (base-60) numeral system. From this derives the modern-day usage of 60 seconds in a minute, 60 minutes in an hour, and 360 (60 × 6) degrees in a circle, as well as the use of minutes and seconds of arc to denote fractions of a degree. Babylonian advances in mathematics were facilitated by the fact that 60 has many divisors: the reciprocal of any integer which is a multiple of divisors of 60 has a finite expansion in base 60. (In decimal arithmetic, only reciprocals of multiples of 2 and 5 have finite decimal expansions.) Also, unlike the Egyptians, Greeks, and Romans, the Babylonians had a true place-value system, where digits written in the left column represented larger values, much as in the decimal system. They lacked, however, an equivalent of the decimal point, and so the place value of a symbol often had to be inferred from the context. Syncopated stage See also: Fundamental theorem of arithmetic and Naive set theory The history of mathematics cannot with certainty be traced back to any school or period before that of the Ionian Greeks, but the subsequent history may be divided into periods, the distinctions between which are tolerably well marked. Greek mathematics, which originated with the study of geometry, tended from its commencement to be deductive and scientific. Since the fourth century AD, Pythagoras has commonly been given credit for discovering the Pythagorean theorem, a theorem in geometry that states that in a right-angled triangle the area of the square on the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares of the other two sides.[note 5] The ancient mathematical texts are available with the prior mentioned Ancient Egyptians notation and with Plimpton 322 (Babylonian mathematics c. 1900 BC). The study of mathematics as a subject in its own right begins in the 6th century BC with the Pythagoreans, who coined the term "mathematics" from the ancient Greek μάθημα (mathema), meaning "subject of instruction".[14] Plato's influence has been especially strong in mathematics and the sciences. He helped to distinguish between pure and applied mathematics by widening the gap between "arithmetic", now called number theory and "logistic", now called arithmetic. Greek mathematics greatly refined the methods (especially through the introduction of deductive reasoning and mathematical rigor in proofs) and expanded the subject matter of mathematics.[15] Aristotle is credited with what later would be called the law of excluded middle. Abstract Mathematics[16] is what treats of magnitude[note 6] or quantity, absolutely and generally conferred, without regard to any species of particular magnitude, such as arithmetic and geometry, In this sense, abstract mathematics is opposed to mixed mathematics, wherein simple and abstract properties, and the relations of quantities primitively considered in mathematics, are applied to sensible objects, and by that means become intermixed with physical considerations, such as in hydrostatics, optics, and navigation.[16] Archimedes is generally considered to be the greatest mathematician of antiquity and one of the greatest of all time.[17][18] He used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, and gave a remarkably accurate approximation of pi.[19] He also defined the spiral bearing his name, formulae for the volumes of surfaces of revolution and an ingenious system for expressing very large numbers. In the historical development of geometry, the steps in the abstraction of geometry were made by the ancient Greeks. Euclid's Elements being the earliest extant documentation of the axioms of plane geometry— though Proclus tells of an earlier axiomatisation by Hippocrates of Chios.[20] Euclid's Elements (c. 300 BC) is one of the oldest extant Greek mathematical treatises[note 7] and consisted of 13 books written in Alexandria; collecting theorems proven by other mathematicians, supplemented by some original work.[note 8] The document is a successful collection of definitions, postulates (axioms), propositions (theorems and constructions), and mathematical proofs of the propositions. Euclid's first theorem is a lemma that possesses properties of prime numbers. The influential thirteen books cover Euclidean geometry, geometric algebra, and the ancient Greek version of algebraic systems and elementary number theory. It was ubiquitous in the Quadrivium and is instrumental in the development of logic, mathematics, and science. Diophantus of Alexandria was author of a series of books called Arithmetica, many of which are now lost. These texts deal with solving algebraic equations. Boethius provided a place for mathematics in the curriculum in the 6th century when he coined the term quadrivium to describe the study of arithmetic, geometry, astronomy, and music. He wrote De institutione arithmetica, a free translation from the Greek of Nicomachus's Introduction to Arithmetic; De institutione musica, also derived from Greek sources; and a series of excerpts from Euclid's Elements. His works were theoretical, rather than practical, and were the basis of mathematical study until the recovery of Greek and Arabic mathematical works.[21][22] Acrophonic and Milesian numeration The Greeks employed Attic numeration,[23] which was based on the system of the Egyptians and was later adapted and used by the Romans. Greek numerals one through four were vertical lines, as in the hieroglyphics. The symbol for five was the Greek letter Π (pi), which is the letter of the Greek word for five, pente. Numbers six through nine were pente with vertical lines next to it. Ten was represented by the letter (Δ) of the word for ten, deka, one hundred by the letter from the word for hundred, etc. The Ionian numeration used their entire alphabet including three archaic letters. The numeral notation of the Greeks, though far less convenient than that now in use, was formed on a perfectly regular and scientific plan,[24] and could be used with tolerable effect as an instrument of calculation, to which purpose the Roman system was totally inapplicable. The Greeks divided the twenty-four letters of their alphabet into three classes, and, by adding another symbol to each class, they had characters to represent the units, tens, and hundreds. (Jean Baptiste Joseph Delambre's Astronomie Ancienne, t. ii.) Α (α) Β (β) Г (γ) Δ (δ) Ε (ε) Ϝ (ϝ) Ζ (ζ) Η (η) θ (θ) Ι (ι) Κ (κ) Λ (λ) Μ (μ) Ν (ν) Ξ (ξ) Ο (ο) Π (π) Ϟ (ϟ) Ρ (ρ) Σ (σ) Τ (τ) Υ (υ) Φ (φ) Χ (χ) Ψ (ψ) Ω (ω) Ϡ (ϡ) 1 2 3 4 5 6 7 8 9 10 20 30 40 50 60 70 80 90 100 200 300 400 500 600 700 800 900 This system appeared in the third century BC, before the letters digamma (Ϝ), koppa (Ϟ), and sampi (Ϡ) became obsolete. When lowercase letters became differentiated from upper case letters, the lower case letters were used as the symbols for notation. Multiples of one thousand were written as the nine numbers with a stroke in front of them: thus one thousand was ",α", two-thousand was ",β", etc. M (for μὐριοι, as in "myriad") was used to multiply numbers by ten thousand. For example, the number 88,888,888 would be written as M,ηωπη*ηωπη[25] Greek mathematical reasoning was almost entirely geometric (albeit often used to reason about non-geometric subjects such as number theory), and hence the Greeks had no interest in algebraic symbols. The great exception was Diophantus of Alexandria, the great algebraist.[26] His Arithmetica was one of the texts to use symbols in equations. It was not completely symbolic, but was much more so than previous books. An unknown number was called s.[27] The square of s was $\Delta ^{y}$; the cube was $K^{y}$; the fourth power was $\Delta ^{y}\Delta $; and the fifth power was $\Delta K^{y}$.[28][note 9] Chinese mathematical notation See also: Chinese numerals The Chinese used numerals that look much like the tally system.[29] Numbers one through four were horizontal lines. Five was an X between two horizontal lines; it looked almost exactly the same as the Roman numeral for ten. Nowadays, the huāmǎ system is only used for displaying prices in Chinese markets or on traditional handwritten invoices. In the history of the Chinese, there were those who were familiar with the sciences of arithmetic, geometry, mechanics, optics, navigation, and astronomy. Mathematics in China emerged independently by the 11th century BC.[30] It is almost certain that the Chinese were acquainted with several geometrical or rather architectural implements;[note 10] with mechanical machines;[note 11] that they knew of the characteristic property of the magnetic needle; and were aware that astronomical events occurred in cycles. Chinese of that time had made attempts to classify or extend the rules of arithmetic or geometry which they knew, and to explain the causes of the phenomena with which they were acquainted beforehand. The Chinese independently developed very large and negative numbers, decimals, a place value decimal system, a binary system, algebra, geometry, and trigonometry. Chinese mathematics made early contributions, including a place value system.[31][32] The geometrical theorem known to the ancient Chinese were acquainted was applicable in certain cases (namely the ratio of sides).[note 12] It is that geometrical theorems which can be demonstrated in the quasi-experimental way of superposition were also known to them. In arithmetic their knowledge seems to have been confined to the art of calculation by means of the swan-pan, and the power of expressing the results in writing. Our knowledge of the early attainments of the Chinese, slight though it is, is more complete than in the case of most of their contemporaries. It is thus instructive, and serves to illustrate the fact, that it can be known a nation may possess considerable skill in the applied arts with but our knowledge of the later mathematics on which those arts are founded can be scarce. Knowledge of Chinese mathematics before 254 BC is somewhat fragmentary, and even after this date the manuscript traditions are obscure. Dates centuries before the classical period are generally considered conjectural by Chinese scholars unless accompanied by verified archaeological evidence. As in other early societies the focus was on astronomy in order to perfect the agricultural calendar, and other practical tasks, and not on establishing formal systems. The Chinese Board of Mathematics duties were confined to the annual preparation of an almanac, the dates and predictions in which it regulated. Ancient Chinese mathematicians did not develop an axiomatic approach, but made advances in algorithm development and algebra. The achievement of Chinese algebra reached its zenith in the 13th century, when Zhu Shijie invented method of four unknowns. As a result of obvious linguistic and geographic barriers, as well as content, Chinese mathematics and that of the mathematics of the ancient Mediterranean world are presumed to have developed more or less independently up to the time when The Nine Chapters on the Mathematical Art reached its final form, while the Writings on Reckoning and Huainanzi are roughly contemporary with classical Greek mathematics. Some exchange of ideas across Asia through known cultural exchanges from at least Roman times is likely. Frequently, elements of the mathematics of early societies correspond to rudimentary results found later in branches of modern mathematics such as geometry or number theory. The Pythagorean theorem for example, has been attested to the time of the Duke of Zhou. Knowledge of Pascal's triangle has also been shown to have existed in China centuries before Pascal,[33] such as by Shen Kuo. The state of trigonometry in China slowly began to change and advance during the Song Dynasty (960–1279), where Chinese mathematicians began to express greater emphasis for the need of spherical trigonometry in calendrical science and astronomical calculations.[34] The polymath Chinese scientist, mathematician and official Shen Kuo (1031–1095) used trigonometric functions to solve mathematical problems of chords and arcs.[34] Sal Restivo writes that Shen's work in the lengths of arcs of circles provided the basis for spherical trigonometry developed in the 13th century by the mathematician and astronomer Guo Shoujing (1231–1316).[35] As the historians L. Gauchet and Joseph Needham state, Guo Shoujing used spherical trigonometry in his calculations to improve the calendar system and Chinese astronomy.[36][37] The mathematical science of the Chinese would incorporate the work and teaching of Arab missionaries with knowledge of spherical trigonometry who had come to China in the course of the thirteenth century. Indian & Arabic numerals and notation See also: Arabic numerals, Hindu–Arabic numeral system, History of the Hindu–Arabic numeral system, and Mathematics in medieval Islam Although the origin of our present system of numerical notation is ancient, there is no doubt that it was in use among the Hindus over two thousand years ago. The algebraic notation of the Indian mathematician, Brahmagupta, was syncopated. Addition was indicated by placing the numbers side by side, subtraction by placing a dot over the subtrahend (the number to be subtracted), and division by placing the divisor below the dividend, similar to our notation but without the bar. Multiplication, evolution, and unknown quantities were represented by abbreviations of appropriate terms.[38] The Hindu–Arabic numeral system and the rules for the use of its operations, in use throughout the world today, likely evolved over the course of the first millennium AD in India and was transmitted to the west via Islamic mathematics.[39][40] Despite their name, Arabic numerals have roots in India. The reason for this misnomer is Europeans saw the numerals used in an Arabic book, Concerning the Hindu Art of Reckoning, by Muhammed ibn-Musa al-Khwarizmi. Al-Khwārizmī wrote several important books on the Hindu–Arabic numerals and on methods for solving equations. His book On the Calculation with Hindu Numerals, written about 825, along with the work of Al-Kindi,[note 13] were instrumental in spreading Indian mathematics and Indian numerals to the West. Al-Khwarizmi did not claim the numerals as Arabic, but over several Latin translations, the fact that the numerals were Indian in origin was lost. The word algorithm is derived from the Latinization of Al-Khwārizmī's name, Algoritmi, and the word algebra from the title of one of his works, Al-Kitāb al-mukhtaṣar fī hīsāb al-ğabr wa’l-muqābala (The Compendious Book on Calculation by Completion and Balancing). Islamic mathematics developed and expanded the mathematics known to Central Asian civilizations.[41] Al-Khwārizmī gave an exhaustive explanation for the algebraic solution of quadratic equations with positive roots,[42] and Al-Khwārizmī was to teach algebra in an elementary form and for its own sake.[43] Al-Khwārizmī also discussed the fundamental method of "reduction" and "balancing", referring to the transposition of subtracted terms to the other side of an equation, that is, the cancellation of like terms on opposite sides of the equation. This is the operation which al-Khwārizmī originally described as al-jabr.[44] His algebra was also no longer concerned "with a series of problems to be resolved, but an exposition which starts with primitive terms in which the combinations must give all possible prototypes for equations, which henceforward explicitly constitute the true object of study." Al-Khwārizmī also studied an equation for its own sake and "in a generic manner, insofar as it does not simply emerge in the course of solving a problem, but is specifically called on to define an infinite class of problems."[45] Al-Karaji, in his treatise al-Fakhri, extends the methodology to incorporate integer powers and integer roots of unknown quantities.[note 14][46] The historian of mathematics, F. Woepcke,[47] praised Al-Karaji for being "the first who introduced the theory of algebraic calculus." Also in the 10th century, Abul Wafa translated the works of Diophantus into Arabic. Ibn al-Haytham would develop analytic geometry. Al-Haytham derived the formula for the sum of the fourth powers, using a method that is readily generalizable for determining the general formula for the sum of any integral powers. Al-Haytham performed an integration in order to find the volume of a paraboloid, and was able to generalize his result for the integrals of polynomials up to the fourth degree.[note 15][48] In the late 11th century, Omar Khayyam would develop algebraic geometry, wrote Discussions of the Difficulties in Euclid,[note 16] and wrote on the general geometric solution to cubic equations. Nasir al-Din Tusi (Nasireddin) made advances in spherical trigonometry. Muslim mathematicians during this period include the addition of the decimal point notation to the Arabic numerals. The modern Arabic numeral symbols used around the world first appeared in Islamic North Africa in the 10th century. A distinctive Western Arabic variant of the Eastern Arabic numerals began to emerge around the 10th century in the Maghreb and Al-Andalus (sometimes called ghubar numerals, though the term is not always accepted), which are the direct ancestor of the modern Arabic numerals used throughout the world.[49] Many Greek and Arabic texts on mathematics were then translated into Latin, which led to further development of mathematics in medieval Europe. In the 12th century, scholars traveled to Spain and Sicily seeking scientific Arabic texts, including al-Khwārizmī's[note 17] and the complete text of Euclid's Elements.[note 18][50][51] One of the European books that advocated using the numerals was Liber Abaci, by Leonardo of Pisa, better known as Fibonacci. Liber Abaci is better known for the mathematical problem Fibonacci wrote in it about a population of rabbits. The growth of the population ended up being a Fibonacci sequence, where a term is the sum of the two preceding terms. Symbolic stage Symbols by popular introduction date Further information: Table of mathematical symbols by introduction date Early arithmetic and multiplication See also: Early modern age, Probability, Statistics, Notation in probability and statistics, History of probability, History of statistics, and Scientific revolution The transition to symbolic algebra, where only symbols are used, can first be seen in the work of Ibn al-Banna' al-Marrakushi (1256–1321) and Abū al-Ḥasan ibn ʿAlī al-Qalaṣādī (1412–1482).[52][53] Al-Qalasādī was the last major medieval Arab algebraist, who improved on the algebraic notation earlier used in the Maghreb by Ibn al-Banna.[54] In contrast to the syncopated notations of their predecessors, Diophantus and Brahmagupta, which lacked symbols for mathematical operations,[55] al-Qalasadi's algebraic notation was the first to have symbols for these functions and was thus "the first steps toward the introduction of algebraic symbolism." He represented mathematical symbols using characters from the Arabic alphabet.[54] The 14th century saw the development of new mathematical concepts to investigate a wide range of problems.[56] The two widely used arithmetic symbols are addition and subtraction, + and −. The plus sign was used by 1360 by Nicole Oresme[57][note 19] in his work Algorismus proportionum.[58] It is thought an abbreviation for "et", meaning "and" in Latin, in much the same way the ampersand sign also began as "et". Oresme at the University of Paris and the Italian Giovanni di Casali independently provided graphical demonstrations of the distance covered by a body undergoing uniformly accelerated motion, asserting that the area under the line depicting the constant acceleration and represented the total distance traveled.[59] The minus sign was used in 1489 by Johannes Widmann in Mercantile Arithmetic or Behende und hüpsche Rechenung auff allen Kauffmanschafft,.[60] Widmann used the minus symbol with the plus symbol, to indicate deficit and surplus, respectively.[61] In Summa de arithmetica, geometria, proportioni e proportionalità,[note 20][62] Luca Pacioli used symbols for plus and minus symbols and contained algebra.[note 21] In the 15th century, Ghiyath al-Kashi computed the value of π to the 16th decimal place. Kashi also had an algorithm for calculating nth roots.[note 22] In 1533, Regiomontanus's table of sines and cosines were published.[63] Scipione del Ferro and Niccolò Fontana Tartaglia discovered solutions for cubic equations. Gerolamo Cardano published them in his 1545 book Ars Magna, together with a solution for the quartic equations, discovered by his student Lodovico Ferrari. The radical symbol[note 23] for square root was introduced by Christoph Rudolff.[note 24] Michael Stifel's important work Arithmetica integra[64] contained important innovations in mathematical notation. In 1556, Niccolò Tartaglia used parentheses for precedence grouping. In 1557 Robert Recorde published The Whetstone of Witte which introduced the equal sign (=), as well as plus and minus signs for the English reader. In 1564, Gerolamo Cardano analyzed games of chance beginning the early stages of probability theory. In 1572 Rafael Bombelli published his L'Algebra in which he showed how to deal with the imaginary quantities that could appear in Cardano's formula for solving cubic equations. Simon Stevin's book De Thiende ('the art of tenths'), published in Dutch in 1585, contained a systematic treatment of decimal notation, which influenced all later work on the real number system. The New algebra (1591) of François Viète introduced the modern notational manipulation of algebraic expressions. For navigation and accurate maps of large areas, trigonometry grew to be a major branch of mathematics. Bartholomaeus Pitiscus coin the word "trigonometry", publishing his Trigonometria in 1595. John Napier is best known as the inventor of logarithms[note 25][65] and made common the use of the decimal point in arithmetic and mathematics.[66][67] After Napier, Edmund Gunter created the logarithmic scales (lines, or rules) upon which slide rules are based, it was William Oughtred who used two such scales sliding by one another to perform direct multiplication and division; and he is credited as the inventor of the slide rule in 1622. In 1631 Oughtred introduced the multiplication sign (×) his proportionality sign,[note 26] and abbreviations sin and cos for the sine and cosine functions.[68] Albert Girard also used the abbreviations 'sin', 'cos' and 'tan' for the trigonometric functions in his treatise. Johannes Kepler was one of the pioneers of the mathematical applications of infinitesimals.[note 27] René Descartes is credited as the father of analytical geometry, the bridge between algebra and geometry,[note 28] crucial to the discovery of infinitesimal calculus and analysis. In the 17th century, Descartes introduced Cartesian co-ordinates which allowed the development of analytic geometry.[note 29] Blaise Pascal influenced mathematics throughout his life. His Traité du triangle arithmétique ("Treatise on the Arithmetical Triangle") of 1653 described a convenient tabular presentation for binomial coefficients.[note 30] Pierre de Fermat and Blaise Pascal would investigate probability.[note 31] John Wallis introduced the infinity symbol.[note 32] He similarly used this notation for infinitesimals.[note 33] In 1657, Christiaan Huygens published the treatise on probability, On Reasoning in Games of Chance.[note 34][69] Johann Rahn introduced the division sign (÷, an obelus variant repurposed) and the therefore sign in 1659. William Jones used π in Synopsis palmariorum mathesios[70] in 1706 because it is the initial letter of the Greek word Perimetron (περιμετρον), which means perimeter in Greek. This usage was popularized in 1737 by Euler. In 1734, Pierre Bouguer used double horizontal bar below the inequality sign.[71] Derivatives notation: Leibniz and Newton See also: Leibniz's notation and Leibniz–Newton calculus controversy Derivative notations • Sir Isaac Newton • Gottfried Wilhelm Leibniz The study of linear algebra emerged from the study of determinants, which were used to solve systems of linear equations. Calculus had two main systems of notation, each created by one of the creators: that developed by Isaac Newton and the notation developed by Gottfried Leibniz. Leibniz's is the notation used most often today. Newton's was simply a dot or dash placed above the function.[note 35] In modern usage, this notation generally denotes derivatives of physical quantities with respect to time, and is used frequently in the science of mechanics. Leibniz, on the other hand, used the letter d as a prefix to indicate differentiation, and introduced the notation representing derivatives as if they were a special type of fraction.[note 36] This notation makes explicit the variable with respect to which the derivative of the function is taken. Leibniz also created the integral symbol.[note 37] The symbol is an elongated S, representing the Latin word Summa, meaning "sum". When finding areas under curves, integration is often illustrated by dividing the area into infinitely many tall, thin rectangles, whose areas are added. Thus, the integral symbol is an elongated s, for sum. High division operators and functions Letters of the alphabet in this time were to be used as symbols of quantity; and although much diversity existed with respect to the choice of letters, there were to be several universally recognized rules in the following history.[24] Here thus in the history of equations the first letters of the alphabet were indicatively known as coefficients, the last letters the unknown terms (an incerti ordinis). In algebraic geometry, again, a similar rule was to be observed, the last letters of the alphabet there denoting the variable or current coordinates. Certain letters, such as $\pi $, $e$, etc., were by universal consent appropriated as symbols of the frequently occurring numbers 3.14159 ..., and 2.7182818 ....,[note 38] etc., and their use in any other acceptation was to be avoided as much as possible.[24] Letters, too, were to be employed as symbols of operation, and with them other previously mentioned arbitrary operation characters. The letters $d$, elongated $S$ were to be appropriated as operative symbols in the differential calculus and integral calculus, $\Delta $ and Σ in the calculus of differences.[24] In functional notation, a letter, as a symbol of operation, is combined with another which is regarded as a symbol of quantity.[24][note 39] Beginning in 1718, Thomas Twinin used the division slash (solidus), deriving it from the earlier Arabic horizontal fraction bar. Pierre-Simon, marquis de Laplace developed the widely used Laplacian differential operator.[note 40] In 1750, Gabriel Cramer developed "Cramer's Rule" for solving linear systems. Euler and prime notations Leonhard Euler was one of the most prolific mathematicians in history, and also a prolific inventor of canonical notation. His contributions include his use of e to represent the base of natural logarithms. It is not known exactly why $e$ was chosen, but it was probably because the four letters of the alphabet were already commonly used to represent variables and other constants. Euler used $\pi $ to represent pi consistently. The use of $\pi $ was suggested by William Jones, who used it as shorthand for perimeter. Euler used $i$ to represent the square root of negative one,[note 41] although he earlier used it as an infinite number.[note 42][note 43] For summation, Euler used sigma, Σ.[note 44] For functions, Euler used the notation $f(x)$ to represent a function of $x$. In 1730, Euler wrote the gamma function.[note 45] In 1736, Euler produced his paper on the Seven Bridges of Königsberg[72] initiating the study of graph theory. The mathematician William Emerson[73] would develop the proportionality sign.[note 46][note 47][74][75] Much later in the abstract expressions of the value of various proportional phenomena, the parts-per notation would become useful as a set of pseudo units to describe small values of miscellaneous dimensionless quantities. Marquis de Condorcet, in 1768, advanced the partial differential sign.[note 48] In 1771, Alexandre-Théophile Vandermonde deduced the importance of topological features when discussing the properties of knots related to the geometry of position. Between 1772 and 1788, Joseph-Louis Lagrange re-formulated the formulas and calculations of Classical "Newtonian" mechanics, called Lagrangian mechanics. The prime symbol for derivatives was also made by Lagrange. But in our opinion truths of this kind should be drawn from notions rather than from notations. — Carl Friedrich Gauss[note 49] Gauss, Hamilton, and Matrix notations At the turn of the 19th century, Carl Friedrich Gauss developed the identity sign for congruence relation and, in Quadratic reciprocity, the integral part. Gauss contributed functions of complex variables, in geometry, and on the convergence of series. He gave the satisfactory proofs of the fundamental theorem of algebra and of the quadratic reciprocity law. Gauss developed the theory of solving linear systems by using Gaussian elimination, which was initially listed as an advancement in geodesy.[76] He would also develop the product sign. Also in this time, Niels Henrik Abel and Évariste Galois[note 50] conducted their work on the solvability of equations, linking group theory and field theory. After the 1800s, Christian Kramp would promote factorial notation during his research in generalized factorial function which applied to non-integers.[77] Joseph Diaz Gergonne introduced the set inclusion signs.[note 51] Peter Gustav Lejeune Dirichlet developed Dirichlet L-functions to give the proof of Dirichlet's theorem on arithmetic progressions and began analytic number theory.[note 52] In 1828, Gauss proved his Theorema Egregium (remarkable theorem in Latin), establishing property of surfaces. In the 1830s, George Green developed Green's function. In 1829. Carl Gustav Jacob Jacobi published Fundamenta nova theoriae functionum ellipticarum with his elliptic theta functions. By 1841, Karl Weierstrass, the "father of modern analysis", elaborated on the concept of absolute value and the determinant of a matrix. Matrix notation would be more fully developed by Arthur Cayley in his three papers, on subjects which had been suggested by reading the Mécanique analytique[78] of Lagrange and some of the works of Laplace. Cayley defined matrix multiplication and matrix inverses. Cayley used a single letter to denote a matrix,[79] thus treating a matrix as an aggregate object. He also realized the connection between matrices and determinants,[80] and wrote "There would be many things to say about this theory of matrices which should, it seems to me, precede the theory of determinants".[81] [... The mathematical quaternion] has, or at least involves a reference to, four dimensions. — William Rowan Hamilton[note 53] William Rowan Hamilton would introduce the nabla symbol[note 54] for vector differentials.[82][83] This was previously used by Hamilton as a general-purpose operator sign.[84] Hamilton reformulated Newtonian mechanics, now called Hamiltonian mechanics. This work has proven central to the modern study of classical field theories such as electromagnetism. This was also important to the development of quantum mechanics.[note 55] In mathematics, he is perhaps best known as the inventor of quaternion notation[note 56] and biquaternions. Hamilton also introduced the word "tensor" in 1846.[85][note 57] James Cockle would develop the tessarines[note 58] and, in 1849, coquaternions. In 1848, James Joseph Sylvester introduced into matrix algebra the term matrix.[note 59] Maxwell, Clifford, and Ricci notations In 1864 James Clerk Maxwell reduced all of the then current knowledge of electromagnetism into a linked set of differential equations with 20 equations in 20 variables, contained in A Dynamical Theory of the Electromagnetic Field.[87] (See Maxwell's equations.) The method of calculation which it is necessary to employ was given by Lagrange, and afterwards developed, with some modifications, by Hamilton's equations. It is usually referred to as Hamilton's principle; when the equations in the original form are used they are known as Lagrange's equations. In 1871 Richard Dedekind called a set of real or complex numbers which is closed under the four arithmetic operations a field. In 1873 Maxwell presented A Treatise on Electricity and Magnetism. In 1878, William Kingdon Clifford published his Elements of Dynamic.[88] Clifford developed split-biquaternions,[note 60] which he called algebraic motors. Clifford obviated quaternion study by separating the dot product and cross product of two vectors from the complete quaternion notation.[note 61] This approach made vector calculus available to engineers and others working in three dimensions and skeptical of the lead–lag effect[note 62] in the fourth dimension.[note 63] The common vector notations are used when working with vectors which are spatial or more abstract members of vector spaces, while angle notation (or phasor notation) is a notation used in electronics. In 1881, Leopold Kronecker defined what he called a "domain of rationality", which is a field extension of the field of rational numbers in modern terms.[89] In 1882, Hüseyin Tevfik Paşa wrote the book titled "Linear Algebra".[90][91] Lord Kelvin's aetheric atom theory (1860s) led Peter Guthrie Tait, in 1885, to publish a topological table of knots with up to ten crossings known as the Tait conjectures. In 1893, Heinrich M. Weber gave the clear definition of an abstract field.[note 64] Tensor calculus was developed by Gregorio Ricci-Curbastro between 1887 and 1896, presented in 1892 under the title absolute differential calculus,[92] and the contemporary usage of "tensor" was stated by Woldemar Voigt in 1898.[93] In 1895, Henri Poincaré published Analysis Situs.[94] In 1897, Charles Proteus Steinmetz would publish Theory and Calculation of Alternating Current Phenomena, with the assistance of Ernst J. Berg.[95] From formula mathematics to tensors The above proposition is occasionally useful. — Bertrand Russell [note 65] In 1895 Giuseppe Peano issued his Formulario mathematico,[96] an effort to digest mathematics into terse text based on special symbols. He would provide a definition of a vector space and linear map. He would also introduce the intersection sign, the union sign, the membership sign (is an element of), and existential quantifier[note 66] (there exists). Peano would pass to Bertrand Russell his work in 1900 at a Paris conference; it so impressed Russell that Russell too was taken with the drive to render mathematics more concisely. The result was Principia Mathematica written with Alfred North Whitehead. This treatise marks a watershed in modern literature where symbol became dominant.[note 67] Ricci-Curbastro and Tullio Levi-Civita popularized the tensor index notation around 1900.[97] Mathematical logic and abstraction Abstraction • Felix Klein • Georg Cantor At the beginning of this period, Felix Klein's "Erlangen program" identified the underlying theme of various geometries, defining each of them as the study of properties invariant under a given group of symmetries. This level of abstraction revealed connections between geometry and abstract algebra. Georg Cantor[note 68] would introduce the aleph symbol for cardinal numbers of transfinite sets.[note 69] His notation for the cardinal numbers was the Hebrew letter $\aleph $ (aleph) with a natural number subscript; for the ordinals he employed the Greek letter ω (omega). This notation is still in use today in ordinal notation of a finite sequence of symbols from a finite alphabet which names an ordinal number according to some scheme which gives meaning to the language. His theory created a great deal of controversy. Cantor would, in his study of Fourier series, consider point sets in Euclidean space. After the turn of the 20th century, Josiah Willard Gibbs would in physical chemistry introduce middle dot for dot product and the multiplication sign for cross products. He would also supply notation for the scalar and vector products, which was introduced in Vector Analysis. In 1904, Ernst Zermelo promotes axiom of choice and his proof of the well-ordering theorem.[98] Bertrand Russell would shortly afterward introduce logical disjunction (OR) in 1906. Also in 1906, Poincaré would publish On the Dynamics of the Electron[99] and Maurice Fréchet introduced metric space.[100] Later, Gerhard Kowalewski and Cuthbert Edmund Cullis[101][102][103] would successively introduce matrices notation, parenthetical matrix and box matrix notation respectively. After 1907, mathematicians[note 70] studied knots from the point of view of the knot group and invariants from homology theory.[note 71] In 1908, Joseph Wedderburn's structure theorems were formulated for finite-dimensional algebras over a field. Also in 1908, Ernst Zermelo proposed "definite" property and the first axiomatic set theory, Zermelo set theory. In 1910 Ernst Steinitz published the influential paper Algebraic Theory of Fields.[note 72][note 73] In 1911, Steinmetz would publish Theory and Calculation of Transient Electric Phenomena and Oscillations. Albert Einstein, in 1916, introduced the Einstein notation[note 74] which summed over a set of indexed terms in a formula, thus exerting notational brevity. Arnold Sommerfeld would create the contour integral sign in 1917. Also in 1917, Dimitry Mirimanoff proposes axiom of regularity. In 1919, Theodor Kaluza would solve general relativity equations using five dimensions, the results would have electromagnetic equations emerge.[104] This would be published in 1921 in "Zum Unitätsproblem der Physik".[105] In 1922, Abraham Fraenkel and Thoralf Skolem independently proposed replacing the axiom schema of specification with the axiom schema of replacement. Also in 1922, Zermelo–Fraenkel set theory was developed. In 1923, Steinmetz would publish Four Lectures on Relativity and Space. Around 1924, Jan Arnoldus Schouten would develop the modern notation and formalism for the Ricci calculus framework during the absolute differential calculus applications to general relativity and differential geometry in the early twentieth century.[note 75][106][107][108] In 1925, Enrico Fermi would describe a system comprising many identical particles that obey the Pauli exclusion principle, afterwards developing a diffusion equation (Fermi age equation). In 1926, Oskar Klein would develop the Kaluza–Klein theory. In 1928, Emil Artin abstracted ring theory with Artinian rings. In 1933, Andrey Kolmogorov introduces the Kolmogorov axioms. In 1937, Bruno de Finetti deduced the "operational subjective" concept. Mathematical symbolism See also: Category theory, Model theory, Table of logic symbols, and Logic alphabet Mathematical abstraction began as a process of extracting the underlying essence of a mathematical concept,[109][110] removing any dependence on real world objects with which it might originally have been connected,[111] and generalizing it so that it has wider applications or matching among other abstract descriptions of equivalent phenomena. Two abstract areas of modern mathematics are category theory and model theory. Bertrand Russell,[112] said, "Ordinary language is totally unsuited for expressing what physics really asserts, since the words of everyday life are not sufficiently abstract. Only mathematics and mathematical logic can say as little as the physicist means to say". Though, one can substituted mathematics for real world objects, and wander off through equation after equation, and can build a concept structure which has no relation to reality.[113] Symbolic logic studies the purely formal properties of strings of symbols. The interest in this area springs from two sources. First, the notation used in symbolic logic can be seen as representing the words used in philosophical logic. Second, the rules for manipulating symbols found in symbolic logic can be implemented on a computing machine. Symbolic logic is usually divided into two subfields, propositional logic and predicate logic. Other logics of interest include temporal logic, modal logic and fuzzy logic. The area of symbolic logic called propositional logic, also called propositional calculus, studies the properties of sentences formed from constants[note 76] and logical operators. The corresponding logical operations are known, respectively, as conjunction, disjunction, material conditional, biconditional, and negation. These operators are denoted as keywords[note 77] and by symbolic notation. Some of the introduced mathematical logic notation during this time included the set of symbols used in Boolean algebra. This was created by George Boole in 1854. Boole himself did not see logic as a branch of mathematics, but it has come to be encompassed anyway. Symbols found in Boolean algebra include $\land $ (AND), $\lor $ (OR), and $\lnot $ (not). With these symbols, and letters to represent different truth values, one can make logical statements such as $a\lor \lnot a=1$, that is "(a is true OR a is not true) is true", meaning it is true that a is either true or not true (i.e. false). Boolean algebra has many practical uses as it is, but it also was the start of what would be a large set of symbols to be used in logic.[note 78] Predicate logic, originally called predicate calculus, expands on propositional logic by the introduction of variables[note 79] and by sentences containing variables, called predicates.[note 80] In addition, predicate logic allows quantifiers.[note 81] With these logic symbols and additional quantifiers from predicate logic,[note 82] valid proofs can be made that are irrationally artificial,[note 83] but syntactical.[note 84] Gödel incompleteness notation See also: Proof sketch for Gödel's first incompleteness theorem To every ω-consistent recursive class κ of formulae there correspond recursive class signs r, such that neither v Gen r nor Neg (v Gen r) belongs to Flg (κ) (where v is the free variable of r). — Kurt Gödel[114] While proving his incompleteness theorems,[note 85] Kurt Gödel created an alternative to the symbols normally used in logic. He used Gödel numbers, which were numbers that represented operations with set numbers, and variables with the prime numbers greater than 10. With Gödel numbers, logic statements can be broken down into a number sequence. Gödel then took this one step farther, taking the n prime numbers and putting them to the power of the numbers in the sequence. These numbers were then multiplied together to get the final product, giving every logic statement its own number.[115][note 86] Early 20th-century notation Abstraction of notation is an ongoing process and the historical development of many mathematical topics exhibits a progression from the concrete to the abstract. Various set notations would be developed for fundamental object sets. Around 1924, David Hilbert and Richard Courant published "Methods of mathematical physics. Partial differential equations".[116] In 1926, Oskar Klein and Walter Gordon proposed the Klein–Gordon equation to describe relativistic particles.[note 87] The first formulation of a quantum theory describing radiation and matter interaction is due to Paul Adrien Maurice Dirac, who, during 1920, was first able to compute the coefficient of spontaneous emission of an atom.[117] In 1928, the relativistic Dirac equation was formulated by Dirac to explain the behavior of the relativistically moving electron.[note 88] Dirac described the quantification of the electromagnetic field as an ensemble of harmonic oscillators with the introduction of the concept of creation and annihilation operators of particles. In the following years, with contributions from Wolfgang Pauli, Eugene Wigner, Pascual Jordan, and Werner Heisenberg, and an elegant formulation of quantum electrodynamics due to Enrico Fermi,[118] physicists came to believe that, in principle, it would be possible to perform any computation for any physical process involving photons and charged particles. In 1931, Alexandru Proca developed the Proca equation (Euler–Lagrange equation)[note 89] for the vector meson theory of nuclear forces and the relativistic quantum field equations. John Archibald Wheeler in 1937 develops S-matrix. Studies by Felix Bloch with Arnold Nordsieck,[119] and Victor Weisskopf,[120] in 1937 and 1939, revealed that such computations were reliable only at a first order of perturbation theory, a problem already pointed out by Robert Oppenheimer.[121] At higher orders in the series infinities emerged, making such computations meaningless and casting serious doubts on the internal consistency of the theory itself. With no solution for this problem known at the time, it appeared that a fundamental incompatibility existed between special relativity and quantum mechanics. In the 1930s, the double-struck capital Z for integer number sets was created by Edmund Landau. Nicolas Bourbaki created the double-struck capital Q for rational number sets. In 1935, Gerhard Gentzen made universal quantifiers. In 1936, Tarski's undefinability theorem is stated by Alfred Tarski and proved.[note 90] In 1938, Gödel proposes the constructible universe in the paper "The Consistency of the Axiom of Choice and of the Generalized Continuum-Hypothesis". André Weil and Nicolas Bourbaki would develop the empty set sign in 1939. That same year, Nathan Jacobson would coin the double-struck capital C for complex number sets. Around the 1930s, Voigt notation[note 91] would be developed for multilinear algebra as a way to represent a symmetric tensor by reducing its order. Schönflies notation[note 92] became one of two conventions used to describe point groups (the other being Hermann–Mauguin notation). Also in this time, van der Waerden notation[122][123] became popular for the usage of two-component spinors (Weyl spinors) in four spacetime dimensions. Arend Heyting would introduce Heyting algebra and Heyting arithmetic. The arrow, e.g., →, was developed for function notation in 1936 by Øystein Ore to denote images of specific elements.[note 93][note 94] Later, in 1940, it took its present form, e.g., f: X → Y, through the work of Witold Hurewicz. Werner Heisenberg, in 1941, proposed the S-matrix theory of particle interactions. Bra–ket notation (Dirac notation) is a standard notation for describing quantum states, composed of angle brackets and vertical bars. It can also be used to denote abstract vectors and linear functionals. It is so called because the inner product (or dot product on a complex vector space) of two states is denoted by a ⟨bra\|ket⟩[note 95] consisting of a left part, ⟨φ\|, and a right part, \|ψ⟩. The notation was introduced in 1939 by Paul Dirac,[124] though the notation has precursors in Grassmann's use of the notation [φ\|ψ] for his inner products nearly 100 years previously.[125] Bra–ket notation is widespread in quantum mechanics: almost every phenomenon that is explained using quantum mechanics—including a large portion of modern physics—is usually explained with the help of bra–ket notation. The notation establishes an encoded abstract representation-independence, producing a versatile specific representation (e.g., x, or p, or eigenfunction base) without much ado, or excessive reliance on, the nature of the linear spaces involved. The overlap expression ⟨φ\|ψ⟩ is typically interpreted as the probability amplitude for the state ψ to collapse into the state ϕ. The Feynman slash notation (Dirac slash notation[126]) was developed by Richard Feynman for the study of Dirac fields in quantum field theory. In 1948, Valentine Bargmann and Eugene Wigner proposed the relativistic Bargmann–Wigner equations to describe free particles and the equations are in the form of multi-component spinor field wavefunctions. In 1950, William Vallance Douglas Hodge presented "The topological invariants of algebraic varieties" at the Proceedings of the International Congress of Mathematicians. Between 1954 and 1957, Eugenio Calabi worked on the Calabi conjecture for Kähler metrics and the development of Calabi–Yau manifolds. In 1957, Tullio Regge formulated the mathematical property of potential scattering in the Schrödinger equation.[note 96] Stanley Mandelstam, along with Regge, did the initial development of the Regge theory of strong interaction phenomenology. In 1958, Murray Gell-Mann and Richard Feynman, along with George Sudarshan and Robert Marshak, deduced the chiral structures of the weak interaction in physics. Geoffrey Chew, along with others, would promote matrix notation for the strong interaction, and the associated bootstrap principle, in 1960. In the 1960s, set-builder notation was developed for describing a set by stating the properties that its members must satisfy. Also in the 1960s, tensors are abstracted within category theory by means of the concept of monoidal category. Later, multi-index notation eliminates conventional notions used in multivariable calculus, partial differential equations, and the theory of distributions, by abstracting the concept of an integer index to an ordered tuple of indices. Modern mathematical notation See also: Approximation theory, Universal property, Tensor algebra, Free algebra, and Abstract algebra In the modern mathematics of special relativity, electromagnetism and wave theory, the d'Alembert operator[note 97][note 98] is the Laplace operator of Minkowski space. The Levi-Civita symbol[note 99] is used in tensor calculus. After the full Lorentz covariance formulations that were finite at any order in a perturbation series of quantum electrodynamics, Sin-Itiro Tomonaga, Julian Schwinger and Richard Feynman were jointly awarded with a Nobel prize in physics in 1965.[127] Their contributions, and those of Freeman Dyson, were about covariant and gauge invariant formulations of quantum electrodynamics that allow computations of observables at any order of perturbation theory. Feynman's mathematical technique, based on his diagrams, initially seemed very different from the field-theoretic, operator-based approach of Schwinger and Tomonaga, but Freeman Dyson later showed that the two approaches were equivalent. Renormalization, the need to attach a physical meaning at certain divergences appearing in the theory through integrals, has subsequently become one of the fundamental aspects of quantum field theory and has come to be seen as a criterion for a theory's general acceptability. Quantum electrodynamics has served as the model and template for subsequent quantum field theories. Peter Higgs, Jeffrey Goldstone, and others, Sheldon Glashow, Steven Weinberg and Abdus Salam independently showed how the weak nuclear force and quantum electrodynamics could be merged into a single electroweak force. In the late 1960s, the particle zoo was composed of the then known elementary particles before the discovery of quarks. A step towards the Standard Model was Sheldon Glashow's discovery, in 1960, of a way to combine the electromagnetic and weak interactions.[128] In 1967, Steven Weinberg[129] and Abdus Salam[130] incorporated the Higgs mechanism[131][132][133] into Glashow's electroweak theory, giving it its modern form. The Higgs mechanism is believed to give rise to the masses of all the elementary particles in the Standard Model. This includes the masses of the W and Z bosons, and the masses of the fermions – i.e. the quarks and leptons. Also in 1967, Bryce DeWitt published his equation under the name "Einstein–Schrödinger equation" (later renamed the "Wheeler–DeWitt equation").[134] In 1969, Yoichiro Nambu, Holger Bech Nielsen, and Leonard Susskind described space and time in terms of strings. In 1970, Pierre Ramond develop two-dimensional supersymmetries. Michio Kaku and Keiji Kikkawa would afterwards formulate string variations. In 1972, Michael Artin, Alexandre Grothendieck, Jean-Louis Verdier propose the Grothendieck universe.[135] After the neutral weak currents caused by Z boson exchange were discovered at CERN in 1973,[136][137][138][139] the electroweak theory became widely accepted and Glashow, Salam, and Weinberg shared the 1979 Nobel Prize in Physics for discovering it. The theory of the strong interaction, to which many contributed, acquired its modern form around 1973–74. With the establishment of quantum chromodynamics, a finalized a set of fundamental and exchange particles, which allowed for the establishment of a "standard model" based on the mathematics of gauge invariance, which successfully described all forces except for gravity, and which remains generally accepted within the domain to which it is designed to be applied. In the late 1970s, William Thurston introduced hyperbolic geometry into the study of knots with the hyperbolization theorem. The orbifold notation system, invented by Thurston, has been developed for representing types of symmetry groups in two-dimensional spaces of constant curvature. In 1978, Shing-Tung Yau deduced that the Calabi conjecture have Ricci flat metrics. In 1979, Daniel Friedan showed that the equations of motions of string theory are abstractions of Einstein equations of General Relativity. The first superstring revolution is composed of mathematical equations developed between 1984 and 1986. In 1984, Vaughan Jones deduced the Jones polynomial and subsequent contributions from Edward Witten, Maxim Kontsevich, and others, revealed deep connections between knot theory and mathematical methods in statistical mechanics and quantum field theory. According to string theory, all particles in the "particle zoo" have a common ancestor, namely a vibrating string. In 1985, Philip Candelas, Gary Horowitz,[140] Andrew Strominger, and Edward Witten would publish "Vacuum configurations for superstrings"[141] Later, the tetrad formalism (tetrad index notation) would be introduced as an approach to general relativity that replaces the choice of a coordinate basis by the less restrictive choice of a local basis for the tangent bundle.[note 102][142] In the 1990s, Roger Penrose would propose Penrose graphical notation (tensor diagram notation) as a, usually handwritten, visual depiction of multilinear functions or tensors.[143] Penrose would also introduce abstract index notation.[note 103] In 1995, Edward Witten suggested M-theory and subsequently used it to explain some observed dualities, initiating the second superstring revolution.[note 104] John Conway would further various notations, including the Conway chained arrow notation, the Conway notation of knot theory, and the Conway polyhedron notation. The Coxeter notation system classifies symmetry groups, describing the angles between with fundamental reflections of a Coxeter group. It uses a bracketed notation, with modifiers to indicate certain subgroups. The notation is named after H. S. M. Coxeter and Norman Johnson more comprehensively defined it. Combinatorial LCF notation[note 105] has been developed for the representation of cubic graphs that are Hamiltonian.[144][145] The cycle notation is the convention for writing down a permutation in terms of its constituent cycles.[146] This is also called circular notation and the permutation called a cyclic or circular permutation.[147] Computers and markup notation See also: Symbolic computation, Symbolic dynamics, Computational complexity theory, Mathematical markup language, MathML, Basic Linear Algebra Subprograms, Numerical linear algebra, List of numerical libraries, List of numerical-analysis software, DOT language, Lisp (programming language), Object-oriented programming, and Earley algorithm In 1931, IBM produces the IBM 601 Multiplying Punch; it is an electromechanical machine that could read two numbers, up to 8 digits long, from a card and punch their product onto the same card.[148] In 1934, Wallace Eckert used a rigged IBM 601 Multiplying Punch to automate the integration of differential equations.[149] In 1936, Alan Turing publishes "On Computable Numbers, With an Application to the Entscheidungsproblem".[150][note 106] John von Neumann, pioneer of the digital computer and of computer science,[note 107] in 1945, writes the incomplete First Draft of a Report on the EDVAC. In 1962, Kenneth E. Iverson developed an integral part notation, which became APL, for manipulating arrays that he taught to his students, and described in his book A Programming Language. In 1970, Edgar F. Codd proposed relational algebra as a relational model of data for database query languages. In 1971, Stephen Cook publishes "The complexity of theorem proving procedures"[151] In the 1970s within computer architecture, Quote notation was developed for a representing number system of rational numbers. Also in this decade, the Z notation (just like the APL language, long before it) uses many non-ASCII symbols, the specification includes suggestions for rendering the Z notation symbols in ASCII and in LaTeX. There are presently various C mathematical functions (Math.h) and numerical libraries. They are libraries used in software development for performing numerical calculations. These calculations can be handled by symbolic executions; analyzing a program to determine what inputs cause each part of a program to execute. Mathematica and SymPy are examples of computational software programs based on symbolic mathematics. Future of mathematical notation Main article: Future of mathematics In the history of mathematical notation, ideographic symbol notation has come full circle with the rise of computer visualization systems. The notations can be applied to abstract visualizations, such as for rendering some projections of a Calabi–Yau manifold. Examples of abstract visualization which properly belong to the mathematical imagination can be found in computer graphics. The need for such models abounds, for example, when the measures for the subject of study are actually random variables and not really ordinary mathematical functions. See also Main relevance Abuse of notation, Well-formed formula, Big O notation (L-notation), Dowker notation, Hungarian notation, Infix notation, Positional notation, Polish notation (Reverse Polish notation), Sign-value notation, History of writing numbers Numbers and quantities List of numbers, Irrational and suspected irrational numbers, γ, ζ(3), √2, √3, √5, φ, ρ, δS, α, e, π, δ, Physical constants, c, ε0, h, G, Greek letters used in mathematics, science, and engineering General relevance Order of operations, Scientific notation (Engineering notation), Actuarial notation Dot notation Chemical notation (Lewis dot notation (Electron dot notation)), Dot-decimal notation Arrow notation Knuth's up-arrow notation, infinitary combinatorics (Arrow notation (Ramsey theory)) Geometries Projective geometry, Affine geometry, Finite geometry Lists and outlines Outline of mathematics (Mathematics history topics and Mathematics topics (Mathematics categories)), Mathematical theories ( First-order theories, Theorems and Disproved mathematical ideas), Mathematical proofs (Incomplete proofs), Mathematical identities, Mathematical series, Mathematics reference tables, Mathematical logic topics, Mathematics-based methods, Mathematical functions, Transforms and Operators, Points in mathematics, Mathematical shapes, Knots (Prime knots and Mathematical knots and links), Inequalities, Mathematical concepts named after places, Mathematical topics in classical mechanics, Mathematical topics in quantum theory, Mathematical topics in relativity, String theory topics, Unsolved problems in mathematics, Mathematical jargon, Mathematical examples, Mathematical abbreviations, List of mathematical symbols Misc. Hilbert's problems, Mathematical coincidence, Chess notation, Line notation, Musical notation (Dotted note), Whyte notation, Dice notation, recursive categorical syntax People Mathematicians (Amateur mathematicians and Female mathematicians), Thomas Bradwardine, Thomas Harriot, Felix Hausdorff, Gaston Julia, Helge von Koch, Paul Lévy, Aleksandr Lyapunov, Benoit Mandelbrot, Lewis Fry Richardson, Wacław Sierpiński, Saunders Mac Lane, Paul Cohen, Gottlob Frege, G. S. Carr, Robert Recorde, Bartel Leendert van der Waerden, G. H. Hardy, E. M. Wright, James R. Newman, Carl Gustav Jacob Jacobi, Roger Joseph Boscovich, Eric W. Weisstein, Mathematical probabilists, Statisticians Notes 1. Or the Middle Ages. 2. Such characters, in fact, are preserved with little alteration in the Roman notation, an account of which may be found in John Leslie's Philosophy of Arithmetic. 3. Number theory is the branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well as the properties of objects made out of integers (e.g., rational numbers) or defined as generalizations of the integers (e.g., algebraic integers). 4. Greek: μή μου τοὺς κύκλους τάραττε 5. That is, $a^{2}+b^{2}=c^{2}$. 6. Magnitude (mathematics), the relative size of an object; Magnitude (vector), a term for the size or length of a vector; Scalar (mathematics), a quantity defined only by its magnitude; Euclidean vector, a quantity defined by both its magnitude and its direction; Order of magnitude, the class of scale having a fixed value ratio to the preceding class. 7. Autolycus' On the Moving Sphere is another ancient mathematical manuscript of the time. 8. Proclus, a Greek mathematician who lived several centuries after Euclid, wrote in his commentary of the Elements: "Euclid, who put together the Elements, collecting many of Eudoxus' theorems, perfecting many of Theaetetus', and also bringing to irrefragable demonstration the things which were only somewhat loosely proved by his predecessors". 9. The expression: $2x^{4}+3x^{3}-4x^{2}+5x-6$ would be written as: SS2 C3 x5 M S4 u6 . 10. such as the rule, square, compasses, water level (reed level), and plumb-bob. 11. such as the wheel and axle 12. The area of the square described on the hypotenuse of a right-angled triangle is equal to the sum of the areas of the squares described on the sides 13. Al-Kindi also introduced cryptanalysis and frequency analysis. 14. Something close to a proof by mathematical induction appears in a book written by Al-Karaji around 1000 AD, who used it to prove the binomial theorem, Pascal's triangle, and the sum of integral cubes. 15. He thus came close to finding a general formula for the integrals of polynomials, but he was not concerned with any polynomials higher than the fourth degree. 16. a book about what he perceived as flaws in Euclid's Elements, especially the parallel postulate 17. translated into Latin by Robert of Chester 18. translated in various versions by Adelard of Bath, Herman of Carinthia, and Gerard of Cremona 19. His own personal use started around 1351. 20. Summa de Arithmetica: Geometria Proportioni et Proportionalita. Tr. Sum of Arithmetic: Geometry in proportions and proportionality. 21. Much of the work originated from Piero della Francesca whom he appropriated and purloined. 22. This was a special case of the methods given many centuries later by Ruffini and Horner. 23. That is, ${\sqrt {~}}$. 24. Because, it is thought, it resembled a lowercase "r" (for "radix"). 25. Published in Description of the Marvelous Canon of Logarithms 26. That is, ∷ 27. see Law of Continuity. 28. Using Cartesian coordinates on the plane, the distance between two points (x1, y1) and (x2, y2) is defined by the formula: $d={\sqrt {(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}}},$ which can be viewed as a version of the Pythagorean theorem. 29. Further steps in abstraction were taken by Lobachevsky, Bolyai, Riemann, and Gauss who generalised the concepts of geometry to develop non-Euclidean geometries. 30. Now called Pascal's triangle. 31. For example, the "problem of points". 32. That is, ${\infty }$. 33. For example, ${\frac {1}{\infty }}.$ 34. Original title, "De ratiociniis in ludo aleae" 35. For example, the derivative of the function x would be written as ${\dot {x}}$. The second derivative of x would be written as ${\ddot {x}}$, etc. 36. For example, the derivative of the function x with respect to the variable t in Leibniz's notation would be written as ${dx \over dt}$. 37. That is, $\int _{-N}^{N}f(x)\,dx$. 38. See also: List of representations of e 39. Thus $f(x)$ denotes the mathematical result of the performance of the operation $f$ upon the subject $x$. If upon this result the same operation were repeated, the new result would be expressed by $f[f(x)]$, or more concisely by $f^{2}(x)$, and so on. The quantity $x$ itself regarded as the result of the same operation $f$ upon some other function; the proper symbol for which is, by analogy, $f^{-1}(x)$. Thus $f$ and $f^{-1}$ are symbols of inverse operations, the former cancelling the effect of the latter on the subject $x$. $f(x)$ and $f^{-1}(x)$ in a similar manner are termed inverse functions. 40. That is, $\Delta f(p)$ 41. That is, ${\sqrt {-1}}$ 42. Today, the symbol created by John Wallis, $\infty $, is used for infinity. 43. As in, $\sum _{n=1}^{\infty }{\frac {1}{n^{2}}}$ 44. Capital-sigma notation uses a symbol that compactly represents summation of many similar terms: the summation symbol, Σ, an enlarged form of the upright capital Greek letter Sigma. This is defined as: $\sum _{i=m}^{n}a_{i}=a_{m}+a_{m+1}+a_{m+2}+\cdots +a_{n-1}+a_{n}.$ Where, i represents the index of summation; ai is an indexed variable representing each successive term in the series; m is the lower bound of summation, and n is the upper bound of summation. The "i = m" under the summation symbol means that the index i starts out equal to m. The index, i, is incremented by 1 for each successive term, stopping when i = n. 45. That is, $n!=\int _{0}^{1}(-\ln s)^{n}\,{\rm {d}}s$. valid for n > 0. 46. That is, ∝ 47. Proportionality is the ratio of one quantity to another, especially the ratio of a part compared to a whole. In a mathematical context, a proportion is the statement of equality between two ratios; See Proportionality (mathematics), the relationship of two variables whose ratio is constant. See also aspect ratio, geometric proportions. 48. The curly d or Jacobi's delta. 49. About the proof of Wilson's theorem. Disquisitiones Arithmeticae (1801) Article 76 50. Galois theory and Galois geometry is named after him. 51. That is, "subset of" and "superset of"; This would later be redeveloped by Ernst Schröder. 52. A science of numbers that uses methods from mathematical analysis to solve problems about the integers. 53. Quoted in Robert Perceval Graves' Life of Sir William Rowan Hamilton (3 volumes, 1882, 1885, 1889) 54. That is, $\nabla $ (or, later called del, ∇) 55. See Hamiltonian (quantum mechanics). 56. That is, $i^{2}=j^{2}=k^{2}=ijk=-1$ 57. Though his use describes something different from what is now meant by a tensor. Namely, the norm operation in a certain type of algebraic system (now known as a Clifford algebra). 58. That is, $t=w+xi+yj+zk,\quad w,x,y,z\in \mathbb {R} $ where $ij=ji=k,\quad i^{2}=-1,\quad j^{2}=+1.$ 59. This is Latin for "womb". 60. That is, $q=w+xi+yj+zk$ 61. Clifford intersected algebra with Hamilton's quaternions by replacing Hermann Grassmann's rule epep = 0 by the rule epep = 1. For more details, see exterior algebra. 62. See: Phasor, Group (mathematics), Signal velocity, Polyphase system, Harmonic oscillator, and RLC series circuit 63. Or the concept of a fourth spatial dimension. See also: Spacetime, the unification of time and space as a four-dimensional continuum; and, Minkowski space, the mathematical setting for special relativity. 64. See also: Mathematic fields and Field extension 65. Comment after the proof that 1+1=2, completed in Principia mathematica, by Alfred North Whitehead ... and Bertrand Russell. Volume II, 1st edition (1912) 66. This raises questions of the pure existence theorems. 67. Peano's Formulario Mathematico, though less popular than Russell's work, continued through five editions. The fifth appeared in 1908 and included 4200 formulas and theorems. 68. Inventor of set theory 69. Transfinite arithmetic is the generalization of elementary arithmetic to infinite quantities like infinite sets; See Transfinite numbers, Transfinite induction, and Transfinite interpolation. See also Ordinal arithmetic. 70. Such as Max Dehn, J. W. Alexander, and others. 71. Such as the Alexander polynomial. 72. (German: Algebraische Theorie der Körper) 73. In this paper Steinitz axiomatically studied the properties of fields and defined many important field theoretic concepts like prime field, perfect field and the transcendence degree of a field extension. 74. The indices range over set {1, 2, 3}, $y=\sum _{i=1}^{3}c_{i}x^{i}=c_{1}x^{1}+c_{2}x^{2}+c_{3}x^{3}$ is reduced by the convention to: $y=c_{i}x^{i}\,.$ Upper indices are not exponents but are indices of coordinates, coefficients or basis vectors. See also: Ricci calculus 75. Ricci calculus constitutes the rules of index notation and manipulation for tensors and tensor fields. See also: Synge J.L.; Schild A. (1949). Tensor Calculus. first Dover Publications 1978 edition. pp. 6–108. 76. Here a logical constant is a symbol in symbolic logic that has the same meaning in all models, such as the symbol "=" for "equals". A constant, in a mathematical context, is a number that arises naturally in mathematics, such as π or e; Such mathematics constant value do not change. It can mean polynomial constant term (the term of degree 0) or the constant of integration, a free parameter arising in integration. Related, the physical constant are a physical quantity generally believed to be universal and unchanging. Programming constants are a values that, unlike a variable, cannot be reassociated with a different value. 77. Though not an index term, keywords are terms that represent information. A keyword is a word with special meaning (this is a semantic definition), while syntactically these are terminal symbols in the phrase grammar. See reserved word for the related concept. 78. Most of these symbols can be found in propositional calculus, a formal system described as ${\mathcal {L}}={\mathcal {L}}\ (\mathrm {A} ,\ \Omega ,\ \mathrm {Z} ,\ \mathrm {I} )$. $\mathrm {A} $ is the set of elements, such as the a in the example with Boolean algebra above. $\Omega $ is the set that contains the subsets that contain operations, such as $\lor $ or $\land $. $\mathrm {Z} $ contains the inference rules, which are the rules dictating how inferences may be logically made, and $\mathrm {I} $ contains the axioms. See also: Basic and Derived Argument Forms. 79. Usually denoted by x, y, z, or other lowercase letters Here a symbols that represents a quantity in a mathematical expression, a mathematical variable as used in many sciences. Variables can be symbolic name associated with a value and whose associated value may be changed, known in computer science as a variable reference. A variable can also be the operationalized way in which the attribute is represented for further data processing (e.g., a logical set of attributes). See also: Dependent and independent variables in statistics. 80. Usually denoted by an uppercase letter followed by a list of variables, such as P(x) or Q(y,z) Here a mathematical logic predicate, a fundamental concept in first-order logic. Grammatical predicates are grammatical components of a sentence. Related is the syntactic predicate in parser technology which are guidelines for the parser process. In computer programming, a branch predication allows a choice to execute or not to execute a given instruction based on the content of a machine register. 81. Representing ALL and EXISTS 82. e.g. ∃ for "there exists" and ∀ for "for all" 83. See also: Dialetheism, Contradiction, and Paradox 84. Related, facetious abstract nonsense describes certain kinds of arguments and methods related to category theory which resembles comical literary non-sequitur devices (not illogical non-sequiturs). 85. Gödel's incompleteness theorems shows that Hilbert's program to find a complete and consistent set of axioms for all mathematics is impossible, giving a contested negative answer to Hilbert's second problem 86. For example, take the statement "There exists a number x such that it is not y". Using the symbols of propositional calculus, this would become: $(\exists x)(x=\lnot y)$. If the Gödel numbers replace the symbols, it becomes:$\{8,4,11,9,8,11,5,1,13,9\}$. There are ten numbers, so the ten prime numbers are found and these are: $\{2,3,5,7,11,13,17,19,23,29\}$. Then, the Gödel numbers are made the powers of the respective primes and multiplied, giving: $2^{8}\times 3^{4}\times 5^{11}\times 7^{9}\times 11^{8}\times 13^{11}\times 17^{5}\times 19^{1}\times 23^{13}\times 29^{9}$. The resulting number is approximately $3.096262735\times 10^{78}$. 87. The Klein–Gordon equation is: ${\frac {1}{c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}\psi -\nabla ^{2}\psi +{\frac {m^{2}c^{2}}{\hbar ^{2}}}\psi =0.$ 88. The Dirac equation in the form originally proposed by Dirac is: $\left(\beta mc^{2}+\sum _{k=1}^{3}\alpha _{k}p_{k}\,c\right)\psi (\mathbf {x} ,t)=i\hbar {\frac {\partial \psi (\mathbf {x} ,t)}{\partial t}}$ where, ψ = ψ(x, t) is the wave function for the electron, x and t are the space and time coordinates, m is the rest mass of the electron, p is the momentum, understood to be the momentum operator in the Schrödinger theory, c is the speed of light, and ħ = h/2π is the reduced Planck constant. 89. That is, $\partial _{\mu }(\partial ^{\mu }A^{\nu }-\partial ^{\nu }A^{\mu })+\left({\frac {mc}{\hbar }}\right)^{2}A^{\nu }=0$ 90. The theorem applies more generally to any sufficiently strong formal system, showing that truth in the standard model of the system cannot be defined within the system. 91. Named to honor Voigt's 1898 work. 92. Named after Arthur Moritz Schoenflies 93. See Galois connections. 94. Oystein Ore would also write "Number Theory and Its History". 95. $\langle \phi \|\psi \rangle $ 96. That the scattering amplitude can be thought of as an analytic function of the angular momentum, and that the position of the poles determine power-law growth rates of the amplitude in the purely mathematical region of large values of the cosine of the scattering angle. 97. That is, $\scriptstyle \Box $ 98. Also known as the d'Alembertian or wave operator. 99. Also known as, "permutation symbol" (see: permutation), "antisymmetric symbol" (see: antisymmetric), or "alternating symbol" 100. Note that "masses" (e.g., the coherent non-definite body shape) of particles are periodically reevaluated by the scientific community. The values may have been adjusted; adjustment by operations carried out on instruments in order that it provides given indications corresponding to given values of the measurand. In engineering, mathematics, and geodesy, the optimal parameter such estimation of a mathematical model so as to best fit a data set. 101. For the consensus, see Particle Data Group. 102. A locally defined set of four linearly independent vector fields called a tetrad 103. His usage of the Einstein summation was in order to offset the inconvenience in describing contractions and covariant differentiation in modern abstract tensor notation, while maintaining explicit covariance of the expressions involved. 104. See also: String theory landscape and Swampland 105. Devised by Joshua Lederberg and extended by Coxeter and Frucht 106. And, in 1938, Turing, A. M. (1938). "On Computable Numbers, with an Application to the Entscheidungsproblem. A Correction". Proceedings of the London Mathematical Society. s2-43: 544–546. doi:10.1112/plms/s2-43.6.544.. 107. Among von Neumann's other contributions include the application of operator theory to quantum mechanics, in the development of functional analysis, and on various forms of operator theory. References and citations General • Florian Cajori (1929) A History of Mathematical Notations, 2 vols. Dover reprint in 1 vol., 1993. ISBN 0-486-67766-4. Citations 1. Florian Cajori. A History of Mathematical Notations: Two Volumes in One. Cosimo, Inc., 1 Dec 2011 2. A Dictionary of Science, Literature, & Art, Volume 2. Edited by William Thomas Brande, George William Cox. Pg 683 3. "Notation – from Wolfram MathWorld". Mathworld.wolfram.com. Retrieved 24 June 2014. 4. Diophantos of Alexandria: A Study in the History of Greek Algebra. By Sir Thomas Little Heath. Pg 77. 5. Mathematics: Its Power and Utility. By Karl J. Smith. Pg 86. 6. The Commercial Revolution and the Beginnings of Western Mathematics in Renaissance Florence, 1300–1500. Warren Van Egmond. 1976. Page 233. 7. Solomon Gandz. "The Sources of al-Khowarizmi's Algebra" 8. Encyclopædia Americana. By Thomas Gamaliel Bradford. Pg 314 9. Mathematical Excursion, Enhanced Edition: Enhanced Webassign Edition By Richard N. Aufmann, Joanne Lockwood, Richard D. Nation, Daniel K. Cleg. Pg 186 10. Mathematics in Egypt and Mesopotamia 11. Boyer, C. B. A History of Mathematics, 2nd ed. rev. by Uta C. Merzbach. New York: Wiley, 1989 ISBN 0-471-09763-2 (1991 pbk ed. ISBN 0-471-54397-7). "Mesopotamia" p. 25. 12. Duncan J. Melville (2003). Third Millennium Chronology, Third Millennium Mathematics. St. Lawrence University. 13. Aaboe, Asger (1998). Episodes from the Early History of Mathematics. New York: Random House. pp. 30–31. 14. Heath (1931). "A Manual of Greek Mathematics". Nature. 128 (3235): 5. Bibcode:1931Natur.128..739T. doi:10.1038/128739a0. S2CID 3994109. 15. Sir Thomas L. Heath, A Manual of Greek Mathematics, Dover, 1963, p. 1: "In the case of mathematics, it is the Greek contribution which it is most essential to know, for it was the Greeks who made mathematics a science." 16. The new encyclopædia; or, Universal dictionary of arts and sciences. By Encyclopaedia Perthensi. Pg 49 17. Calinger, Ronald (1999). A Contextual History of Mathematics. Prentice-Hall. p. 150. ISBN 0-02-318285-7. Shortly after Euclid, compiler of the definitive textbook, came Archimedes of Syracuse (ca. 287 212 BC), the most original and profound mathematician of antiquity. 18. "Archimedes of Syracuse". The MacTutor History of Mathematics archive. January 1999. Retrieved 9 June 2008. 19. O'Connor, J.J.; Robertson, E.F. (February 1996). "A history of calculus". University of St Andrews. Archived from the original on 15 July 2007. Retrieved 7 August 2007. 20. "Proclus' Summary". Gap.dcs.st-and.ac.uk. Archived from the original on 23 September 2015. Retrieved 24 June 2014. 21. Caldwell, John (1981) "The De Institutione Arithmetica and the De Institutione Musica", pp. 135–54 in Margaret Gibson, ed., Boethius: His Life, Thought, and Influence, (Oxford: Basil Blackwell). 22. Folkerts, Menso, "Boethius" Geometrie II, (Wiesbaden: Franz Steiner Verlag, 1970). 23. Mathematics and Measurement By Oswald Ashton Wentworth Dilk. Pg 14 24. A dictionary of science, literature and art, ed. by W.T. Brande. Pg 683 25. Boyer, Carl B. A History of Mathematics, 2nd edition, John Wiley & Sons, Inc., 1991. 26. Diophantine Equations. Submitted by: Aaron Zerhusen, Chris Rakes, & Shasta Meece. MA 330-002. Dr. Carl Eberhart. 16 February 1999. 27. A History of Greek Mathematics: From Aristarchus to Diophantus. By Sir Thomas Little Heath. Pg 456 28. A History of Greek Mathematics: From Aristarchus to Diophantus. By Sir Thomas Little Heath. Pg 458 29. The American Mathematical Monthly, Volume 16. Pg 131 30. "Overview of Chinese mathematics". Groups.dcs.st-and.ac.uk. Retrieved 24 June 2014. 31. George Gheverghese Joseph, The Crest of the Peacock: Non-European Roots of Mathematics, Penguin Books, London, 1991, pp.140—148 32. Georges Ifrah, Universalgeschichte der Zahlen, Campus, Frankfurt/New York, 1986, pp.428—437 33. "Frank J. Swetz and T. I. Kao: Was Pythagoras Chinese?". Psupress.psu.edu. Retrieved 24 June 2014. 34. Needham, Joseph (1986). Science and Civilization in China: Volume 3, Mathematics and the Sciences of the Heavens and the Earth. Taipei: Caves Books, Ltd. 35. Sal Restivo 36. Needham, Joseph (1986). Science and Civilization in China: Volume 3, Mathematics and the Sciences of the Heavens and the Earth. Taipei: Caves Books, Ltd. 37. Marcel Gauchet, 151. 38. Boyer, C. B. A History of Mathematics, 2nd ed. rev. by Uta C. Merzbach. New York: Wiley, 1989 ISBN 0-471-09763-2 (1991 pbk ed. ISBN 0-471-54397-7). "China and India" p. 221. (cf., "he was the first one to give a general solution of the linear Diophantine equation ax + by = c, where a, b, and c are integers. [...] It is greatly to the credit of Brahmagupta that he gave all integral solutions of the linear Diophantine equation, whereas Diophantus himself had been satisfied to give one particular solution of an indeterminate equation. Inasmuch as Brahmagupta used some of the same examples as Diophantus, we see again the likelihood of Greek influence in India – or the possibility that they both made use of a common source, possibly from Babylonia. It is interesting to note also that the algebra of Brahmagupta, like that of Diophantus, was syncopated. Addition was indicated by juxtaposition, subtraction by placing a dot over the subtrahend, and division by placing the divisor below the dividend, as in our fractional notation but without the bar. The operations of multiplication and evolution (the taking of roots), as well as unknown quantities, were represented by abbreviations of appropriate words.") 39. Robert Kaplan, "The Nothing That Is: A Natural History of Zero", Allen Lane/The Penguin Press, London, 1999 40. ""The ingenious method of expressing every possible number using a set of ten symbols (each symbol having a place value and an absolute value) emerged in India. The idea seems so simple nowadays that its significance and profound importance is no longer appreciated. Its simplicity lies in the way it facilitated calculation and placed arithmetic foremost amongst useful inventions. the importance of this invention is more readily appreciated when one considers that it was beyond the two greatest men of Antiquity, Archimedes and Apollonius." – Pierre-Simon Laplace". History.mcs.st-and.ac.uk. Retrieved 24 June 2014. 41. A.P. Juschkewitsch, "Geschichte der Mathematik im Mittelalter", Teubner, Leipzig, 1964 42. Boyer, C. B. A History of Mathematics, 2nd ed. rev. by Uta C. Merzbach. New York: Wiley, 1989 ISBN 0-471-09763-2 (1991 pbk ed. ISBN 0-471-54397-7). "The Arabic Hegemony" p. 230. (cf., "The six cases of equations given above exhaust all possibilities for linear and quadratic equations having positive root. So systematic and exhaustive was al-Khwārizmī's exposition that his readers must have had little difficulty in mastering the solutions.") 43. Gandz and Saloman (1936), The sources of Khwarizmi's algebra, Osiris i, pp. 263–77: "In a sense, Khwarizmi is more entitled to be called "the father of algebra" than Diophantus because Khwarizmi is the first to teach algebra in an elementary form and for its own sake, Diophantus is primarily concerned with the theory of numbers". 44. Boyer, C. B. A History of Mathematics, 2nd ed. rev. by Uta C. Merzbach. New York: Wiley, 1989 ISBN 0-471-09763-2 (1991 pbk ed. ISBN 0-471-54397-7). "The Arabic Hegemony" p. 229. (cf., "It is not certain just what the terms al-jabr and muqabalah mean, but the usual interpretation is similar to that implied in the translation above. The word al-jabr presumably meant something like "restoration" or "completion" and seems to refer to the transposition of subtracted terms to the other side of an equation; the word muqabalah is said to refer to "reduction" or "balancing" – that is, the cancellation of like terms on opposite sides of the equation.") 45. Rashed, R.; Armstrong, Angela (1994). The Development of Arabic Mathematics. Springer. pp. 11–12. ISBN 0-7923-2565-6. OCLC 29181926. 46. Victor J. Katz (1998). History of Mathematics: An Introduction, pp. 255–59. Addison-Wesley. ISBN 0-321-01618-1. 47. F. Woepcke (1853). Extrait du Fakhri, traité d'Algèbre par Abou Bekr Mohammed Ben Alhacan Alkarkhi. Paris. 48. Katz, Victor J. (1995). "Ideas of Calculus in Islam and India". Mathematics Magazine. 68 (3): 163–74. doi:10.1080/0025570X.1995.11996307. 49. 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The Mathematical Principles of Natural Philosophy, Volume 1. By Sir Isaac Newton, John Machin. Pg 12. 112. In The Scientific Outlook (1931) 113. Mathematics simplified and made attractive: or, The laws of motion explained. By Thomas Fisher. Pg 15. (cf. But an abstraction not founded upon, and not consonant with Nature and (Logical) Truth, would be a falsity, an insanity.) 114. Proposition VI, On Formally Undecidable Propositions in Principia Mathematica and Related Systems I (1931) 115. Casti, John L. 5 Golden Rules. New York: MJF Books, 1996. 116. Gr. Methoden Der Mathematischen Physik 117. P.A.M. Dirac (1927). "The Quantum Theory of the Emission and Absorption of Radiation". Proceedings of the Royal Society of London A. 114 (767): 243–265. Bibcode:1927RSPSA.114..243D. doi:10.1098/rspa.1927.0039. 118. E. Fermi (1932). "Quantum Theory of Radiation". Reviews of Modern Physics. 4 (1): 87–132. Bibcode:1932RvMP....4...87F. doi:10.1103/RevModPhys.4.87. 119. F. Bloch; A. Nordsieck (1937). "Note on the Radiation Field of the Electron". Physical Review. 52 (2): 54–59. Bibcode:1937PhRv...52...54B. doi:10.1103/PhysRev.52.54. 120. V. F. Weisskopf (1939). "On the Self-Energy and the Electromagnetic Field of the Electron". Physical Review. 56 (1): 72–85. Bibcode:1939PhRv...56...72W. doi:10.1103/PhysRev.56.72. 121. R. Oppenheimer (1930). "Note on the Theory of the Interaction of Field and Matter". Physical Review. 35 (5): 461–477. Bibcode:1930PhRv...35..461O. doi:10.1103/PhysRev.35.461. 122. Van der Waerden B.L. (1929). "Spinoranalyse". Nachr. Ges. Wiss. Göttingen Math.-Phys. 1929: 100–109. 123. Veblen O. (1933). "Geometry of two-component Spinors". Proc. Natl. Acad. Sci. USA. 19 (4): 462–474. Bibcode:1933PNAS...19..462V. doi:10.1073/pnas.19.4.462. PMC 1086023. PMID 16577541. 124. Dirac, P.A.M. (1939). "A new notation for quantum mechanics". Mathematical Proceedings of the Cambridge Philosophical Society. 35 (3): 416–418. Bibcode:1939PCPS...35..416D. doi:10.1017/S0305004100021162. S2CID 121466183. 125. H. Grassmann (1862). Extension Theory. History of Mathematics Sources. American Mathematical Society, London Mathematical Society, 2000 translation by Lloyd C. Kannenberg. 126. Weinberg, Steven (1964), The quantum theory of fields, Volume 2, Cambridge University Press, 1995, p. 358, ISBN 0-521-55001-7 127. "The Nobel Prize in Physics 1965". Nobel Foundation. Retrieved 9 October 2008. 128. S.L. Glashow (1961). "Partial-symmetries of weak interactions". Nuclear Physics. 22 (4): 579–588. Bibcode:1961NucPh..22..579G. doi:10.1016/0029-5582(61)90469-2. 129. S. Weinberg (1967). "A Model of Leptons". Physical Review Letters. 19 (21): 1264–1266. Bibcode:1967PhRvL..19.1264W. doi:10.1103/PhysRevLett.19.1264. 130. A. Salam (1968). N. Svartholm (ed.). Elementary Particle Physics: Relativistic Groups and Analyticity. Eighth Nobel Symposium. Stockholm: Almquvist and Wiksell. p. 367. 131. F. Englert; R. Brout (1964). "Broken Symmetry and the Mass of Gauge Vector Mesons". Physical Review Letters. 13 (9): 321–323. Bibcode:1964PhRvL..13..321E. doi:10.1103/PhysRevLett.13.321. 132. P.W. Higgs (1964). "Broken Symmetries and the Masses of Gauge Bosons". Physical Review Letters. 13 (16): 508–509. Bibcode:1964PhRvL..13..508H. doi:10.1103/PhysRevLett.13.508. 133. G.S. Guralnik; C.R. Hagen; T.W.B. Kibble (1964). "Global Conservation Laws and Massless Particles". Physical Review Letters. 13 (20): 585–587. Bibcode:1964PhRvL..13..585G. doi:10.1103/PhysRevLett.13.585. 134. http://www.physics.drexel.edu/~vkasli/phys676/Notes%20for%20a%20brief%20history%20of%20quantum%20gravity%20-%20Carlo%20Rovelli.pdf 135. Bourbaki, Nicolas (1972). "Univers". In Artin, Michael; Grothendieck, Alexandre; Jean-Louis Verdier (eds.). Séminaire de Géométrie Algébrique du Bois Marie - 1963-64 - Théorie des topos et cohomologie étale des schémas - (SGA 4) - vol. 1 (Lecture notes in mathematics 269) (in French). Berlin; New York: Springer-Verlag. pp. 185–217. 136. F.J. Hasert; et al. (1973). "Search for elastic muon-neutrino electron scattering". Physics Letters B. 46 (1): 121. Bibcode:1973PhLB...46..121H. doi:10.1016/0370-2693(73)90494-2. 137. F.J. Hasert; et al. (1973). "Observation of neutrino-like interactions without muon or electron in the gargamelle neutrino experiment". Physics Letters B. 46 (1): 138. Bibcode:1973PhLB...46..138H. doi:10.1016/0370-2693(73)90499-1. 138. F.J. Hasert; et al. (1974). "Observation of neutrino-like interactions without muon or electron in the Gargamelle neutrino experiment". Nuclear Physics B. 73 (1): 1. Bibcode:1974NuPhB..73....1H. doi:10.1016/0550-3213(74)90038-8. 139. D. Haidt (4 October 2004). "The discovery of the weak neutral currents". CERN Courier. Retrieved 8 May 2008. 140. "Mainpage". 141. Candelas, P. (1985). "Vacuum configurations for superstrings". Nuclear Physics B. 258: 46–74. Bibcode:1985NuPhB.258...46C. doi:10.1016/0550-3213(85)90602-9. 142. De Felice, F.; Clarke, C.J.S. (1990), Relativity on Curved Manifolds, p. 133 143. "Quantum invariants of knots and 3-manifolds" by V. G. Turaev (1994), page 71 144. Pisanski, Tomaž; Servatius, Brigitte (2013), "2.3.2 Cubic graphs and LCF notation", Configurations from a Graphical Viewpoint, Springer, p. 32, ISBN 9780817683641 145. Frucht, R. (1976), "A canonical representation of trivalent Hamiltonian graphs", Journal of Graph Theory, 1 (1): 45–60, doi:10.1002/jgt.3190010111 146. Fraleigh 2002:89; Hungerford 1997:230 147. Dehn, Edgar. Algebraic Equations, Dover. 1930:19 148. "The IBM 601 Multiplying Punch". Columbia.edu. Retrieved 24 June 2014. 149. "Interconnected Punched Card Equipment". Columbia.edu. 24 October 1935. Retrieved 24 June 2014. 150. Proceedings of the London Mathematical Society 42 (2) 151. Cook, Stephen (1971). "The complexity of theorem proving procedures". Proceedings of the Third Annual ACM Symposium on Theory of Computing. pp. 151–158. doi:10.1145/800157.805047. ISBN 9781450374644. S2CID 7573663. Further reading General • A Short Account of the History of Mathematics. By Walter William Rouse Ball. • A Primer of the History of Mathematics. By Walter William Rouse Ball. • A History of Elementary Mathematics: With Hints on Methods of Teaching. By Florian Cajori. • A History of Elementary Mathematics. By Florian Cajori. • A History of Mathematics. By Florian Cajori. • A Short History of Greek Mathematics. By James Gow. • On the Development of Mathematical Thought During the Nineteenth Century. By John Theodore Merz. • A New Mathematical and Philosophical Dictionary. By Peter Barlow. • Historical Introduction to Mathematical Literature. By George Abram Miller • A Brief History of Mathematics. By Karl Fink, Wooster Woodruff Beman, David Eugene Smith • History of Modern Mathematics. By David Eugene Smith. • History of modern mathematics. By David Eugene Smith, Mansfield Merriman. Other • Principia Mathematica, Volume 1 & Volume 2. By Alfred North Whitehead, Bertrand Russell. • The Mathematical Principles of Natural Philosophy, Volume 1, Issue 1. By Sir Isaac Newton, Andrew Motte, William Davis, John Machin, William Emerson. • General investigations of curved surfaces of 1827 and 1825. By Carl Friedrich Gaus. External links • Mathematical Notation: Past and Future • History of Mathematical Notation • Earliest Uses of Mathematical Notation • Finger counting. files.chem.vt.edu. • Some Common Mathematical Symbols and Abbreviations (with History). Isaiah Lankham, Bruno Nachtergaele, Anne Schilling. 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Mathematics: The Loss of Certainty Mathematics: The Loss of Certainty is a book by Morris Kline on the developing perspectives within mathematical cultures throughout the centuries.[1] Mathematics: The Loss of Certainty AuthorMorris Kline PublisherOxford University Press Publication date 1980 Pages366 ISBN0-19-502754-X OCLC6042956 Followed byMathematics and the Search for Knowledge This book traces the history of how new results in mathematics have provided surprises to mathematicians through the ages. Examples include how 19th century mathematicians were surprised by the discovery of non-Euclidean geometry and how Godel's incompleteness theorem disappointed many logicians. Kline furthermore discusses the close relation of some of the most prominent mathematicians such as Newton and Leibniz to God. He believes that Newton's religious interests were the true motivation of his mathematical and scientific work. He quotes Newton from a letter to Reverend Richard Bentley of December 10, 1692: When I wrote my treatise about our system The Mathematical Principles of Natural Philosophy, I had an eye on such principles as might work with considering men for the belief in a Deity; and nothing can rejoice me more than to find it useful for that purpose. He also believes Leibniz regarded science as a religious mission which scientists were duty bound to undertake. Kline quotes Leibniz from an undated letter of 1699 or 1700: It seems to me that the principal goal of the whole of mankind must be the knowledge and development of the wonders of God, and that this is the reason that God gave him the empire of the globe. Kline also argues that the attempt to establish a universally acceptable, logically sound body of mathematics has failed. He believes that most mathematicians today do not work on applications. Instead they continue to produce new results in pure mathematics at an ever-increasing pace. Criticism In the reviews of this book, a number of specialists, paying tribute to the author's outlook, accuse him of biased emotionality, dishonesty and incompetence. In particular, Raymond G. Ayoub in The American Mathematical Monthly[2] writes: For centuries, Euclidean geometry seemed to be a good model of space. The results were and still are used effectively in astronomy and in navigation. When it was subjected to the close scrutiny of formalism, it was found to have weaknesses and it is interesting to observe that, this time, it was the close scrutiny of the formalism that led to the discovery (some would say invention) of non-Euclidean geometry. (It was several years later that a satisfactory Euclidean model was devised.) This writer fails to see why this discovery was, in the words of Kline, a "debacle". Is it not, on the contrary, a great triumph?... Professor Kline does not deal honestly with his readers. He is a learned man and knows perfectly well that many mathematical ideas created in abstracto have found significant application in the real world. He chooses to ignore this fact, acknowledged by even the most fanatic opponents of mathematics. He does this to support an untenable dogma. One is reminded of the story of the court jester to Louis XIV: the latter had written a poem and asked the jester his opinion. "Your majesty is capable of anything. Your majesty has set out to write doggerel and your majesty has succeeded." On balance, such, alas, must be said of this book. John Corcoran in Mathematical Reviews:[3] The overall purpose of the book is to advance as a philosophy of mathematics a mentalistic pragmatism which exalts "applied mathematics" and denigrates both "pure mathematics" and foundational studies. Although its thesis is predicated in part on the deep foundational achievements of twentieth century logicians, the basic philosophy is a close cousin of various philosophies which were influential in the nineteenth century. Moreover, as can be seen from the above-listed ideas, the author's grasp of twentieth century logic is not reliable. Accordingly he finds it surprising (p. 322, 323) that Hilbert, Gödel, Church, members of the Bourbaki school, and other "leaders in the work on foundations affirm that the mathematical concepts and properties exist in some objective sense and that they can be apprehended by human minds". His only argument against the Platonistic realism of the mathematicians just mentioned is based on his own failure to make the distinction between (human) error and (mathematical) falsehood (p. 324)... The author does not seem to realize that in order to have knowledge it is not necessary to be infallible, nor does he recognize that loss of certainty is not the same as loss of truth. The philosophical and the foundational aspects of the author's argument are woven into a comprehensive survey and interpretation of the history of mathematics. One could hope that the argument would be somewhat redeemed by sound historical work, but this is not so. Two of the periods most important for the author's viewpoint are both interpreted inconsistently. (a) In some passages the author admits the obvious truth that experience and observation played a key role in the development of classical Greek mathematics (pp. 9, 18, 24, 167). But in other passages, he alleges that classical Greek mathematicians scorned experience and observation, founding their theories on "self-evident truths" (pp. 17, 20, 21, 22, 29, 95, 307). (b) In some passages the author portrays the beginning of the nineteenth century as a time of widespread confidence in the soundness of mathematics (pp. 6, 68, 78, 103, 173), but in other passages he describes this period as a time of intellectual turmoil wherein mathematicians entertained grave doubts about the basis of their science (pp. 152, 153, 170, 308)... One can only regret the philosophical, foundational, and historical inadequacies which vitiate the main argument and which tend to distract attention from the many sound and fascinating observations and insights provided by the book. Amy Dahan in Revue d'histoire des sciences:[4] Quant aux derniers chapitres sur les grandes tendances des mathématiques contemporaines, ils sont franchement décevants, assez superficiels. Il n'y a pas d'analyse de la mathématique contemporaine (grande période structuraliste, retour au « concret », flux entre les mathématiques et la physique, etc. Scott Weinstein in ETC: A Review of General Semantics:[5] Professor Kline's book is a lively account of a fascinating subject. Its conclusions are, however, overdrawn and in many cases unjustified. The lesson to be learned from twentieth century foundational research is not that mathematics is in a sorry state, but rather the extent to which deep philosophical issues about mathematics may be illuminated, if not settled, by mathematics itself. Gödel's theorems do indeed intimate that there may be limits to what we can come to know in mathematics, but they also demonstrate through themselves the great heights to which human reason can ascend through mathematical thought. Ian Stewart in Educational Studies in Mathematics:[6] This book is firmly in the tradition that we have come to expect from this author; and my reaction to it is much like my reaction to its predecessors: I think three quarters of it is superb, and the other quarter is outrageous nonsense; and the reason is that Morris Kline really doesn't understand what today's mathematics is about, although he has an enviable grasp of yesterday's... Morris Kline has said elsewhere that he considers the crowning achievement of twentieth-century mathematics to be the Godel theorem. I don't agree: the Gddel theorem, astonishing and deep as it is, had little effect on the mainstream of real mathematical development. It didn't actually lead into anything new and powerful except more theorems of the same kind. It affected how mathematicians thought about what they were doing; but its effect on what they actually did is close to zero. Compare this to the rise of topology: fifty years of apparently introverted efforts by mathematicians, largely ignoring applied science; polished and perfected and developed into a body of technique of immense and still largely unrealised power; and within the last decade becoming important in virtually every field of applied science: engineering, physics, chemistry, numerical analysis. Topology has far more claim to be the crowning achievement of this century. But Morris Kline can see only the introversion. It doesn't seem to occur to him that a mathematical problem may require concentrated contemplation of mathematics, rather than the problem to which one hopes to apply the resulting theory, to obtain a satisfactory solution. But if I want to cut down an apple tree, and my saw is too blunt, no amount of contemplation of the tree will sharpen it... There is good mathematics; there is bad mathematics. There are mathematicians who are totally uninterested in science, who are building tools that science will find indispensable. There are mathematicians passionately interested in science, and building tools for specific use there, whose work will become as obsolete as the Zeppelin or the electronic valve. The path from discovery to utility is a rabbit-warren of false ends: mathematics for its own sake has had, and wil continue to have, its place in the scheme of things. And, after all, the isolation of the topologist who knows no physics is no worse than that of the physicist who knows no topology. Today's science requires specialization from its individuals: the collective activity of scientists as a whole is where the links are forged. If only Morris Kline showed some inkling of the nature of this process, I would take his arguments more seriously. But his claim that mathematics has gone into decline is one based too much on ignorance, and his arguments are tawdry in comparison to the marvellous, shining vigour of today's mathematics. I too would like to see more overt recognition by mathematicians of the importance of scientific problems; but to miss the fact that they do splendid work even in this apparent isolation is to lose the battle before it has begun. Bibliography • Morris Kline, Mathematics: The Loss of Certainty, Oxford University Press, 1980 ISBN 0-19-502754-X Notes 1. John Little (1981) Review:Mathematics: The Loss of Certainty, New Scientist January 15, 1981, link from Google Books 2. Raymond G. Ayoub, The American Mathematical Monthly, Vol. 89, No. 9 (Nov., 1982), pp. 715–717 3. John Corcoran, Mathematical Reviews, MR584068 (82e:03013). 4. Amy Dahan-Dalmédico, Revue d'histoire des sciences, Vol. 36, No. 3/4 (JUILLET-DÉCEMBRE 1983), pp. 356–358. 5. Scott Weinstein, ETC: A Review of General Semantics, Vol. 38, No. 4 (Winter 1981), pp. 425–430 6. Ian Stewart, Educational Studies in Mathematics, Vol. 13, No. 4 (Nov., 1982), pp. 446–447 Further reading • "Review of Mathematics: The Loss of Certainty". The Wilson Quarterly. 5 (2): 160–161. 1981-01-01. JSTOR 40256113. • Weinstein, Scott (1981-01-01). Kline, Morris; Kleine (eds.). "THE LOSS OF CERTAINTY". ETC: A Review of General Semantics. 38 (4): 425–430. JSTOR 42575575. • Long, Calvin T. (1981-01-01). "Review of MATHEMATICS: The Loss of Certainty (L)". The Mathematics Teacher. 74 (3): 234–235. JSTOR 27962408. • Boas, R. P. (1981-01-01). Kline, Morris (ed.). "Nevertheless, Let's Get on with the Job". The Two-Year College Mathematics Journal. 12 (2): 141–142. doi:10.2307/3027376. JSTOR 3027376. • Guberman, J. (1983-01-01). "Review of Mathematics: The Loss of Certainty". Leonardo. 16 (4): 328–328. doi:10.2307/1574971. JSTOR 1574971. • Stewart, Ian (1982-01-01). "Review of Mathematics, The Loss of Certainty". Educational Studies in Mathematics. 13 (4): 446–447. JSTOR 3482328. • Dahan-Dalmédico, Amy (1983-01-01). "Review of Mathematics, The Loss of Certainty". Revue d'histoire des sciences. 36 (3/4): 356–358. JSTOR 23632221. • Quadling, Douglas (1981-01-01). "Review of Mathematics: The Loss of Certainty". The Mathematical Gazette. 65 (434): 300–301. doi:10.2307/3616614. JSTOR 3616614. • Robles, J. A. (1981-01-01). "Review of Mathematics, the Loss of Certainty". Crítica: Revista Hispanoamericana de Filosofía. 13 (39): 87–91. JSTOR 40104258. • Ayoub, Raymond G. (1982-01-01). "Review of Mathematics: The Loss of Certainty". The American Mathematical Monthly. 89 (9): 715–717. doi:10.2307/2975679. JSTOR 2975679.	Wikipedia
The Man Who Loved Only Numbers The Man Who Loved Only Numbers is a biography of the famous mathematician Paul Erdős written by Paul Hoffman. The book was first published on July 15, 1998, by Hyperion Books as a hardcover edition. A paperback edition appeared in 1999. The book is, in the words of the author, "a work in oral history based on the recollections of Erdős, his collaborators and their spouses". The book was a bestseller in the United Kingdom and has been published in 15 different languages. The book won the 1999 Rhône-Poulenc Prize, beating many distinguished and established writers, including E. O. Wilson.[2] Not to be confused with the PBS Nova episode "The Man Who Loved Numbers" (Season 15, Ep 19), about Ramanujan. The Man Who Loved Only Numbers Front cover AuthorPaul Hoffman Original titleThe Man Who Loved Only Numbers: The Story of Paul Erdős and the Search for Mathematical Truth CountryUnited States LanguageEnglish GenreBiography PublishedJuly 15, 1998 PublisherHyperion Books Media typePrint Pages301 pp. ISBN978-0786863624 How the book came about Hoffman received an assignment by The Atlantic Monthly in 1987 to profile Erdős, which won the National Magazine Award for feature writing. After this, Hoffman followed Erdős on his travels for the last 10 years of his life learning about his exceedingly unusual life and interviewing his numerous collaborators in the process of writing this book. Content A large part of the book concerns Erdős, but a lot of it is about other mathematicians, past and present, including Ronald Graham, Carl Friedrich Gauss, Srinivasa Ramanujan, and G.H. Hardy.[3] In the book, Erdős enjoys listening to Hardy when he speaks about Ramanujan. Hoffman also tries to give examples of what mathematics is and why he views it as important, and why many mathematicians such as Erdős devote their whole lives to mathematics. It also contains some history of Europe and the United States of Erdős's time. The book, on the whole, portrays Erdős in a favourable light, pointing out his many endearing qualities, like his childlike simplicity, his generosity and altruistic nature, his kindness and gentleness towards children. However, it also attempts to illustrate his helplessness in doing mundane tasks, the difficulties faced by those close to him because of his eccentricities, and his stubborn and frustrating behaviour. Erdős's nursing of Jon Folkman Main article: Jon Folkman Hoffman reports the following anecdote, which displays Erdős's single-minded devotion to his friends and mathematics. In the late 1960s, the young mathematician Jon Folkman was diagnosed as having advanced brain cancer. During Folkman's hospitalization, he was visited repeatedly by Ronald Graham and Paul Erdős. After his brain surgery, Folkman was despairing that he had lost his mathematical skills. As soon as Folkman received Graham and Erdős at the hospital, Erdős challenged Folkman with mathematical problems, helping to rebuild his confidence.[1] Hoffman notes that Folkman's recovery was short-lived. Notwithstanding his ability to solve the problems posed by Erdős, Folkman purchased a gun and killed himself. Folkman's supervisor at RAND, Delbert Ray Fulkerson, blamed himself for failing to notice suicidal behaviors in Folkman. Years later Fulkerson also killed himself.[1] Writing style The book is mostly written without much technical detail and can be read by anyone without a mathematical background. Hoffman does give some relatively simple examples of mathematical problems throughout the book (like Cantor's diagonal argument) to illustrate some of the ideas in modern mathematics. Notes 1. Hoffman, Paul (1998), The Man Who Loved Only Numbers: The Story of Paul Erdős and the Search for Mathematical Truth, Hyperion, pp. 109–110, ISBN 978-0-7868-6362-4. 2. Prizes for Science Books previous winners and shortlists, The Royal Society website 3. Alexander, James (September 27, 1998). "Planning an Infinite Stay". The New York Times. Retrieved 6 May 2022. References • Hoffman, Paul (1997). Archimedes' Revenge: The Joys and Perils of Mathematics. Ballantine Books. ISBN 0-449-00089-3.	Wikipedia
The Mathematical Experience The Mathematical Experience (1981) is a book by Philip J. Davis and Reuben Hersh that discusses the practice of modern mathematics from a historical and philosophical perspective. The book discusses the psychology of mathematicians, and gives examples of famous proofs and outstanding problems. It goes on to speculate about what a proof really means, in relationship to actual truth. Other topics include mathematics in education and some of the math that occurs in computer science. The Mathematical Experience AuthorPhilip J. Davis and Reuben Hersh CountryUnited States LanguageEnglish GenreMathematics Philosophy History PublisherSpringer Publication date 1981 Media typePrint (Hardcover) Pages487pp (Second Edition) The first paperback edition won a U.S. National Book Award in Science.[1][lower-alpha 1] It is cited by some mathematicians as influential in their decision to continue their studies in graduate school; and has been hailed as a classic of mathematical literature.[2] On the other hand, Martin Gardner disagreed with some of the authors' philosophical opinions.[3] A new edition, published in 1995, includes exercises and problems, making the book more suitable for classrooms. There is also The Companion Guide to The Mathematical Experience, Study Edition. Both were co-authored with Elena Marchisotto.[4] Davis and Hersh wrote a follow-up book, Descartes' Dream: The World According to Mathematics (Harcourt, 1986), and each has written other books with related themes, such as Mathematics And Common Sense: A Case of Creative Tension by Davis and What is Mathematics, Really? by Hersh. Notes 1. This was the 1983 award for paperback Science. From 1980 to 1983 in National Book Award history there were dual hardcover and paperback awards in most categories, and several nonfiction subcategories including General Nonfiction. Most of the paperback award-winners were reprints, including this one. References 1. "National Book Awards – 1983". National Book Foundation. Retrieved 2012-03-07. 2. jkauzlar (perhaps James Joseph Kauzlarich?) (18 September 2002). "MathForge.net--Power Tools for Online Mathematics". Archived from the original on 2006-10-02. One of the classics of mathematical literature, The Mathematical Experience, by Philip J Davis and Rueben Hersh, remains pertinent and fulfills its lofty ambitions even 20 years past its 1981 publication. 3. Gardner, Martin (August 13, 1981). "Is Mathematics for Real?". New York Review of Books: 37–40. 4. Reviews of the 1995 edition: • Burgess, J. P.; Ernest, P. (June 1997), Philosophia Mathematica, 5 (2): 175–188, doi:10.1093/philmat/5.2.173{{citation}}: CS1 maint: untitled periodical (link) • Bultheel, A. (1997), "Review", Bulletin of the Belgian Mathematical Society, 4 (5): 706–707 • Millett, Kenneth C. (November 1997), "Review" (PDF), Notices of the American Mathematical Society, 44 (10): 1316–1318 • Wilders, Richard J. (March 2012), "Review", MAA Reviews External links At Wikiversity, you can learn more and teach others about The Mathematical Experience at the School of The Mathematical Experience. • Book Review of the 1995 edition, by Kenneth C. Millett at the American Mathematical Society. • The Mathematical Experience from the Internet Archive Major mathematics areas • History • Timeline • Future • Outline • Lists • Glossary Foundations • Category theory • Information theory • Mathematical logic • Philosophy of mathematics • Set theory • Type theory Algebra • Abstract • Commutative • Elementary • Group theory • Linear • Multilinear • Universal • Homological Analysis • Calculus • Real analysis • Complex analysis • Hypercomplex analysis • Differential equations • Functional analysis • Harmonic analysis • Measure theory Discrete • Combinatorics • Graph theory • Order theory Geometry • Algebraic • Analytic • Arithmetic • Differential • Discrete • Euclidean • Finite Number theory • Arithmetic • Algebraic number theory • Analytic number theory • Diophantine geometry Topology • General • Algebraic • Differential • Geometric • Homotopy theory Applied • Engineering mathematics • Mathematical biology • Mathematical chemistry • Mathematical economics • Mathematical finance • Mathematical physics • Mathematical psychology • Mathematical sociology • Mathematical statistics • Probability • Statistics • Systems science • Control theory • Game theory • Operations research Computational • Computer science • Theory of computation • Computational complexity theory • Numerical analysis • Optimization • Computer algebra Related topics • Mathematicians • lists • Informal mathematics • Films about mathematicians • Recreational mathematics • Mathematics and art • Mathematics education • Mathematics portal • Category • Commons • WikiProject	Wikipedia
8 8 (eight) is the natural number following 7 and preceding 9. ← 7 8 9 → −1 0 1 2 3 4 5 6 7 8 9 → • List of numbers • Integers ← 0 10 20 30 40 50 60 70 80 90 → Cardinaleight Ordinal8th (eighth) Numeral systemoctal Factorization23 Divisors1, 2, 4, 8 Greek numeralΗ´ Roman numeralVIII, viii Greek prefixocta-/oct- Latin prefixocto-/oct- Binary10002 Ternary223 Senary126 Octal108 Duodecimal812 Hexadecimal816 Greekη (or Η) Arabic, Kurdish, Persian, Sindhi, Urdu٨ Amharic፰ Bengali৮ Chinese numeral八,捌 Devanāgarī८ Kannada೮ Malayalam൮ Telugu౮ Tamil௮ Hebrewח Khmer៨ Thai๘ ArmenianԸ ը Etymology English eight, from Old English eahta, æhta, Proto-Germanic ahto is a direct continuation of Proto-Indo-European oḱtṓ(w)-, and as such cognate with Greek ὀκτώ and Latin octo-, both of which stems are reflected by the English prefix oct(o)-, as in the ordinal adjective octaval or octavary, the distributive adjective is octonary. The adjective octuple (Latin octu-plus) may also be used as a noun, meaning "a set of eight items"; the diminutive octuplet is mostly used to refer to eight siblings delivered in one birth. The Semitic numeral is based on a root θmn-, whence Akkadian smn-, Arabic ṯmn-, Hebrew šmn- etc. The Chinese numeral, written 八 (Mandarin: bā; Cantonese: baat), is from Old Chinese priāt-, ultimately from Sino-Tibetan b-r-gyat or b-g-ryat which also yielded Tibetan brgyat. It has been argued that, as the cardinal number 7 is the highest number of items that can universally be cognitively processed as a single set, the etymology of the numeral eight might be the first to be considered composite, either as "twice four" or as "two short of ten", or similar. The Turkic words for "eight" are from a Proto-Turkic stem sekiz, which has been suggested as originating as a negation of eki "two", as in "without two fingers" (i.e., "two short of ten; two fingers are not being held up");[1] this same principle is found in Finnic kakte-ksa, which conveys a meaning of "two before (ten)". The Proto-Indo-European reconstruction *oḱtṓ(w)- itself has been argued as representing an old dual, which would correspond to an original meaning of "twice four". Proponents of this "quaternary hypothesis" adduce the numeral 9, which might be built on the stem new-, meaning "new" (indicating the beginning of a "new set of numerals" after having counted to eight).[2] Evolution of the Arabic digit The modern digit 8, like all modern Arabic numerals other than zero, originates with the Brahmi numerals. The Brahmi digit for eight by the 1st century was written in one stroke as a curve └┐ looking like an uppercase H with the bottom half of the left line and the upper half of the right line removed. However, the digit for eight used in India in the early centuries of the Common Era developed considerable graphic variation, and in some cases took the shape of a single wedge, which was adopted into the Perso-Arabic tradition as ٨ (and also gave rise to the later Devanagari form ८); the alternative curved glyph also existed as a variant in Perso-Arabic tradition, where it came to look similar to our digit 5. The digits as used in Al-Andalus by the 10th century were a distinctive western variant of the glyphs used in the Arabic-speaking world, known as ghubār numerals (ghubār translating to "sand table"). In these digits, the line of the 5-like glyph used in Indian manuscripts for eight came to be formed in ghubār as a closed loop, which was the 8-shape that became adopted into European use in the 10th century.[3] Just as in most modern typefaces, in typefaces with text figures the character for the digit 8 usually has an ascender, as, for example, in . The infinity symbol ∞, described as a "sideways figure eight", is unrelated to the digit 8 in origin; it is first used (in the mathematical meaning "infinity") in the 17th century, and it may be derived from the Roman numeral for "one thousand" CIƆ, or alternatively from the final Greek letter, ω. In mathematics Eight is the third composite number, lying between the fourth prime number (7) and the fourth composite number (9). 8 is the first non-unitary cube prime of the form p3. With proper divisors 1, 2, and 4, it is the third power of two (23). 8 is the first number which is neither prime nor semiprime and the only nonzero perfect power that is one less than another perfect power, by Mihăilescu's Theorem. • 8 is the first proper Leyland number of the form xy + yx, where in its case x and y both equal 2.[4] • 8 is the sum between the first pair of twin-primes (3, 5), and the only twin-prime sum that is not a multiple of 3 or 12. • 8 is the sixth Fibonacci number and the first even, non-prime Fibonacci number. It is also the only positive Fibonacci number aside from 1 that is a perfect cube.[5] • 8 is the third refactorable number, as it has exactly four positive divisors, and 4 is one of them. • 8 is the only composite number with a prime aliquot sum of 7 (1 + 2 + 4)[6] that is part of the aliquot sequence (8, 7, 1, 0). • 8 is the first number to be the aliquot sum of two numbers: the discrete semiprime 10, and the squared prime 49. • 8 is the difference between 53 and 61, which are the two smallest prime numbers that do not divide the order of any sporadic group. Sphenic numbers always have exactly eight divisors.[7] A polygon with eight sides is an octagon.[8] The sides and span of a regular octagon, or truncated square, are in 1 : 1 + √2 silver ratio, and its circumscribing square has a side and diagonal length ratio of 1 : √2; with both the silver ratio and the square root of two intimately interconnected through Pell numbers, where in particular the quotient of successive Pell numbers generates rational approximations for coordinates of a regular octagon.[9][10] With a central angle of 45 degrees and an internal angle of 135 degrees, a regular octagon can fill a plane-vertex with a regular triangle and a regular icositetragon, as well as tessellate two-dimensional space alongside squares in the truncated square tiling. This tiling is one of eight Archimedean tilings that are semi-regular, or made of more than one type of regular polygon, and the only tiling that can admit a regular octagon.[11] The Ammann–Beenker tiling is a nonperiodic tesselation of prototiles that feature prominent octagonal silver eightfold symmetry, that is the two-dimensional orthographic projection of the four-dimensional 8-8 duoprism.[12] In number theory, figurate numbers representing octagons are called octagonal numbers.[13] A cube is a regular polyhedron with eight vertices that also forms the cubic honeycomb, the only regular honeycomb in three-dimensional space.[14] Through various truncation operations, the cubic honeycomb generates eight other convex uniform honeycombs under the cubic group ${\tilde {C}}_{3}$.[15] The octahedron, with eight equilateral triangles as faces, is the dual polyhedron to the cube and one of eight convex deltahedra.[16][17] The stella octangula, or eight-pointed star, is the only stellation with octahedral symmetry. It has eight triangular faces alongside eight vertices that forms a cubic faceting, composed of two self-dual tetrahedra that makes it the simplest of five regular compounds. The cuboctahedron, on the other hand, is a rectified cube or rectified octahedron, and one of only two convex quasiregular polyhedra. It contains eight equilateral triangular faces, whose first stellation is the cube-octahedron compound.[18][19] There are also eight uniform polyhedron compounds made purely of octahedra, including the regular compound of five octahedra, and an infinite amount of polyhedron compounds made only of octahedra as triangular antiprisms (UC22 and UC23, with p = 3 and q = 1). The truncated tetrahedron is the simplest Archimedean solid, made of four triangles and four hexagons, the hexagonal prism, which classifies as an irregular octahedron and parallelohedron, is able to tessellate space as a three-dimensional analogue of the hexagon, and the gyrobifastigium, with four square faces and four triangular faces, is the only Johnson solid that is able to tessellate space. The truncated octahedron, also a parallelohedron, is the permutohedron of order four, with eight hexagonal faces alongside six squares is likewise the only Archimedean solid that can generate a honeycomb on its own. A tesseract or 8-cell is the four-dimensional analogue of the cube. It is one of six regular polychora, with a total of eight cubical cells, hence its name. Its dual figure is the analogue of the octahedron, with twice the amount of cells and simply termed the 16-cell, that is the orthonormal basis of vectors in four dimensions. Whereas a tesseractic honeycomb is self-dual, a 16-cell honeycomb is dual to a 24-cell honeycomb that is made of 24-cells. The 24-cell is also regular, and made purely of octahedra whose vertex arrangement represents the ring of Hurwitz integral quaternions. Both the tesseract and the 16-cell can fit inside a 24-cell, and in a 24-cell honeycomb, eight 24-cells meet at a vertex. Also, the Petrie polygon of the tesseract and the 16-cell is a regular octagon. Vertex-transitive semiregular polytopes whose facets are finite exist up through the 8th dimension. In the third dimension, they include the Archimedean solids and the infinite family of uniform prisms and antiprisms, while in the fourth dimension, only the rectified 5-cell, the rectified 600-cell, and the snub 24-cell are semiregular polytopes. For dimensions five through eight, the demipenteract and the k21 polytopes 221, 321, and 421 are the only semiregular (Gosset) polytopes. Collectively, the k21 family of polytopes contains eight figures that are rooted in the triangular prism, which is the simplest semiregular polytope that is made of three cubes and two equilateral triangles. It also includes one of only three semiregular Euclidean honeycombs: the affine 521 honeycomb that represents the arrangement of vertices of the eight-dimensional $\mathrm {E} _{8}$ lattice, and made of 421 facets. The culminating figure is the ninth-dimensional 621 honeycomb, which is the only affine semiregular paracompact hyperbolic honeycomb with infinite facets and vertex figures in the k21 family. There are no other finite semiregular polytopes or honeycombs in dimensions n > 8. The octonions are a hypercomplex normed division algebra that are an extension of the complex numbers. They are realized in eight dimensions, where they have an isotopy group over the real numbers that is spin group Spin(8), the unique such group that exhibits a phenomenon of triality. As a double cover of special orthogonal group SO(8), Spin(8) contains the special orthogonal Lie algebra D4 as its Dynkin diagram, whose order-three outer automorphism is isomorphic to the symmetric group S3, giving rise to its triality. Over finite fields, the eight-dimensional Steinberg group 3D4(q3) is simple, and one of sixteen such groups in the classification of finite simple groups. As is Lie algebra E8, whose complex form in 248 dimensions is the largest of five exceptional Lie algebras that include E7 and E6, which are held inside E8. The smallest such algebra is G2, that is the automorphism group of the octonions. In mathematical physics, special unitary group SO(3) has an eight-dimensional adjoint representation whose colors are ascribed gauge symmetries that represent the vectors of the eight gluons in the Standard Model. The $\mathrm {E} _{8}$ lattice Γ8 is the smallest positive even unimodular lattice. As a lattice, it holds the optimal structure for the densest packing of 240 spheres in eight dimensions, whose lattice points also represent the root system of Lie group E8. This honeycomb arrangement is shared by a unique complex tessellation of Witting polytopes, also with 240 vertices. Each complex Witting polytope is made of Hessian polyhedral cells that have Möbius–Kantor polygons as faces, each with eight vertices and eight complex equilateral triangles as edges, whose Petrie polygons form regular octagons. In general, positive even unimodular lattices only exist in dimensions proportional to eight. In the 16th dimension, there are two such lattices : Γ8 ⊕ Γ8 and Γ16, while in the 24th dimension there are precisely twenty-four such lattices that are called the Niemeier lattices, the most important being the Leech lattice, which can be constructed using the octonions as well as with three copies of the ring of icosians that are isomorphic to the $\mathrm {E} _{8}$ lattice.[20][21] The order of the smallest non-abelian group all of whose subgroups are normal is 8. The number 8 is involved with a number of interesting mathematical phenomena related to the notion of Bott periodicity. If $O(\infty )$ is the direct limit of the inclusions of real orthogonal groups $O(1)\hookrightarrow O(2)\hookrightarrow \ldots \hookrightarrow O(k)\hookrightarrow \ldots $, the following holds: $\pi _{k+8}(O(\infty ))\cong \pi _{k}(O(\infty ))$. Clifford algebras also display a periodicity of 8.[22] For example, the algebra Cl(p + 8,q) is isomorphic to the algebra of 16 by 16 matrices with entries in Cl(p,q). We also see a period of 8 in the K-theory of spheres and in the representation theory of the rotation groups, the latter giving rise to the 8 by 8 spinorial chessboard. All of these properties are closely related to the properties of the octonions, which occupy the highest possible dimension for a normed division algebra. List of basic calculations Multiplication 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 8 × x 8 16 24 32 40 48 56 64 72 80 88 96 104 112 120 Division 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 8 ÷ x 8 4 2.6 2 1.6 1.3 1.142857 1 0.8 0.8 0.72 0.6 0.615384 0.571428 0.53 x ÷ 8 0.125 0.25 0.375 0.5 0.625 0.75 0.875 1 1.125 1.25 1.375 1.5 1.625 1.75 1.875 Exponentiation 1 2 3 4 5 6 7 8 9 10 11 12 13 8x 8 64 512 4096 32768 262144 2097152 16777216 134217728 1073741824 8589934592 68719476736 549755813888 x8 1 256 6561 65536 390625 1679616 5764801 16777216 43046721 100000000 214358881 429981696 815730721 In other bases A number is divisible by 8 if its last three digits, when written in decimal, are also divisible by 8, or its last three digits are 0 when written in binary. 8 is the base of the octal number system, which is mostly used with computers.[23] In octal, one digit represents three bits. In modern computers, a byte is a grouping of eight bits, also called an octet. In science Physics • In nuclear physics, the second magic number.[24] • In particle physics, the eightfold way is used to classify sub-atomic particles.[25] • In statistical mechanics, the eight-vertex model has 8 possible configurations of arrows at each vertex.[26] Astronomy • Messier object M8, a magnitude 5.0 nebula in the constellation of Sagittarius.[27] • The New General Catalogue object NGC 8, a double star in the constellation Pegasus. • Since the demotion of Pluto to a dwarf planet on 24 August 2006, in our Solar System, eight of the bodies orbiting the Sun are considered to be planets. Chemistry • The atomic number of oxygen.[28] • The most stable allotrope of a sulfur molecule is made of eight sulfur atoms arranged in a rhombic form.[29] • The maximum number of electrons that can occupy a valence shell (except for transitional elements), see the octet rule. • The red pigment lycopene consists of eight isoprene units.[30] Geology • A disphenoid crystal is bounded by eight scalene triangles arranged in pairs. A ditetragonal prism in the tetragonal crystal system has eight similar faces whose alternate interfacial angles only are equal. Biology • All spiders, and more generally all arachnids, have eight legs.[31] Orb-weaver spiders of the cosmopolitan family Areneidae have eight similar eyes.[32] • The octopus and its cephalopod relatives in genus Argonauta have eight arms (tentacles). • Compound coelenterates of the subclass or order Alcyonaria have polyps with eight-branched tentacles and eight septa.[33] • Sea anemones of genus Edwardsia have eight mesenteries.[34] • Animals of phylum Ctenophora swim by means of eight meridional bands of transverse ciliated plates, each plate representing a row of large modified cilia.[35] • The eight-spotted forester (genus Alypia, family Zygaenidae) is a diurnal moth having black wings with brilliant white spots. • The ascus in fungi of the class Ascomycetes, following nuclear fusion, bears within it typically eight ascospores.[36] • Herbs of genus Coreopsis (tickseed) have showy flower heads with involucral bracts in two distinct series of eight each. • In human adult dentition there are eight teeth in each quadrant.[37] The eighth tooth is the so-called wisdom tooth. • There are eight cervical nerves on each side in man and most mammals.[38] Psychology • There are eight Jungian cognitive functions, according to the MBTI models by John Beebe and Linda Berens.[39] • Timothy Leary identified a hierarchy of eight levels of consciousness. In technology • A byte is eight bits.[40] • Many (mostly historic) computer architectures are eight-bit, among them the Nintendo Entertainment System. • Standard-8 and Super-8 are 8 mm film formats.[41] • Video8, Hi8 and Digital8 are related 8 mm video formats.[42] • On most phones, the 8 key is associated with the letters T, U, and V, but on the BlackBerry Pearl it is the key for B and N. • An eight may refer to an eight-cylinder engine or automobile.[43] A V8 engine is an internal combustion engine with eight cylinders configured in two banks (rows) of four forming a "V" when seen from the end. • A figure-eight knot (so named for its configuration) is a kind of stopper knot.[44] • The number eight written in parentheses is the code for the musical note in Windows Live Messenger. • In a seven-segment display, when an 8 is illuminated, all the display bulbs are on. In measurement • The SI prefix for 10008 is yotta (Y), and for its reciprocal, yocto (y). • In liquid measurement (United States customary units), there are eight fluid ounces in a cup, eight pints in a gallon and eight tablespoonfuls in a gill.[45] • There are eight furlongs in a mile.[46] • The clove, an old English unit of weight, was equal to eight pounds when measuring cheese.[47] • An eight may be an article of clothing of the eighth size. • Force eight is the first wind strength attributed to a gale on the Beaufort scale when announced on a Shipping Forecast.[48] In culture Currency • Sailors and civilians alike from the 1500s onward referred to evenly divided parts of the Spanish dollar as "pieces of eight", or "bits". Architecture • Various types of buildings are usually eight-sided (octagonal), such as single-roomed gazebos and multi-roomed pagodas (descended from stupas; see religion section below). • Eight caulicoles rise out of the leafage in a Corinthian capital, ending in leaves that support the volutes. Hinduism • Also known as Ashtha, Aṣṭa, or Ashta in Sanskrit, it is the number of wealth and abundance. • The goddess of wealth and prosperity, Lakshmi, has eight forms known as Ashta Lakshmi and worshipped as: "Maha-lakshmi, Dhana-lakshmi, Dhanya-lakshmi, Gaja-lakshmi, Santana-lakshmi, Veera-lakshmi, Vijaya-lakshmi and Vidhya-lakshmi "[49] • There are eight nidhi, or seats of wealth, according to Hinduism. • There are eight guardians of the directions known as Astha-dikpalas.[50] • There are eight Hindu monasteries established by the saint Madhvacharya in Udupi, India popularly known as the Ashta Mathas of Udupi.[51] Buddhism • The Dharmacakra, a Buddhist symbol, has eight spokes.[52] The Buddha's principal teaching—the Four Noble Truths—ramifies as the Noble Eightfold Path and the Buddha emphasizes the importance of the eight attainments or jhanas. • In Mahayana Buddhism, the branches of the Eightfold Path are embodied by the Eight Great Bodhisattvas: (Manjusri, Vajrapani, Avalokiteśvara, Maitreya, Ksitigarbha, Nivaranavishkambhi, Akasagarbha, and Samantabhadra).[53] These are later (controversially) associated with the Eight Consciousnesses according to the Yogacara school of thought: consciousness in the five senses, thought-consciousness, self-consciousness, and unconsciousness-"consciousness" or "store-house consciousness" (alaya-vijñana). The "irreversible" state of enlightenment, at which point a Bodhisattva goes on "autopilot", is the Eight Ground or bhūmi. In general, "eight" seems to be an auspicious number for Buddhists, e.g., the "eight auspicious symbols" (the jewel-encrusted parasol; the goldfish (always shown as a pair, e.g., the glyph of Pisces); the self-replenishing amphora; the white kamala lotus-flower; the white conch; the eternal (Celtic-style, infinitely looping) knot; the banner of imperial victory; the eight-spoked wheel that guides the ship of state, or that symbolizes the Buddha's teaching). Similarly, Buddha's birthday falls on the 8th day of the 4th month of the Chinese calendar. Judaism • The religious rite of brit milah (commonly known as circumcision) is held on a baby boy's eighth day of life.[54] • Hanukkah is an eight-day Jewish holiday that starts on the 25th day of Kislev.[55] • Shemini Atzeret (Hebrew: "Eighth Day of Assembly") is a one-day Jewish holiday immediately following the seven-day holiday of Sukkot.[56] Christianity • The spiritual Eighth Day, because the number 7 refers to the days of the week (which repeat themselves). • The number of Beatitudes.[57] • 1 Peter 3:20 states that there were eight people on Noah's Ark.[58] • The Antichrist is the eighth king in the Book of Revelation.[59] Islam • In Islam, eight is the number of angels carrying the throne of Allah in heaven.[60] • The number of gates of heaven Taoism • Ba Gua[61] • Ba Xian[62] • Ba Duan Jin Other • In Wicca, there are eight Sabbats, festivals, seasons, or spokes in the Wheel of the Year.[63] • In Ancient Egyptian mythology, the Ogdoad represents the eight primordial deities of creation.[64] • In Scientology there are eight dynamics of existence.[65]: 39 • There is also the Ogdoad in Gnosticism.[66] As a lucky number • The number eight is considered to be a lucky number in Chinese and other Asian cultures.[67] Eight (八; accounting 捌; pinyin bā) is considered a lucky number in Chinese culture because it sounds like the word meaning to generate wealth (發(T) 发(S); Pinyin: fā). Property with the number 8 may be valued greatly by Chinese. For example, a Hong Kong number plate with the number 8 was sold for $640,000.[68] The opening ceremony of the Summer Olympics in Beijing started at 8 seconds and 8 minutes past 8 pm (local time) on 8 August 2008.[69] • In Pythagorean numerology (a pseudoscience) the number 8 represents victory, prosperity and overcoming. • Eight (八, hachi, ya) is also considered a lucky number in Japan, but the reason is different from that in Chinese culture.[70] Eight gives an idea of growing prosperous, because the letter (八) broadens gradually. • The Japanese thought of eight (や, ya) as a holy number in the ancient times. The reason is less well-understood, but it is thought that it is related to the fact they used eight to express large numbers vaguely such as manyfold (やえはたえ, Yae Hatae) (literally, eightfold and twentyfold), many clouds (やくも, Yakumo) (literally, eight clouds), millions and millions of Gods (やおよろずのかみ, Yaoyorozu no Kami) (literally, eight millions of Gods), etc. It is also guessed that the ancient Japanese gave importance to pairs, so some researchers guess twice as four (よ, yo), which is also guessed to be a holy number in those times because it indicates the world (north, south, east, and west) might be considered a very holy number. • In numerology, 8 is the number of building, and in some theories, also the number of destruction. In astrology • In astrology, Scorpio is the 8th astrological sign of the Zodiac.[71] • In the Middle Ages, 8 was the number of "unmoving" stars in the sky, and symbolized the perfection of incoming planetary energy. In music and dance • A note played for one-eighth the duration of a whole note is called an eighth note, or quaver.[72] • An octave, the interval between two musical notes with the same letter name (where one has double the frequency of the other), is so called because there are eight notes between the two on a standard major or minor diatonic scale, including the notes themselves and without chromatic deviation.[73] The ecclesiastical modes are ascending diatonic musical scales of eight notes or tones comprising an octave. • There are eight notes in the octatonic scale. • There are eight musicians in a double quartet or an octet.[74] Both terms may also refer to a musical composition for eight voices or instruments.[75] • Caledonians is a square dance for eight, resembling the quadrille. • Albums with the number eight in their title include 8 by the Swedish band Arvingarna, 8 by the American rock band Incubus,[76] The Meaning of 8 by Minnesota indie rock band Cloud Cult and 8ight by Anglo-American singer-songwriter Beatie Wolfe.[77] • Dream Theater's eighth album Octavarium contains many different references to the number 8, including the number of songs and various aspects of the music and cover artwork. • "Eight maids a-milking" is the gift on the eighth day of Christmas in the carol "The Twelve Days of Christmas".[78] • The 8-track cartridge is a musical recording format. • "#8" is the stage name of Slipknot vocalist Corey Taylor. • "Too Many Eights" is a song by Athens, Georgia's Supercluster.[79] • Eight Seconds, a Canadian musical group popular in the 1980s with their most notable song "Kiss You (When It's Dangerous)".[80] • "Eight Days a Week" is a #1 single for the music group the Beatles.[81] • Figure 8 is the fifth studio album by singer-songwriter Elliott Smith, released in the year 2000,[82] an album released by Julia Darling in 1999,[83] and an album released by Outasight in 2011.[84] • Ming Hao from the k-pop group Seventeen goes by the name "The8".[85] • "8 (circle)" is the eighth song on the album 22, A Million by the American band Bon Iver.[86] • "8" is the eighth song on the album When We All Fall Asleep, Where Do We Go? by Billie Eilish.[87] In film and television • 8 Guys is a 2003 short film written and directed by Dane Cook. • 8 Man (or Eightman): 1963 Japanese manga and anime superhero. • 8 Mile is a 2002 film directed by Curtis Hanson.[88] • 8 mm is a 1999 film directed by Joel Schumacher.[89] • 8 Women (Original French title: 8 femmes) is a 2001 film directed by François Ozon.[90] • Eight Below is a 2006 film directed by Frank Marshall.[91] • Eight Legged Freaks is a 2002 film directed by Ellory Elkayem.[92] • Eight Men Out is a 1988 film directed by John Sayles.[93] • Jennifer Eight, also known as Jennifer 8, is a 1992 film written and directed by Bruce Robinson.[94] • Eight Is Enough is an American television comedy-drama series. • In Stargate SG-1 and Stargate Atlantis, dialing an 8-chevron address will open a wormhole to another galaxy. • The Hateful Eight is a 2015 American western mystery film written and directed by Quentin Tarantino.[95] • Kate Plus 8 is an American reality television show.[96] • Ocean's 8 is an American heist comedy film directed by Gary Ross.[97] In sports and other games • Eight-ball pool is played with a cue ball and 15 numbered balls, the black ball numbered 8 being the middle and most important one, as the winner is the player or side that legally pockets it after first pocketing its numerical group of 7 object balls (for other meanings see Eight ball (disambiguation)). • In chess, each side has eight pawns and the board is made of 64 squares arranged in an eight by eight lattice. The eight queens puzzle is a challenge to arrange eight queens on the board so that none can capture any of the others. • In the game of eights or Crazy Eights, each successive player must play a card either of the same suit or of the same rank as that played by the preceding player, or may play an eight and call for any suit. The object is to get rid of all one's cards first. • In association football, the number 8 has historically been the number of the Central Midfielder. • In Australian rules football, the top eight teams at the end of the Australian Football League regular season qualify for the finals series (i.e. playoffs). • In baseball: • The center fielder is designated as number 8 for scorekeeping purposes. • The Men's College World Series, the final phase of the NCAA Division I tournament, features eight teams. • In rugby union, the only position without a proper name is the Number 8, a forward position. • In rugby league: • Most competitions (though not the Super League, which uses static squad numbering) use a position-based player numbering system in which one of the two starting props wears the number 8. • The Australia-based National Rugby League has its own 8-team finals series, similar but not identical in structure to that of the Australian Football League. • In rowing, an "eight" refers to a sweep-oar racing boat with a crew of eight rowers plus a coxswain.[98] • In the 2008 Games of the XXIX Olympiad held in Beijing, the official opening was on 08/08/08 at 8:08:08 p.m. CST. • In rock climbing, climbers frequently use the figure-eight knot to tie into their harnesses. • The Women's College World Series, the final phase of the NCAA Division I softball tournament, like its men's counterpart in baseball, features eight teams. • In curling an 8 point 'Eight Ender' is a perfect end. Each team delivers 8 Stones per end. In foods • Nestlé sells a brand of chocolates filled with peppermint-flavoured cream called After Eight, referring to the time 8 p.m.[99] • There are eight vegetables in V8 juice. In literature • Eights may refer to octosyllabic, usually iambic, lines of verse. • The drott-kvaett, an Old Icelandic verse, consisted of a stanza of eight regular lines.[100] • In Terry Pratchett's Discworld series, eight is a magical number[101] and is considered taboo. Eight is not safe to be said by wizards on the Discworld and is the number of Bel-Shamharoth. Also, there are eight days in a Disc week and eight colours in a Disc spectrum, the eighth one being octarine. • Lewis Carroll's poem The Hunting of the Snark has 8 "fits" (cantos), which is noted in the full name "The Hunting of the Snark – An Agony, in Eight Fits".[102] • Eight apparitions appear to Macbeth in Act 4 scene 1 of Shakespeare's Macbeth as representations of the eight descendants of Banquo. In slang • An "eighth" is a common measurement of marijuana, meaning an eighth of an ounce. It is also a common unit of sale for psilocybin mushrooms. • Avril Lavigne's song "Sk8er Boi" uses this convention in the title. • The Housing Choice Voucher Program, operated by the United States Department of Housing and Urban Development, is commonly referred to as the Section 8 program, as this was the original section of the Act which instituted the program.[103] • In Colombia and Venezuela, "volverse un ocho" (meaning to tie oneself in a figure 8) refers to getting in trouble or contradicting oneself. • In China, "8" is used in chat speak as a term for parting. This is due to the closeness in pronunciation of "8" (bā) and the English word "bye". Other uses • A figure 8 is the common name of a geometric shape, often used in the context of sports, such as skating.[104] Figure-eight turns of a rope or cable around a cleat, pin, or bitt are used to belay something.[105] See also • The Magical Number Seven, Plus or Minus Two • List of highways numbered 8 References 1. Etymological Dictionary of Turkic Languages: Common Turkic and Interturkic stems starting with letters «L», «M», «N», «P», «S», Vostochnaja Literatura RAS, 2003, 241f. (altaica.ru Archived 31 October 2007 at the Wayback Machine) 2. the hypothesis is discussed critically (and rejected as "without sufficient support") by Werner Winter, 'Some thought about Indo-European numerals' in: Jadranka Gvozdanović (ed.), Indo-European Numerals, Walter de Gruyter, 1992, 14f. 3. Georges Ifrah, The Universal History of Numbers: From Prehistory to the Invention of the Computer transl. David Bellos et al. London: The Harvill Press (1998): 395, Fig. 24.68. 4. Sloane, N. J. A. (ed.). "Sequence A076980 (Leyland numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. 5. Bryan Bunch, The Kingdom of Infinite Number. New York: W. H. Freeman & Company (2000): 88 6. Sloane, N. J. A. (ed.). "Sequence A001065 (Sum of proper divisors (or aliquot parts) of n: sum of divisors of n that are less than n.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 8 August 2023. 7. Weisstein, Eric W. "Sphenic Number". mathworld.wolfram.com. Retrieved 7 August 2020. ...then every sphenic number n=pqr has precisely eight positive divisors 8. Weisstein, Eric W. "Octagon". mathworld.wolfram.com. Retrieved 7 August 2020. 9. Bicknell, Marjorie (1975). "A primer on the Pell sequence and related sequences". Fibonacci Quarterly. 13 (4): 345–349. MR 0387173. 10. Knuth, Donald E. (1994). "Leaper graphs". The Mathematical Gazette. 78 (483): 283. arXiv:math.CO/9411240. Bibcode:1994math.....11240K. doi:10.2307/3620202. JSTOR 3620202. S2CID 16856513. 11. Weisstein, Eric W. "Regular Octagon". mathworld.wolfram.com. Retrieved 25 June 2022. 12. Katz, A (1995). "Matching rules and quasiperiodicity: the octagonal tilings". In Axel, F.; Gratias, D. (eds.). Beyond quasicrystals. Springer. pp. 141–189. doi:10.1007/978-3-662-03130-8_6. ISBN 978-3-540-59251-8. 13. Deza, Elena; Deza, Michel (2012). Figurate Numbers. World Scientific. pp. 39, 40, 92, 151. ISBN 9789814355483.. 14. Weisstein, Eric W. "Cube". mathworld.wolfram.com. Retrieved 7 August 2020. 15. Branko Grünbaum (1994). "Uniform tilings of 3-space". Geombinatorics. 4 (2): 49–56. 16. Freudenthal, H; van der Waerden, B. L. (1947), "Over een bewering van Euclides ("On an Assertion of Euclid")", Simon Stevin (in Dutch), 25: 115–128 17. Roger Kaufman. "The Convex Deltahedra And the Allowance of Coplanar Faces". The Kaufman Website. Retrieved 25 June 2022. 18. Weisstein, Eric W. "Cuboctahedron". mathworld.wolfram.com. Retrieved 25 June 2022. 19. Coxeter, H.S.M. (1973) [1948]. Regular Polytopes (3rd ed.). New York: Dover. pp. 18–19. 20. Wilson, Robert A. (2009). "Octonions and the Leech lattice". Journal of Algebra. 322 (6): 2186–2190. doi:10.1016/j.jalgebra.2009.03.021. MR 2542837. 21. Conway, John H.; Sloane, N. J. A. (1988). "Algebraic Constructions for Lattices". Sphere Packings, Lattices and Groups. New York, NY: Springer. doi:10.1007/978-1-4757-2016-7. eISSN 2196-9701. ISBN 978-1-4757-2016-7. 22. Lounesto, Pertti (3 May 2001). Clifford Algebras and Spinors. Cambridge University Press. p. 216. ISBN 978-0-521-00551-7. ...Clifford algebras, contains or continues with two kinds of periodicities of 8... 23. Weisstein, Eric W. "Octal". mathworld.wolfram.com. Retrieved 7 August 2020. 24. Ilangovan, K. (10 June 2019). Nuclear Physics. MJP Publisher. p. 30. 25. Gell-Mann, M. (15 March 1961). THE EIGHTFOLD WAY: A THEORY OF STRONG INTERACTION SYMMETRY (Technical report). OSTI 4008239. 26. Baxter, R. J. (5 April 1971). "Eight-Vertex Model in Lattice Statistics". Physical Review Letters. 26 (14): 832–833. Bibcode:1971PhRvL..26..832B. doi:10.1103/PhysRevLett.26.832. 27. "Messier Object 8". www.messier.seds.org. Retrieved 7 August 2020. 28. Thomas, Mary Ann (15 August 2004). Oxygen. The Rosen Publishing Group, Inc. p. 12. ISBN 978-1-4042-0159-0. Knowing that oxygen has an atomic number of 8, 29. Choppin, Gregory R.; Johnsen, Russell H. (1972). Introductory chemistry. Addison-Wesley Pub. Co. p. 366. ISBN 978-0-201-01022-0. under normal conditions the most stable allotropic form (Fig. 23-8a). Sulfur molecules within the crystal consist of puckered rings of eight sulfur atoms linked by single... 30. Puri, Basant; Hall, Anne (16 December 1998). Phytochemical Dictionary: A Handbook of Bioactive Compounds from Plants, Second Edition. CRC Press. p. 810. ISBN 978-0-203-48375-6. The chemical structure of lycopene consists of a long chain of eight isoprene units joined head to tail 31. Parker, Barbara Keevil (28 December 2006). Ticks. Lerner Publications. p. 7. ISBN 978-0-8225-6464-5. Arachnids have eight legs 32. Jackman, J. A. (1997). A Field Guide to Spiders & Scorpions of Texas. Gulf Publishing Company. p. 70. ISBN 978-0-87719-264-0. Araneids have eight eyes 33. Fisher, James; Huxley, Julian (1961). The Doubleday Pictorial Library of Nature: Earth, Plants, Animals. Doubleday. p. 311. Polyps with eight branched tentacles and eight septa 34. Bourne, Gilbert Charles (1911). "Anthozoa" . In Chisholm, Hugh (ed.). Encyclopædia Britannica. Vol. 02 (11th ed.). Cambridge University Press. pp. 97–105, see page 100. Zoantharia.....It is not known whether all the eight mesenteries of Edwardsia are developed simultaneously or not, but in the youngest form which has been studied all the eight mesenteries were present 35. The Century Dictionary and Cyclopedia: A work of Universal Reference in all Departments of Knowledge with a New Atlas of the World. 1906. p. 1384. ...are radially symmetrical, and swim by means of eight meridional ciliated bands, ... 36. Parrish, Fred K. (1975). Keys to Water Quality Indicative Organisms of the Southeastern United States. Environmental Protection Agency, Office of Research and Development, Environmental Monitoring and Support Laboratory, Biological Methods Branch, Aquatics Biology Section. p. 11. ... the ascospores, are borne in sac like structures termed asci. The ascus usually contains eight as cospores,... 37. Dofka, Charline M. (1996). Competency Skills for the Dental Assistant. Cengage Learning. p. 83. ISBN 978-0-8273-6685-5. ...In each quadrant of the permanent set of teeth (dentition), there are eight teeth 38. Quain, Jones (1909). Quain's Elements of Anatomy. Longmans, Green, & Company. p. 52. These eight pairs are usually reckoned as eight cervical nerves ... 39. Beebe, John (17 June 2016). Energies and Patterns in Psychological Type: The reservoir of consciousness. Routledge. p. 124. ISBN 978-1-317-41366-0. Linda Berens used the term 'cognitive processes' (1999) to refer to the eight types of consciousness that Jung discovered. 40. "Definition of byte \| Dictionary.com". www.dictionary.com. Retrieved 8 August 2020. 41. Kindem, Gorham; PhD, Robert B. Musburger (21 August 2012). Introduction to Media Production: The Path to Digital Media Production. CRC Press. p. 320. ISBN 978-1-136-05322-1. There used to be two 8 mm formats: standard 8 mm and Super-8 mm. 42. The Library of Congress Veterans History Project: Field Kit : Conducting and Preserving Interviews. Veterans History Project, American Folklife Center, Library of Congress. 2008. p. 15. Betacam SX 8 mm Hi8, Digital8, Video8 DVD-Video"; 43. "Definition of eight \| Dictionary.com". www.dictionary.com. Retrieved 8 August 2020. 44. Griffiths, Garth (1971). Boating in Canada: Practical Piloting and Seamanship. University of Toronto Press. p. 32. ISBN 978-0-8020-1817-5. First is a stopper knot, the figure of eight, ... 45. The Milwaukee Cook Book. Press of Houtkamp Printing. 1907. 46. "Definition of furlong \| Dictionary.com". www.dictionary.com. Retrieved 8 August 2020. 47. "Definition of clove \| Dictionary.com". www.dictionary.com. Retrieved 8 August 2020. 48. Fairhall, David; Peyton, Mike (17 May 2013). Pass Your Yachtmaster. A&C Black. ISBN 978-1-4081-5627-8. Gale warnings will be given if mean wind speeds of force 8 (34–40 knots) 49. Hatcher, Brian A. (5 October 2015). Hinduism in the Modern World. Routledge. ISBN 978-1-135-04630-9. a group manifestation of eight forms 50. Jeyaraj, Daniel (23 September 2004). Genealogy of the South Indian Deities: An English Translation of Bartholomäus Ziegenbalg's Original German Manuscript with a Textual Analysis and Glossary. Routledge. p. 168. ISBN 978-1-134-28703-1. He is one of the eight guardians of the world 51. Ramachandran, Nirmala (2000). Hindu Heritage. Stamford Lake Publication. p. 72. ISBN 978-955-8733-09-7. The temple has eight monasteries, founded by Madhvacharya 52. Issitt, Micah; Main, Carlyn (16 September 2014). Hidden Religion: The Greatest Mysteries and Symbols of the World's Religious Beliefs: The Greatest Mysteries and Symbols of the World's Religious Beliefs. ABC-CLIO. p. 186. ISBN 978-1-61069-478-0. The dharmachakra is typically depicted with eight spokes, 53. Hay, Jeff (6 March 2009). World Religions. Greenhaven Publishing LLC. p. 61. ISBN 978-0-7377-4627-3. The focus of ordinary believers' religious life is on following a relevant version of the Eightfold Path ... 54. Rosten, Leo (14 April 2010). The New Joys of Yiddish: Completely Updated. Potter/Ten Speed/Harmony/Rodale. p. 48. ISBN 978-0-307-56604-1. Brit Milah is observed on a boy's eighth day of life 55. Ross, Kathy (1 August 2012). Crafts for Hanukkah. Millbrook Press. p. 7. ISBN 978-0-7613-6836-6. Hanukkah is an eight-day Jewish holiday 56. Axelrod, Cantor Matt (24 December 2013). Your Guide to the Jewish Holidays: From Shofar to Seder. Rowman & Littlefield. p. 58. ISBN 978-0-7657-0990-5. Shemini Atzeret—literally, "the eighth day of assembly" 57. "CATHOLIC ENCYCLOPEDIA: The Eight Beatitudes". www.newadvent.org. Retrieved 9 August 2020. 58. Akintola, Olufolahan Olatoye (2011). Nations of the World…How They Evolved!: Families and Nations That Came Out of Ham. Hilldew View International Limited. p. 8. ISBN 978-0-9569702-2-0. These eight souls in Noah's ark were the survivors... 59. Livingstone (2001). Life Application New Testament Commentary. Tyndale House Publishers, Inc. ISBN 978-0-8423-7066-0. 60. Mahmutćehajić, Rusmir (2011). Maintaining the Sacred Center: The Bosnian City of Stolac. World Wisdom, Inc. p. 201. ISBN 978-1-935493-91-4. ... at the last, eight Angels will carry the Throne... 61. Little, Stephen; Eichman, Shawn; Shipper, Kristofer; Ebrey, Patricia Buckley (1 January 2000). Taoism and the Arts of China. University of California Press. p. 139. ISBN 978-0-520-22785-9. Evidence for the early use of the Eight Trigrams in a religious Taoist... 62. Ho, Peter Kwok Man; Kwok, Man-Ho; O'Brien, Joanne (1990). The Eight Immortals of Taoism: Legends and Fables of Popular Taoism. Meridian. p. 7. ISBN 978-0-452-01070-3. ...famous Eight Immortals of China... 63. Zimmermann, Denise; Gleason, Katherine; Liguana, Miria (2006). The Complete Idiot's Guide to Wicca and Witchcraft. Penguin. p. 172. ISBN 978-1-59257-533-6. There are eight Sabbats 64. Remler, Pat (2010). Egyptian Mythology, A to Z. Infobase Publishing. p. 79. ISBN 978-1-4381-3180-1. ...of the gods of the Ogdoad, or the eight deities of the Egyptian creation... 65. Wallis, Roy (1977). The Road to Total Freedom: A Sociological Analysis of Scientology. Columbia University Press. ISBN 0231042000. OL 4596322M. 66. David, Fideler (1993). Jesus Christ, Sun of God: Ancient Cosmology and Early Christian Symbolism. Quest Books. p. 128. ISBN 978-0-8356-0696-7. 67. Ang, Swee Hoon (1997). "Chinese consumers' perception of alpha-numeric brand names". Journal of Consumer Marketing. 14 (3): 220–233. doi:10.1108/07363769710166800. Archived from the original on 5 December 2011. 68. Steven C. Bourassa; Vincent S. Peng (1999). "Hedonic Prices and House Numbers: The Influence of Feng Shui" (PDF). International Real Estate Review. 2 (1): 79–93. Archived from the original (PDF) on 13 April 2015. Retrieved 11 May 2011. 69. "Olympics opening ceremony: China makes its point with greatestshow". the Guardian. 8 August 2008. Retrieved 29 November 2022. 70. Jefkins, Frank (6 December 2012). Modern Marketing Communications. Springer Science & Business Media. p. 36. ISBN 978-94-011-6868-7. ...eight being a lucky number in Japanese. 71. "Definition of SCORPIO". www.merriam-webster.com. Retrieved 10 August 2020. 72. "Definition of eighth note \| Dictionary.com". www.dictionary.com. Retrieved 9 August 2020. 73. "Definition of OCTAVE". www.merriam-webster.com. Retrieved 9 August 2020. a tone or note that is eight steps above or below another note or tone 74. "Definition of octet \| Dictionary.com". www.dictionary.com. Retrieved 9 August 2020. a company of eight singers or musicians. 75. "Definition of octet \| Dictionary.com". www.dictionary.com. Retrieved 9 August 2020. a musical composition for eight voices or instruments. 76. "Incubus Premiere New Song "Glitterbomb", Detail New Album "8"". Theprp.com. 17 March 2017. Retrieved 9 August 2020. 77. Beatie Wolfe-8ight, retrieved 9 August 2020 78. Tribble, Mimi (2004). 300 Ways to Make the Best Christmas Ever!: Decorations, Carols, Crafts & Recipes for Every Kind of Christmas Tradition. Sterling Publishing Company, Inc. ISBN 978-1-4027-1685-0. 79. Too Many Eights – Supercluster \| Song Info \| AllMusic, retrieved 10 August 2020 80. "Eight Seconds \| Biography & History". AllMusic. Retrieved 10 August 2020. 81. Eight Days a Week – The Beatles \| Song Info \| AllMusic, retrieved 10 August 2020 82. Figure 8 – Elliott Smith \| Songs, Reviews, Credits \| AllMusic, retrieved 10 August 2020 83. Figure 8 – Julia Darling \| Songs, Reviews, Credits \| AllMusic, retrieved 10 August 2020 84. Figure 8 – Outasight \| Songs, Reviews, Credits \| AllMusic, retrieved 10 August 2020 85. "The8 \| Credits". AllMusic. Retrieved 10 August 2020. 86. 22, A Million – Bon Iver \| Songs, Reviews, Credits \| AllMusic, retrieved 10 August 2020 87. When We All Fall Asleep, Where Do We Go? - Billie Eilish \| Songs, Reviews, Credits \| AllMusic, retrieved 10 August 2020 88. 8 Mile (2002) – Curtis Hanson \| Synopsis, Characteristics, Moods, Themes and Related \| AllMovie, retrieved 10 August 2020 89. 8MM (1999) – Joel Schumacher \| Synopsis, Characteristics, Moods, Themes and Related \| AllMovie, retrieved 10 August 2020 90. 8 Women (2001) – François Ozon \| Synopsis, Characteristics, Moods, Themes and Related \| AllMovie, retrieved 10 August 2020 91. Eight Below (2006) – Bruce Hendricks, Frank Marshall \| Synopsis, Characteristics, Moods, Themes and Related \| AllMovie, retrieved 10 August 2020 92. Eight Legged Freaks (2002) – Ellory Elkayem \| Synopsis, Characteristics, Moods, Themes and Related \| AllMovie, retrieved 10 August 2020 93. Eight Men Out (1988) – John Sayles \| Synopsis, Characteristics, Moods, Themes and Related \| AllMovie, retrieved 10 August 2020 94. Jennifer Eight (1992) – Bruce Robinson \| Synopsis, Characteristics, Moods, Themes and Related \| AllMovie, retrieved 10 August 2020 95. The Hateful Eight (2015) – Quentin Tarantino \| Synopsis, Characteristics, Moods, Themes and Related \| AllMovie, retrieved 10 August 2020 96. Jon & Kate Plus 8 (2007) - \| Synopsis, Characteristics, Moods, Themes and Related \| AllMovie, retrieved 10 August 2020 97. Ocean's 8 (2018) – Sandra Bullock \| Synopsis, Characteristics, Moods, Themes and Related \| AllMovie, retrieved 13 January 2023 98. "Definition of EIGHT". www.merriam-webster.com. Retrieved 10 August 2020. 99. "Buy After Eight® Online \| Nestlé Family ME". www.nestle-family.com. Retrieved 10 August 2020. 100. "Definition of DROTT-KVAETT". www.merriam-webster.com. Retrieved 10 August 2020. 101. Collins, Robert; Latham, Robert (1988). Science Fiction & Fantasy Book Review Annual. Meckler. p. 289. ISBN 978-0-88736-249-1. 102. "The Hunting of the Snark". www.gutenberg.org. Retrieved 10 August 2020. 103. "CT Housing Choice Voucher Program". www.cthcvp.org. Retrieved 11 August 2020. Welcome to the Housing Choice Voucher Program (also known as Section 8) 104. Boys' Life. Boy Scouts of America, Inc. 1931. p. 20. lunge forward upon this skate in a left outside forward circle, in just the reverse of your right outside forward circle, until you complete a figure 8. 105. Day, Cyrus Lawrence (1986). The Art of Knotting & Splicing. Naval Institute Press. p. 231. ISBN 978-0-87021-062-4. To make a line temporarily fast by winding it, figure – eight fashion, round a cleat, a belaying pin, or a pair of bitts. External links • The Octonions, John C. 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3 3 (three) is a number, numeral and digit. It is the natural number following 2 and preceding 4, and is the smallest odd prime number and the only prime preceding a square number. It has religious or cultural significance in many societies. ← 2 3 4 → −1 0 1 2 3 4 5 6 7 8 9 → • List of numbers • Integers ← 0 10 20 30 40 50 60 70 80 90 → Cardinalthree Ordinal3rd (third) Numeral systemternary Factorizationprime Prime2nd Divisors1, 3 Greek numeralΓ´ Roman numeralIII, iii Greek prefixtri- Latin prefixtre-/ter- Binary112 Ternary103 Senary36 Octal38 Duodecimal312 Hexadecimal316 Arabic, Kurdish, Persian, Sindhi, Urdu٣ Bengali, Assamese৩ Chinese三，弎，叄 Devanāgarī३ Ge'ez፫ Greekγ (or Γ) Hebrewג Japanese三/参 Khmer៣ Malayalam൩ Tamil௩ Telugu౩ Kannada೩ Thai๓ N'Ko߃ Lao໓ GeorgianႢ/ⴂ/გ (Gani) Evolution of the Arabic digit The use of three lines to denote the number 3 occurred in many writing systems, including some (like Roman and Chinese numerals) that are still in use. That was also the original representation of 3 in the Brahmic (Indian) numerical notation, its earliest forms aligned vertically.[1] However, during the Gupta Empire the sign was modified by the addition of a curve on each line. The Nāgarī script rotated the lines clockwise, so they appeared horizontally, and ended each line with a short downward stroke on the right. In cursive script, the three strokes were eventually connected to form a glyph resembling a ⟨3⟩ with an additional stroke at the bottom: ३. The Indian digits spread to the Caliphate in the 9th century. The bottom stroke was dropped around the 10th century in the western parts of the Caliphate, such as the Maghreb and Al-Andalus, when a distinct variant ("Western Arabic") of the digit symbols developed, including modern Western 3. In contrast, the Eastern Arabs retained and enlarged that stroke, rotating the digit once more to yield the modern ("Eastern") Arabic digit "٣".[2] In most modern Western typefaces, the digit 3, like the other decimal digits, has the height of a capital letter, and sits on the baseline. In typefaces with text figures, on the other hand, the glyph usually has the height of a lowercase letter "x" and a descender: "". In some French text-figure typefaces, though, it has an ascender instead of a descender. A common graphic variant of the digit three has a flat top, similar to the letter Ʒ (ezh). This form is sometimes used to prevent falsifying a 3 as an 8. It is found on UPC-A barcodes and standard 52-card decks. Mathematics 3 is the second smallest prime number and the first odd prime number. It is the first unique prime, such that the period length value of 1 of the decimal expansion of its reciprocal, 0.333..., is unique. 3 is a twin prime with 5, and a cousin prime with 7, and the only known number $n$ such that $n$! − 1 and $n$! + 1 are prime, as well as the only prime number $p$ such that $p$ − 1 yields another prime number, 2. A triangle is made of three sides. It is the smallest non-self-intersecting polygon and the only polygon not to have proper diagonals. When doing quick estimates, 3 is a rough approximation of π, 3.1415..., and a very rough approximation of e, 2.71828... 3 is the first Mersenne prime, as well as the second Mersenne prime exponent and the second double Mersenne prime exponent, for 7 and 127, respectively. 3 is also the first of five known Fermat primes, which include 5, 17, 257, and 65537. It is the second Fibonacci prime (and the second Lucas prime), the second Sophie Germain prime, the third Harshad number in base 10, and the second factorial prime, as it is equal to 2! + 1. 3 is the second and only prime triangular number, and Gauss proved that every integer is the sum of at most 3 triangular numbers. 3 is the number of non-collinear points needed to determine a plane, a circle, and a parabola. Three is the only prime which is one less than a perfect square. Any other number which is $n^{2}$ − 1 for some integer $n$ is not prime, since it is ($n$ − 1)($n$ + 1). This is true for 3 as well (with $n$ = 2), but in this case the smaller factor is 1. If $n$ is greater than 2, both $n$ − 1 and $n$ + 1 are greater than 1 so their product is not prime. A natural number is divisible by three if the sum of its digits in base 10 is divisible by 3. For example, the number 21 is divisible by three (3 times 7) and the sum of its digits is 2 + 1 = 3. Because of this, the reverse of any number that is divisible by three (or indeed, any permutation of its digits) is also divisible by three. For instance, 1368 and its reverse 8631 are both divisible by three (and so are 1386, 3168, 3186, 3618, etc.). See also Divisibility rule. This works in base 10 and in any positional numeral system whose base divided by three leaves a remainder of one (bases 4, 7, 10, etc.). Three of the five Platonic solids have triangular faces – the tetrahedron, the octahedron, and the icosahedron. Also, three of the five Platonic solids have vertices where three faces meet – the tetrahedron, the hexahedron (cube), and the dodecahedron. Furthermore, only three different types of polygons comprise the faces of the five Platonic solids – the triangle, the square, and the pentagon. There are only three distinct 4×4 panmagic squares. According to Pythagoras and the Pythagorean school, the number 3, which they called triad, is the noblest of all digits, as it is the only number to equal the sum of all the terms below it, and the only number whose sum with those below equals the product of them and itself.[3] There are three finite convex uniform polytope groups in three dimensions, aside from the infinite families of prisms and antiprisms: the tetrahedral group, the octahedral group, and the icosahedral group. In dimensions $n$ ⩾ 5, there are only three regular polytopes: the $n$-simplexes, $n$-cubes, and $n$-orthoplexes. In dimensions $n$ ⩾ 9, the only three uniform polytope families, aside from the numerous infinite proprismatic families, are the $\mathrm {A} _{n}$ simplex, $\mathrm {B} _{n}$ cubic, and $\mathrm {D} _{n}$ demihypercubic families. For paracompact hyperbolic honeycombs, there are three groups in dimensions 6 and 9, or equivalently of ranks 7 and 10, with no other forms in higher dimensions. Of the final three groups, the largest and most important is ${\bar {T}}_{9}$, that is associated with an important Kac–Moody Lie algebra $\mathrm {E} _{10}$.[4] The trisection of the angle was one of the three famous problems of antiquity. Numeral systems There is some evidence to suggest that early man may have used counting systems which consisted of "One, Two, Three" and thereafter "Many" to describe counting limits. Early peoples had a word to describe the quantities of one, two, and three but any quantity beyond was simply denoted as "Many". This is most likely based on the prevalence of this phenomenon among people in such disparate regions as the deep Amazon and Borneo jungles, where western civilization's explorers have historical records of their first encounters with these indigenous people.[5] List of basic calculations Multiplication 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 50 100 1000 10000 3 × x 3 6 9 12 15 18 21 24 27 30 33 36 39 42 45 48 51 54 57 60 63 66 69 72 75 150 300 3000 30000 Division 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 3 ÷ x 3 1.5 1 0.75 0.6 0.5 0.428571 0.375 0.3 0.3 0.27 0.25 0.230769 0.2142857 0.2 0.1875 0.17647058823529411 0.16 0.157894736842105263 0.15 x ÷ 3 0.3 0.6 1 1.3 1.6 2 2.3 2.6 3 3.3 3.6 4 4.3 4.6 5 5.3 5.6 6 6.3 6.6 Exponentiation 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 3x 3 9 27 81 243 729 2187 6561 19683 59049 177147 531441 1594323 4782969 14348907 43046721 129140163 387420489 1162261467 3486784401 x3 1 8 27 64 125 216 343 512 729 1000 1331 1728 2197 2744 3375 4096 4913 5832 6859 8000 Science • The Roman numeral III stands for giant star in the Yerkes spectral classification scheme. • Three is the atomic number of lithium. • Three is the ASCII code of "End of Text". • Three is the number of dimensions that humans can perceive. Humans perceive the universe to have three spatial dimensions, but some theories, such as string theory, suggest there are more. • Three is the number of elementary fermion generations according to the Standard Model of particle physics. • The triangle, a polygon with three edges and three vertices, is the most stable physical shape. For this reason it is widely utilized in construction, engineering and design.[6] • The ability of the human eye to distinguish colors is based upon the varying sensitivity of different cells in the retina to light of different wavelengths. Humans being trichromatic, the retina contains three types of color receptor cells, or cones. • There are three primary colors in the additive and subtractive models. Protoscience • In European alchemy, the three primes (Latin: tria prima) were salt (), sulfur () and mercury ().[7][8] • The three doshas (weaknesses) and their antidotes are the basis of Ayurvedic medicine in India. Pseudoscience • Three is the symbolic representation for Mu, Augustus Le Plongeon's and James Churchward's lost continent.[9] Philosophy • Philosophers such as Aquinas, Kant, Hegel, C. S. Peirce, and Karl Popper have made threefold divisions, or trichotomies, which have been important in their work. • Hegel's dialectic of Thesis + Antithesis = Synthesis creates three-ness from two-ness. Religion Many world religions contain triple deities or concepts of trinity, including: • The Hindu Trimurti • The Hindu Tridevi • The Three Jewels of Buddhism • The Three Pure Ones of Taoism • The Christian Holy Trinity • The Triple Goddess of Wicca Christianity • The threefold office of Christ is a Christian doctrine which states that Christ performs the functions of prophet, priest, and king. • The ministry of Jesus lasted approximately three years.[10] • During the Agony in the Garden, Christ asked three times for the cup to be taken from him. • Jesus rose from the dead on the third day after his death. • The devil tempted Jesus three times. • Saint Peter thrice denied Jesus and thrice affirmed his faith in Jesus. • The Magi – wise men who were astronomers/astrologers from Persia – gave Jesus three gifts.[11][12] • There are three Synoptic Gospels and three epistles of John. • Paul the Apostle went blind for three days after his conversion to Christianity. Judaism • Noah had three sons: Ham, Shem and Japheth • The Three Patriarchs: Abraham, Isaac and Jacob • The prophet Balaam beat his donkey three times. • The prophet Jonah spent three days and nights in the belly of a large fish • Three divisions of the Written Torah: Torah (Five Books of Moses), Nevi'im (Prophets), Ketuvim (Writings)[13] • Three divisions of the Jewish people: Kohen, Levite, Yisrael • Three daily prayers: Shacharit, Mincha, Maariv • Three Shabbat meals • Shabbat ends when three stars are visible in the night sky[14] • Three Pilgrimage Festivals: Passover, Shavuot, Sukkot • Three matzos on the Passover Seder table[15] • The Three Weeks, a period of mourning bridging the fast days of Seventeenth of Tammuz and Tisha B'Av • Three cardinal sins for which a Jew must die rather than transgress: idolatry, murder, sexual immorality[16] • Upsherin, a Jewish boy's first haircut at age 3[17] • A Beth din is composed of three members • Potential converts are traditionally turned away three times to test their sincerity[18] • In the Jewish mystical tradition of the Kabbalah, it is believed that the soul consists of three parts, with the highest being neshamah ("breath"), the middle being ruach ("wind" or "spirit") and the lowest being nefesh ("repose").[19] Sometimes the two elements of Chayah ("life" or "animal") and Yechidah ("unit") are additionally mentioned. • In the Kabbalah, the Tree of Life (Hebrew: Etz ha-Chayim, עץ החיים) refers to a latter 3-pillar diagrammatic representation of its central mystical symbol, known as the 10 Sephirot. Islam • The three core principles in Shia tradition: Tawhid (Oneness of God), Nabuwwa (Concept of Prophethood), Imama (Concept of Imam) Buddhism • The Triple Bodhi (ways to understand the end of birth) are Budhu, Pasebudhu, and Mahaarahath. • The Three Jewels, the three things that Buddhists take refuge in. Shinto • The Imperial Regalia of Japan of the sword, mirror, and jewel. Daoism • The Three Treasures (Chinese: 三寶; pinyin: sānbǎo; Wade–Giles: san-pao), the basic virtues in Taoism. • The Three Dantians • Three Lines of a Trigram • Three Sovereigns: Heaven Fu Xi (Hand – Head – 3º Eye), Humanity Shen Nong (Unit 69), Hell Nüwa (Foot – Abdomen – Umbiculus). Hinduism • The Trimurti: Brahma the Creator, Vishnu the Preserver, and Shiva the Destroyer. • The three Gunas found in Samkhya school of Hindu philosophy.[20] • The three paths to salvation in the Bhagavad Gita named Karma Yoga, Bhakti Yoga and Jnana Yoga. Zoroastrianism • The three virtues of Humata, Hukhta and Huvarshta (Good Thoughts, Good Words and Good Deeds) are a basic tenet in Zoroastrianism. Norse mythology Three is a very significant number in Norse mythology, along with its powers 9 and 27. • Prior to Ragnarök, there will be three hard winters without an intervening summer, the Fimbulwinter. • Odin endured three hardships upon the World Tree in his quest for the runes: he hanged himself, wounded himself with a spear, and suffered from hunger and thirst. • Bor had three sons, Odin, Vili, and Vé. Other religions • The Wiccan Rule of Three. • The Triple Goddess: Maiden, Mother, Crone; the three fates. • The sons of Cronus: Zeus, Poseidon, and Hades. • The Slavic god Triglav has three heads. Esoteric tradition • The Theosophical Society has three conditions of membership. • Gurdjieff's Three Centers and the Law of Three. • Liber AL vel Legis, the central scripture of the religion of Thelema, consists of three chapters, corresponding to three divine narrators respectively: Nuit, Hadit and Ra-Hoor-Khuit. • The Triple Greatness of Hermes Trismegistus is an important theme in Hermeticism. As a lucky or unlucky number Three (三, formal writing: 叁, pinyin sān, Cantonese: saam1) is considered a good number in Chinese culture because it sounds like the word "alive" (生 pinyin shēng, Cantonese: saang1), compared to four (四, pinyin: sì, Cantonese: sei1), which sounds like the word "death" (死 pinyin sǐ, Cantonese: sei2). Counting to three is common in situations where a group of people wish to perform an action in synchrony: Now, on the count of three, everybody pull! Assuming the counter is proceeding at a uniform rate, the first two counts are necessary to establish the rate, and the count of "three" is predicted based on the timing of the "one" and "two" before it. Three is likely used instead of some other number because it requires the minimal amount counts while setting a rate. There is another superstition that it is unlucky to take a third light, that is, to be the third person to light a cigarette from the same match or lighter. This superstition is sometimes asserted to have originated among soldiers in the trenches of the First World War when a sniper might see the first light, take aim on the second and fire on the third. The phrase "Third time's the charm" refers to the superstition that after two failures in any endeavor, a third attempt is more likely to succeed. This is also sometimes seen in reverse, as in "third man [to do something, presumably forbidden] gets caught". Luck, especially bad luck, is often said to "come in threes".[21] Sports • In American and Canadian football, a field goal is worth three points. • In association football: • For purposes of league standings, since the mid-1990s almost all leagues have awarded three points for a win. • A team that wins three trophies in a season is said to have won a treble. • A player who scores three goals in a match is said to have scored a hat-trick. • In baseball: • A batter strikes out upon the third strike in any single batting appearance. • Each team's half of an inning ends once the defense has recorded three outs (unless the home team has a walk-off hit in the ninth inning or any extra inning). • In scorekeeping, "3" denotes the first baseman. • In basketball: • Three points are awarded for a basket made from behind a designated arc on the floor. • The "3 position" is the small forward. • In bowling, three strikes bowled consecutively is known as a "turkey". • In cricket, a bowler who is credited with dismissals of batsmen on three consecutive deliveries has achieved a "hat-trick". • In Gaelic games (Gaelic football for men and women, hurling, and camogie), three points are awarded for a goal, scored when the ball passes underneath the crossbar and between the goal posts. • In ice hockey: • Scoring three goals is called a "hat trick" (usually not hyphenated in North America). • A team will typically have three forwards on the ice at any given time. • In professional wrestling, a pin is when one holds the opponent's shoulders against the mat for a count of three. • In rugby union: • A successful penalty kick for goal or drop goal is worth three points. • In the French variation of the bonus points system, a team receives a bonus point in the league standings if it wins a match while scoring at least three more tries than its opponent. • The starting tighthead prop wears the jersey number 3. • In rugby league: • One of the two starting centres wears the jersey number 3. (An exception to this rule is the Super League, which uses static squad numbering.) • A "threepeat" is a term for winning three consecutive championships. • A triathlon consists of three events: swimming, bicycling, and running. • In many sports a competitor or team is said to win a Triple Crown if they win three particularly prestigious competitions. • In volleyball, once the ball is served, teams are allowed to touch the ball three times before being required to return the ball to the other side of the court, with the definition of "touch" being slightly different between indoor and beach volleyball. Film • A number of film versions of the novel The Three Musketeers by Alexandre Dumas: (1921, 1933, 1948, 1973, 1992, 1993 and 2011). • 3 Days of the Condor (1975), starring Robert Redford, Faye Dunaway, Cliff Robertson, and Max von Sydow. • Three Amigos (1986), comedy film starring Steve Martin, Chevy Chase, and Martin Short. • Three Kings (1999), starring George Clooney, Mark Wahlberg, Ice Cube, and Spike Jonze. • 3 Days to Kill (2014), starring Kevin Costner. • Three Billboards Outside Ebbing, Missouri (2017), starring Frances McDormand, Woody Harrelson, Sam Rockwell. See also • Cube (algebra) – (3 superscript) • Third • Triad • Rule of three • List of highways numbered 3 References 1. Smith, David Eugene; Karpinski, Louis Charles (1911). The Hindu-Arabic numerals. Boston; London: Ginn and Company. pp. 27–29, 40–41. 2. Georges Ifrah, The Universal History of Numbers: From Prehistory to the Invention of the Computer transl. David Bellos et al. London: The Harvill Press (1998): 393, Fig. 24.63 3. Priya Hemenway (2005), Divine Proportion: Phi In Art, Nature, and Science, Sterling Publishing Company Inc., pp. 53–54, ISBN 1-4027-3522-7 4. Allcock, Daniel (May 2018). "Prenilpotent Pairs in the E10 root lattice" (PDF). Mathematical Proceedings of the Cambridge Philosophical Society. 164 (3): 473–483. Bibcode:2018MPCPS.164..473A. doi:10.1017/S0305004117000287. S2CID 8547735. Archived (PDF) from the original on 2022-11-03. Retrieved 2022-11-03. "The details of the previous section were E10-specific, but the same philosophy looks likely to apply to the other symmetrizable hyperbolic root systems...it seems valuable to give an outline of how the calculations would go", regarding E10 as a model example of symmetrizability of other root hyperbolic En systems. 5. Gribbin, Mary; Gribbin, John R.; Edney, Ralph; Halliday, Nicholas (2003). Big numbers. Cambridge: Wizard. ISBN 1840464313. 6. "Most stable shape- triangle". Maths in the city. Retrieved February 23, 2015. 7. Eric John Holmyard. Alchemy. 1995. p.153 8. Walter J. Friedlander. The golden wand of medicine: a history of the caduceus symbol in medicine. 1992. p.76-77 9. Churchward, James (1931). "The Lost Continent of Mu – Symbols, Vignettes, Tableaux and Diagrams". Biblioteca Pleyades. Archived from the original on 2015-07-18. Retrieved 2016-03-15. 10. "HUG 31, ff. 017r-v, inc. CF ad CE = CF ad CV". Codices Hugeniani Online. doi:10.1163/2468-0303-cohu_31-015. 11. "Encyclopaedia Britannica". Lexikon des Gesamten Buchwesens Online (in German). doi:10.1163/9789004337862_lgbo_com_050367. 12. T. E. T. (25 January 1877). "The Encyclopaedia Britannica". Nature. XV (378): 269–271. Archived from the original on 24 July 2020. Retrieved 12 July 2019. 13. Marcus, Rabbi Yossi (2015). "Why are many things in Judaism done three times?". Ask Moses. Archived from the original on 2 April 2015. Retrieved 16 March 2015. 14. "Shabbat". Judaism 101. 2011. Archived from the original on 29 June 2009. Retrieved 16 March 2015. 15. Kitov, Eliyahu (2015). "The Three Matzot". Chabad.org. Archived from the original on 24 March 2015. Retrieved 16 March 2015. 16. Kaplan, Rabbi Aryeh (28 August 2004). "Judaism and Martyrdom". Aish.com. Archived from the original on 20 March 2015. Retrieved 16 March 2015. 17. "The Basics of the Upsherin: A Boy's First Haircut". Chabad.org. 2015. Archived from the original on 22 March 2015. Retrieved 16 March 2015. 18. "The Conversion Process". Center for Conversion to Judaism. Archived from the original on 23 February 2021. Retrieved 16 March 2015. 19. Kaplan, Aryeh. "The Soul Archived 2015-02-24 at the Wayback Machine". Aish. From The Handbook of Jewish Thought (Vol. 2, Maznaim Publishing. Reprinted with permission.) September 4, 2004. Retrieved February 24, 2015. 20. James G. Lochtefeld, Guna, in The Illustrated Encyclopedia of Hinduism: A-M, Vol. 1, Rosen Publishing, ISBN 978-0-8239-3179-8, page 265 21. See "bad Archived 2009-03-02 at the Wayback Machine" in the Oxford Dictionary of Phrase and Fable, 2006, via Encyclopedia.com. • Wells, D. The Penguin Dictionary of Curious and Interesting Numbers London: Penguin Group. (1987): 46–48 External links Look up three in Wiktionary, the free dictionary. Wikimedia Commons has media related to 3 (number). • Tricyclopedic Book of Threes by Michael Eck • Threes in Human Anatomy by John A. McNulty • Grime, James. "3 is everywhere". Numberphile. Brady Haran. Archived from the original on 2013-05-14. Retrieved 2013-04-13. • The Number 3 • The Positive Integer 3 • Prime curiosities: 3 Integers 0s • 0 • 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 • 31 • 32 • 33 • 34 • 35 • 36 • 37 • 38 • 39 • 40 • 41 • 42 • 43 • 44 • 45 • 46 • 47 • 48 • 49 • 50 • 51 • 52 • 53 • 54 • 55 • 56 • 57 • 58 • 59 • 60 • 61 • 62 • 63 • 64 • 65 • 66 • 67 • 68 • 69 • 70 • 71 • 72 • 73 • 74 • 75 • 76 • 77 • 78 • 79 • 80 • 81 • 82 • 83 • 84 • 85 • 86 • 87 • 88 • 89 • 90 • 91 • 92 • 93 • 94 • 95 • 96 • 97 • 98 • 99 100s • 100 • 101 • 102 • 103 • 104 • 105 • 106 • 107 • 108 • 109 • 110 • 111 • 112 • 113 • 114 • 115 • 116 • 117 • 118 • 119 • 120 • 121 • 122 • 123 • 124 • 125 • 126 • 127 • 128 • 129 • 130 • 131 • 132 • 133 • 134 • 135 • 136 • 137 • 138 • 139 • 140 • 141 • 142 • 143 • 144 • 145 • 146 • 147 • 148 • 149 • 150 • 151 • 152 • 153 • 154 • 155 • 156 • 157 • 158 • 159 • 160 • 161 • 162 • 163 • 164 • 165 • 166 • 167 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8-cubic honeycomb The 8-cubic honeycomb or octeractic honeycomb is the only regular space-filling tessellation (or honeycomb) in Euclidean 8-space. 8-cubic honeycomb (no image) TypeRegular 8-honeycomb Uniform 8-honeycomb FamilyHypercube honeycomb Schläfli symbol{4,36,4} {4,35,31,1} t0,8{4,36,4} {∞}(8) Coxeter-Dynkin diagrams 8-face type{4,36} 7-face type{4,35} 6-face type{4,34} 5-face type{4,33} 4-face type{4,32} Cell type{4,3} Face type{4} Face figure{4,3} (octahedron) Edge figure8 {4,3,3} (16-cell) Vertex figure256 {4,36} (8-orthoplex) Coxeter group[4,36,4] Dualself-dual Propertiesvertex-transitive, edge-transitive, face-transitive, cell-transitive It is analogous to the square tiling of the plane and to the cubic honeycomb of 3-space, and the tesseractic honeycomb of 4-space. There are many different Wythoff constructions of this honeycomb. The most symmetric form is regular, with Schläfli symbol {4,36,4}. Another form has two alternating hypercube facets (like a checkerboard) with Schläfli symbol {4,35,31,1}. The lowest symmetry Wythoff construction has 256 types of facets around each vertex and a prismatic product Schläfli symbol {∞}(8). Related honeycombs The [4,36,4], , Coxeter group generates 511 permutations of uniform tessellations, 271 with unique symmetry and 270 with unique geometry. The expanded 8-cubic honeycomb is geometrically identical to the 8-cubic honeycomb. The 8-cubic honeycomb can be alternated into the 8-demicubic honeycomb, replacing the 8-cubes with 8-demicubes, and the alternated gaps are filled by 8-orthoplex facets. Quadrirectified 8-cubic honeycomb A quadrirectified 8-cubic honeycomb, , contains all trirectified 8-orthoplex facets and is the Voronoi tessellation of the D8* lattice. Facets can be identically colored from a doubled ${\tilde {C}}_{8}$×2, [[4,36,4]] symmetry, alternately colored from ${\tilde {C}}_{8}$, [4,36,4] symmetry, three colors from ${\tilde {B}}_{8}$, [4,35,31,1] symmetry, and 4 colors from ${\tilde {D}}_{8}$, [31,1,34,31,1] symmetry. See also • List of regular polytopes References • Coxeter, H.S.M. Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8 p. 296, Table II: Regular honeycombs • Kaleidoscopes: Selected Writings of H. S. M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 • (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45] Fundamental convex regular and uniform honeycombs in dimensions 2–9 Space Family ${\tilde {A}}_{n-1}$ ${\tilde {C}}_{n-1}$ ${\tilde {B}}_{n-1}$ ${\tilde {D}}_{n-1}$ ${\tilde {G}}_{2}$ / ${\tilde {F}}_{4}$ / ${\tilde {E}}_{n-1}$ E2 Uniform tiling {3[3]} δ3 hδ3 qδ3 Hexagonal E3 Uniform convex honeycomb {3[4]} δ4 hδ4 qδ4 E4 Uniform 4-honeycomb {3[5]} δ5 hδ5 qδ5 24-cell honeycomb E5 Uniform 5-honeycomb {3[6]} δ6 hδ6 qδ6 E6 Uniform 6-honeycomb {3[7]} δ7 hδ7 qδ7 222 E7 Uniform 7-honeycomb {3[8]} δ8 hδ8 qδ8 133 • 331 E8 Uniform 8-honeycomb {3[9]} δ9 hδ9 qδ9 152 • 251 • 521 E9 Uniform 9-honeycomb {3[10]} δ10 hδ10 qδ10 E10 Uniform 10-honeycomb {3[11]} δ11 hδ11 qδ11 En-1 Uniform (n-1)-honeycomb {3[n]} δn hδn qδn 1k2 • 2k1 • k21	Wikipedia
Chaos theory Chaos theory is an interdisciplinary area of scientific study and branch of mathematics focused on underlying patterns and deterministic laws of dynamical systems that are highly sensitive to initial conditions, and were once thought to have completely random states of disorder and irregularities.[1] Chaos theory states that within the apparent randomness of chaotic complex systems, there are underlying patterns, interconnection, constant feedback loops, repetition, self-similarity, fractals, and self-organization.[2] The butterfly effect, an underlying principle of chaos, describes how a small change in one state of a deterministic nonlinear system can result in large differences in a later state (meaning that there is sensitive dependence on initial conditions).[3] A metaphor for this behavior is that a butterfly flapping its wings in Texas can cause a tornado in Brazil.[4][5][6] Small differences in initial conditions, such as those due to errors in measurements or due to rounding errors in numerical computation, can yield widely diverging outcomes for such dynamical systems, rendering long-term prediction of their behavior impossible in general.[7] This can happen even though these systems are deterministic, meaning that their future behavior follows a unique evolution[8] and is fully determined by their initial conditions, with no random elements involved.[9] In other words, the deterministic nature of these systems does not make them predictable.[10][11] This behavior is known as deterministic chaos, or simply chaos. The theory was summarized by Edward Lorenz as:[12] Chaos: When the present determines the future, but the approximate present does not approximately determine the future. Chaotic behavior exists in many natural systems, including fluid flow, heartbeat irregularities, weather, and climate.[13][14][8] It also occurs spontaneously in some systems with artificial components, such as road traffic.[2] This behavior can be studied through the analysis of a chaotic mathematical model, or through analytical techniques such as recurrence plots and Poincaré maps. Chaos theory has applications in a variety of disciplines, including meteorology,[8] anthropology,[15] sociology, environmental science, computer science, engineering, economics, ecology, and pandemic crisis management.[16][17] The theory formed the basis for such fields of study as complex dynamical systems, edge of chaos theory, and self-assembly processes. Introduction Chaos theory concerns deterministic systems whose behavior can, in principle, be predicted. Chaotic systems are predictable for a while and then 'appear' to become random. The amount of time for which the behavior of a chaotic system can be effectively predicted depends on three things: how much uncertainty can be tolerated in the forecast, how accurately its current state can be measured, and a time scale depending on the dynamics of the system, called the Lyapunov time. Some examples of Lyapunov times are: chaotic electrical circuits, about 1 millisecond; weather systems, a few days (unproven); the inner solar system, 4 to 5 million years.[18] In chaotic systems, the uncertainty in a forecast increases exponentially with elapsed time. Hence, mathematically, doubling the forecast time more than squares the proportional uncertainty in the forecast. This means, in practice, a meaningful prediction cannot be made over an interval of more than two or three times the Lyapunov time. When meaningful predictions cannot be made, the system appears random.[19] Chaos theory is a method of qualitative and quantitative analysis to investigate the behavior of dynamic systems that cannot be explained and predicted by single data relationships, but must be explained and predicted by whole, continuous data relationships. Chaotic dynamics In common usage, "chaos" means "a state of disorder".[20][21] However, in chaos theory, the term is defined more precisely. Although no universally accepted mathematical definition of chaos exists, a commonly used definition, originally formulated by Robert L. Devaney, says that to classify a dynamical system as chaotic, it must have these properties:[22] 1. it must be sensitive to initial conditions, 2. it must be topologically transitive, 3. it must have dense periodic orbits. In some cases, the last two properties above have been shown to actually imply sensitivity to initial conditions.[23][24] In the discrete-time case, this is true for all continuous maps on metric spaces.[25] In these cases, while it is often the most practically significant property, "sensitivity to initial conditions" need not be stated in the definition. If attention is restricted to intervals, the second property implies the other two.[26] An alternative and a generally weaker definition of chaos uses only the first two properties in the above list.[27] Sensitivity to initial conditions Main article: Butterfly effect Sensitivity to initial conditions means that each point in a chaotic system is arbitrarily closely approximated by other points that have significantly different future paths or trajectories. Thus, an arbitrarily small change or perturbation of the current trajectory may lead to significantly different future behavior.[2] Sensitivity to initial conditions is popularly known as the "butterfly effect", so-called because of the title of a paper given by Edward Lorenz in 1972 to the American Association for the Advancement of Science in Washington, D.C., entitled Predictability: Does the Flap of a Butterfly's Wings in Brazil set off a Tornado in Texas?.[28] The flapping wing represents a small change in the initial condition of the system, which causes a chain of events that prevents the predictability of large-scale phenomena. Had the butterfly not flapped its wings, the trajectory of the overall system could have been vastly different. As suggested in Lorenz's book entitled The Essence of Chaos, published in 1993,[5] "sensitive dependence can serve as an acceptable definition of chaos". In the same book, Lorenz defined the butterfly effect as: "The phenomenon that a small alteration in the state of a dynamical system will cause subsequent states to differ greatly from the states that would have followed without the alteration." The above definition is consistent with the sensitive dependence of solutions on initial conditions (SDIC). An idealized skiing model was developed to illustrate the sensitivity of time-varying paths to initial positions.[5] A predictability horizon can be determined before the onset of SDIC (i.e., prior to significant separations of initial nearby trajectories).[29] A consequence of sensitivity to initial conditions is that if we start with a limited amount of information about the system (as is usually the case in practice), then beyond a certain time, the system would no longer be predictable. This is most prevalent in the case of weather, which is generally predictable only about a week ahead.[30] This does not mean that one cannot assert anything about events far in the future—only that some restrictions on the system are present. For example, we know that the temperature of the surface of the earth will not naturally reach 100 °C (212 °F) or fall below −130 °C (−202 °F) on earth (during the current geologic era), but we cannot predict exactly which day will have the hottest temperature of the year. In more mathematical terms, the Lyapunov exponent measures the sensitivity to initial conditions, in the form of rate of exponential divergence from the perturbed initial conditions.[31] More specifically, given two starting trajectories in the phase space that are infinitesimally close, with initial separation $\delta \mathbf {Z} _{0}$, the two trajectories end up diverging at a rate given by $\|\delta \mathbf {Z} (t)\|\approx e^{\lambda t}\|\delta \mathbf {Z} _{0}\|,$ where $t$ is the time and $\lambda $ is the Lyapunov exponent. The rate of separation depends on the orientation of the initial separation vector, so a whole spectrum of Lyapunov exponents can exist. The number of Lyapunov exponents is equal to the number of dimensions of the phase space, though it is common to just refer to the largest one. For example, the maximal Lyapunov exponent (MLE) is most often used, because it determines the overall predictability of the system. A positive MLE is usually taken as an indication that the system is chaotic.[8] In addition to the above property, other properties related to sensitivity of initial conditions also exist. These include, for example, measure-theoretical mixing (as discussed in ergodic theory) and properties of a K-system.[11] Non-periodicity A chaotic system may have sequences of values for the evolving variable that exactly repeat themselves, giving periodic behavior starting from any point in that sequence. However, such periodic sequences are repelling rather than attracting, meaning that if the evolving variable is outside the sequence, however close, it will not enter the sequence and in fact, will diverge from it. Thus for almost all initial conditions, the variable evolves chaotically with non-periodic behavior. Topological mixing Topological mixing (or the weaker condition of topological transitivity) means that the system evolves over time so that any given region or open set of its phase space eventually overlaps with any other given region. This mathematical concept of "mixing" corresponds to the standard intuition, and the mixing of colored dyes or fluids is an example of a chaotic system. Topological mixing is often omitted from popular accounts of chaos, which equate chaos with only sensitivity to initial conditions. However, sensitive dependence on initial conditions alone does not give chaos. For example, consider the simple dynamical system produced by repeatedly doubling an initial value. This system has sensitive dependence on initial conditions everywhere, since any pair of nearby points eventually becomes widely separated. However, this example has no topological mixing, and therefore has no chaos. Indeed, it has extremely simple behavior: all points except 0 tend to positive or negative infinity. Topological transitivity A map $f:X\to X$ is said to be topologically transitive if for any pair of non-empty open sets $U,V\subset X$, there exists $k>0$ such that $f^{k}(U)\cap V\neq \emptyset $. Topological transitivity is a weaker version of topological mixing. Intuitively, if a map is topologically transitive then given a point x and a region V, there exists a point y near x whose orbit passes through V. This implies that it is impossible to decompose the system into two open sets.[32] An important related theorem is the Birkhoff Transitivity Theorem. It is easy to see that the existence of a dense orbit implies topological transitivity. The Birkhoff Transitivity Theorem states that if X is a second countable, complete metric space, then topological transitivity implies the existence of a dense set of points in X that have dense orbits.[33] Density of periodic orbits For a chaotic system to have dense periodic orbits means that every point in the space is approached arbitrarily closely by periodic orbits.[32] The one-dimensional logistic map defined by x → 4 x (1 – x) is one of the simplest systems with density of periodic orbits. For example, ${\tfrac {5-{\sqrt {5}}}{8}}$ → ${\tfrac {5+{\sqrt {5}}}{8}}$ → ${\tfrac {5-{\sqrt {5}}}{8}}$ (or approximately 0.3454915 → 0.9045085 → 0.3454915) is an (unstable) orbit of period 2, and similar orbits exist for periods 4, 8, 16, etc. (indeed, for all the periods specified by Sharkovskii's theorem).[34] Sharkovskii's theorem is the basis of the Li and Yorke[35] (1975) proof that any continuous one-dimensional system that exhibits a regular cycle of period three will also display regular cycles of every other length, as well as completely chaotic orbits. Strange attractors Some dynamical systems, like the one-dimensional logistic map defined by x → 4 x (1 – x), are chaotic everywhere, but in many cases chaotic behavior is found only in a subset of phase space. The cases of most interest arise when the chaotic behavior takes place on an attractor, since then a large set of initial conditions leads to orbits that converge to this chaotic region.[36] An easy way to visualize a chaotic attractor is to start with a point in the basin of attraction of the attractor, and then simply plot its subsequent orbit. Because of the topological transitivity condition, this is likely to produce a picture of the entire final attractor, and indeed both orbits shown in the figure on the right give a picture of the general shape of the Lorenz attractor. This attractor results from a simple three-dimensional model of the Lorenz weather system. The Lorenz attractor is perhaps one of the best-known chaotic system diagrams, probably because it is not only one of the first, but it is also one of the most complex, and as such gives rise to a very interesting pattern that, with a little imagination, looks like the wings of a butterfly. Unlike fixed-point attractors and limit cycles, the attractors that arise from chaotic systems, known as strange attractors, have great detail and complexity. Strange attractors occur in both continuous dynamical systems (such as the Lorenz system) and in some discrete systems (such as the Hénon map). Other discrete dynamical systems have a repelling structure called a Julia set, which forms at the boundary between basins of attraction of fixed points. Julia sets can be thought of as strange repellers. Both strange attractors and Julia sets typically have a fractal structure, and the fractal dimension can be calculated for them. Coexisting attractors In contrast to single type chaotic solutions, recent studies using Lorenz models [40][41] have emphasized the importance of considering various types of solutions. For example, coexisting chaotic and non-chaotic may appear within the same model (e.g., the double pendulum system) using the same modeling configurations but different initial conditions. The findings of attractor coexistence, obtained from classical and generalized Lorenz models,[37][38][39] suggested a revised view that "the entirety of weather possesses a dual nature of chaos and order with distinct predictability", in contrast to the conventional view of "weather is chaotic". Minimum complexity of a chaotic system Discrete chaotic systems, such as the logistic map, can exhibit strange attractors whatever their dimensionality. Universality of one-dimensional maps with parabolic maxima and Feigenbaum constants $\delta =4.669201...$,$\alpha =2.502907...$[42][43] is well visible with Tahn map proposed as a toy model for discrete laser dynamics: $x\rightarrow Gx(1-\mathrm {tanh} (x))$, where $x$ stands for electric field amplitude, $G$[44] is laser gain as bifurcation parameter. The gradual increase of $G$ at interval $[0,\infty )$ changes dynamics from regular to chaotic one[45] with qualitatively the same bifurcation diagram as those for logistic map. In contrast, for continuous dynamical systems, the Poincaré–Bendixson theorem shows that a strange attractor can only arise in three or more dimensions. Finite-dimensional linear systems are never chaotic; for a dynamical system to display chaotic behavior, it must be either nonlinear or infinite-dimensional. The Poincaré–Bendixson theorem states that a two-dimensional differential equation has very regular behavior. The Lorenz attractor discussed below is generated by a system of three differential equations such as: ${\begin{aligned}{\frac {\mathrm {d} x}{\mathrm {d} t}}&=\sigma y-\sigma x,\\{\frac {\mathrm {d} y}{\mathrm {d} t}}&=\rho x-xz-y,\\{\frac {\mathrm {d} z}{\mathrm {d} t}}&=xy-\beta z.\end{aligned}}$ where $x$, $y$, and $z$ make up the system state, $t$ is time, and $\sigma $, $\rho $, $\beta $ are the system parameters. Five of the terms on the right hand side are linear, while two are quadratic; a total of seven terms. Another well-known chaotic attractor is generated by the Rössler equations, which have only one nonlinear term out of seven. Sprott[46] found a three-dimensional system with just five terms, that had only one nonlinear term, which exhibits chaos for certain parameter values. Zhang and Heidel[47][48] showed that, at least for dissipative and conservative quadratic systems, three-dimensional quadratic systems with only three or four terms on the right-hand side cannot exhibit chaotic behavior. The reason is, simply put, that solutions to such systems are asymptotic to a two-dimensional surface and therefore solutions are well behaved. While the Poincaré–Bendixson theorem shows that a continuous dynamical system on the Euclidean plane cannot be chaotic, two-dimensional continuous systems with non-Euclidean geometry can exhibit chaotic behavior.[49] Perhaps surprisingly, chaos may occur also in linear systems, provided they are infinite dimensional.[50] A theory of linear chaos is being developed in a branch of mathematical analysis known as functional analysis. The above elegant set of three ordinary differential equations has been referred to as the three-dimensional Lorenz model.[51] Since 1963, higher-dimensional Lorenz models have been developed in numerous studies[52][53][37][38] for examining the impact of an increased degree of nonlinearity, as well as its collective effect with heating and dissipations, on solution stability. Infinite dimensional maps The straightforward generalization of coupled discrete maps[54] is based upon convolution integral which mediates interaction between spatially distributed maps: $\psi _{n+1}({\vec {r}},t)=\int K({\vec {r}}-{\vec {r}}^{,},t)f[\psi _{n}({\vec {r}}^{,},t)]d{\vec {r}}^{,}$, where kernel $K({\vec {r}}-{\vec {r}}^{,},t)$ is propagator derived as Green function of a relevant physical system,[55] $f[\psi _{n}({\vec {r}},t)]$ might be logistic map alike $\psi \rightarrow G\psi [1-\tanh(\psi )]$ or complex map. For examples of complex maps the Julia set $f[\psi ]=\psi ^{2}$ or Ikeda map $\psi _{n+1}=A+B\psi _{n}e^{i(\|\psi _{n}\|^{2}+C)}$ may serve. When wave propagation problems at distance $L=ct$ with wavelength $\lambda =2\pi /k$ are considered the kernel $K$ may have a form of Green function for Schrödinger equation:.[56][57] $K({\vec {r}}-{\vec {r}}^{,},L)={\frac {ik\exp[ikL]}{2\pi L}}\exp[{\frac {ik\|{\vec {r}}-{\vec {r}}^{,}\|^{2}}{2L}}]$. Jerk systems In physics, jerk is the third derivative of position, with respect to time. As such, differential equations of the form $J\left({\overset {...}{x}},{\ddot {x}},{\dot {x}},x\right)=0$ are sometimes called jerk equations. It has been shown that a jerk equation, which is equivalent to a system of three first order, ordinary, non-linear differential equations, is in a certain sense the minimal setting for solutions showing chaotic behavior. This motivates mathematical interest in jerk systems. Systems involving a fourth or higher derivative are called accordingly hyperjerk systems.[58] A jerk system's behavior is described by a jerk equation, and for certain jerk equations, simple electronic circuits can model solutions. These circuits are known as jerk circuits. One of the most interesting properties of jerk circuits is the possibility of chaotic behavior. In fact, certain well-known chaotic systems, such as the Lorenz attractor and the Rössler map, are conventionally described as a system of three first-order differential equations that can combine into a single (although rather complicated) jerk equation. Another example of a jerk equation with nonlinearity in the magnitude of $x$ is: ${\frac {\mathrm {d} ^{3}x}{\mathrm {d} t^{3}}}+A{\frac {\mathrm {d} ^{2}x}{\mathrm {d} t^{2}}}+{\frac {\mathrm {d} x}{\mathrm {d} t}}-\|x\|+1=0.$ Here, A is an adjustable parameter. This equation has a chaotic solution for A=3/5 and can be implemented with the following jerk circuit; the required nonlinearity is brought about by the two diodes: In the above circuit, all resistors are of equal value, except $R_{A}=R/A=5R/3$, and all capacitors are of equal size. The dominant frequency is $1/2\pi RC$. The output of op amp 0 will correspond to the x variable, the output of 1 corresponds to the first derivative of x and the output of 2 corresponds to the second derivative. Similar circuits only require one diode[59] or no diodes at all.[60] See also the well-known Chua's circuit, one basis for chaotic true random number generators.[61] The ease of construction of the circuit has made it a ubiquitous real-world example of a chaotic system. Spontaneous order Under the right conditions, chaos spontaneously evolves into a lockstep pattern. In the Kuramoto model, four conditions suffice to produce synchronization in a chaotic system. Examples include the coupled oscillation of Christiaan Huygens' pendulums, fireflies, neurons, the London Millennium Bridge resonance, and large arrays of Josephson junctions.[62] History An early proponent of chaos theory was Henri Poincaré. In the 1880s, while studying the three-body problem, he found that there can be orbits that are nonperiodic, and yet not forever increasing nor approaching a fixed point.[63][64][65] In 1898, Jacques Hadamard published an influential study of the chaotic motion of a free particle gliding frictionlessly on a surface of constant negative curvature, called "Hadamard's billiards".[66] Hadamard was able to show that all trajectories are unstable, in that all particle trajectories diverge exponentially from one another, with a positive Lyapunov exponent. Chaos theory began in the field of ergodic theory. Later studies, also on the topic of nonlinear differential equations, were carried out by George David Birkhoff,[67] Andrey Nikolaevich Kolmogorov,[68][69][70] Mary Lucy Cartwright and John Edensor Littlewood,[71] and Stephen Smale.[72] Except for Smale, these studies were all directly inspired by physics: the three-body problem in the case of Birkhoff, turbulence and astronomical problems in the case of Kolmogorov, and radio engineering in the case of Cartwright and Littlewood. Although chaotic planetary motion had not been observed, experimentalists had encountered turbulence in fluid motion and nonperiodic oscillation in radio circuits without the benefit of a theory to explain what they were seeing. Despite initial insights in the first half of the twentieth century, chaos theory became formalized as such only after mid-century, when it first became evident to some scientists that linear theory, the prevailing system theory at that time, simply could not explain the observed behavior of certain experiments like that of the logistic map. What had been attributed to measure imprecision and simple "noise" was considered by chaos theorists as a full component of the studied systems. In 1959 Boris Valerianovich Chirikov proposed a criterion for the emergence of classical chaos in Hamiltonian systems (Chirikov criterion). He applied this criterion to explain some experimental results on plasma confinement in open mirror traps.[73][74] This is regarded as the very first physical theory of chaos, which succeeded in explaining a concrete experiment. And Boris Chirikov himself is considered as a pioneer in classical and quantum chaos.[75][76][77] The main catalyst for the development of chaos theory was the electronic computer. Much of the mathematics of chaos theory involves the repeated iteration of simple mathematical formulas, which would be impractical to do by hand. Electronic computers made these repeated calculations practical, while figures and images made it possible to visualize these systems. As a graduate student in Chihiro Hayashi's laboratory at Kyoto University, Yoshisuke Ueda was experimenting with analog computers and noticed, on November 27, 1961, what he called "randomly transitional phenomena". Yet his advisor did not agree with his conclusions at the time, and did not allow him to report his findings until 1970.[78][79] Edward Lorenz was an early pioneer of the theory. His interest in chaos came about accidentally through his work on weather prediction in 1961.[13] Lorenz and his collaborator Ellen Fetter[80] were using a simple digital computer, a Royal McBee LGP-30, to run weather simulations. They wanted to see a sequence of data again, and to save time they started the simulation in the middle of its course. They did this by entering a printout of the data that corresponded to conditions in the middle of the original simulation. To their surprise, the weather the machine began to predict was completely different from the previous calculation. They tracked this down to the computer printout. The computer worked with 6-digit precision, but the printout rounded variables off to a 3-digit number, so a value like 0.506127 printed as 0.506. This difference is tiny, and the consensus at the time would have been that it should have no practical effect. However, Lorenz discovered that small changes in initial conditions produced large changes in long-term outcome.[81] Lorenz's discovery, which gave its name to Lorenz attractors, showed that even detailed atmospheric modeling cannot, in general, make precise long-term weather predictions. In 1963, Benoit Mandelbrot, studying information theory, discovered that noise in many phenomena (including stock prices and telephone circuits) was patterned like a Cantor set, a set of points with infinite roughness and detail [82] Mandelbrot described both the "Noah effect" (in which sudden discontinuous changes can occur) and the "Joseph effect" (in which persistence of a value can occur for a while, yet suddenly change afterwards).[83][84] In 1967, he published "How long is the coast of Britain? Statistical self-similarity and fractional dimension", showing that a coastline's length varies with the scale of the measuring instrument, resembles itself at all scales, and is infinite in length for an infinitesimally small measuring device.[85] Arguing that a ball of twine appears as a point when viewed from far away (0-dimensional), a ball when viewed from fairly near (3-dimensional), or a curved strand (1-dimensional), he argued that the dimensions of an object are relative to the observer and may be fractional. An object whose irregularity is constant over different scales ("self-similarity") is a fractal (examples include the Menger sponge, the Sierpiński gasket, and the Koch curve or snowflake, which is infinitely long yet encloses a finite space and has a fractal dimension of circa 1.2619). In 1982, Mandelbrot published The Fractal Geometry of Nature, which became a classic of chaos theory.[86] In December 1977, the New York Academy of Sciences organized the first symposium on chaos, attended by David Ruelle, Robert May, James A. Yorke (coiner of the term "chaos" as used in mathematics), Robert Shaw, and the meteorologist Edward Lorenz. The following year Pierre Coullet and Charles Tresser published "Itérations d'endomorphismes et groupe de renormalisation", and Mitchell Feigenbaum's article "Quantitative Universality for a Class of Nonlinear Transformations" finally appeared in a journal, after 3 years of referee rejections.[43][87] Thus Feigenbaum (1975) and Coullet & Tresser (1978) discovered the universality in chaos, permitting the application of chaos theory to many different phenomena. In 1979, Albert J. Libchaber, during a symposium organized in Aspen by Pierre Hohenberg, presented his experimental observation of the bifurcation cascade that leads to chaos and turbulence in Rayleigh–Bénard convection systems. He was awarded the Wolf Prize in Physics in 1986 along with Mitchell J. Feigenbaum for their inspiring achievements.[88] In 1986, the New York Academy of Sciences co-organized with the National Institute of Mental Health and the Office of Naval Research the first important conference on chaos in biology and medicine. There, Bernardo Huberman presented a mathematical model of the eye tracking dysfunction among people with schizophrenia.[89] This led to a renewal of physiology in the 1980s through the application of chaos theory, for example, in the study of pathological cardiac cycles. In 1987, Per Bak, Chao Tang and Kurt Wiesenfeld published a paper in Physical Review Letters[90] describing for the first time self-organized criticality (SOC), considered one of the mechanisms by which complexity arises in nature. Alongside largely lab-based approaches such as the Bak–Tang–Wiesenfeld sandpile, many other investigations have focused on large-scale natural or social systems that are known (or suspected) to display scale-invariant behavior. Although these approaches were not always welcomed (at least initially) by specialists in the subjects examined, SOC has nevertheless become established as a strong candidate for explaining a number of natural phenomena, including earthquakes, (which, long before SOC was discovered, were known as a source of scale-invariant behavior such as the Gutenberg–Richter law describing the statistical distribution of earthquake sizes, and the Omori law[91] describing the frequency of aftershocks), solar flares, fluctuations in economic systems such as financial markets (references to SOC are common in econophysics), landscape formation, forest fires, landslides, epidemics, and biological evolution (where SOC has been invoked, for example, as the dynamical mechanism behind the theory of "punctuated equilibria" put forward by Niles Eldredge and Stephen Jay Gould). Given the implications of a scale-free distribution of event sizes, some researchers have suggested that another phenomenon that should be considered an example of SOC is the occurrence of wars. These investigations of SOC have included both attempts at modelling (either developing new models or adapting existing ones to the specifics of a given natural system), and extensive data analysis to determine the existence and/or characteristics of natural scaling laws. In the same year, James Gleick published Chaos: Making a New Science, which became a best-seller and introduced the general principles of chaos theory as well as its history to the broad public.[92] Initially the domain of a few, isolated individuals, chaos theory progressively emerged as a transdisciplinary and institutional discipline, mainly under the name of nonlinear systems analysis. Alluding to Thomas Kuhn's concept of a paradigm shift exposed in The Structure of Scientific Revolutions (1962), many "chaologists" (as some described themselves) claimed that this new theory was an example of such a shift, a thesis upheld by Gleick. The availability of cheaper, more powerful computers broadens the applicability of chaos theory. Currently, chaos theory remains an active area of research,[93] involving many different disciplines such as mathematics, topology, physics,[94] social systems,[95] population modeling, biology, meteorology, astrophysics, information theory, computational neuroscience, pandemic crisis management,[16][17] etc. Lorenz's pioneering contributions to chaotic modeling Throughout his career, Professor Lorenz authored a total of 61 research papers, out of which 58 were solely authored by him.[96] Commencing with the 1960 conference in Japan, Lorenz embarked on a journey of developing diverse models aimed at uncovering the SDIC and chaotic features. A recent review of Lorenz's model [97]progression spanning from 1960 to 2008 revealed his adeptness at employing varied physical systems to illustrate chaotic phenomena. These systems encompassed Quasi-geostrophic systems, the Conservative Vorticity Equation, the Rayleigh-Bénard Convection Equations, and the Shallow Water Equations. Moreover, Lorenz can be credited with the early application of the logistic map to explore chaotic solutions, a milestone he achieved ahead of his colleagues (e.g. Lorenz 1964[98] ). A popular but inaccurate analogy for chaos The sensitive dependence on initial conditions (i.e., butterfly effect) has been illustrated using the following folklore:[92] For want of a nail, the shoe was lost. For want of a shoe, the horse was lost. For want of a horse, the rider was lost. For want of a rider, the battle was lost. For want of a battle, the kingdom was lost. And all for the want of a horseshoe nail. Based on the above, many people mistakenly believe that the impact of a tiny initial perturbation monotonically increases with time and that any tiny perturbation can eventually produce a large impact on numerical integrations. However, in 2008, Lorenz stated that he did not feel that this verse described true chaos but that it better illustrated the simpler phenomenon of instability and that the verse implicitly suggests that subsequent small events will not reverse the outcome (Lorenz, 2008 [99]). Based on the analysis, the verse only indicates divergence, not boundedness.[6] Boundedness is important for the finite size of a butterfly pattern.[6][99][100] In a recent study,[101] the characteristic of the aforementioned verse was recently denoted as "finite-time sensitive dependence". Applications Although chaos theory was born from observing weather patterns, it has become applicable to a variety of other situations. Some areas benefiting from chaos theory today are geology, mathematics, biology, computer science, economics,[103][104][105] engineering,[106][107] finance,[108][109][110][111][112] meteorology, philosophy, anthropology,[15] physics,[113][114][115] politics,[116][117] population dynamics,[118] and robotics. A few categories are listed below with examples, but this is by no means a comprehensive list as new applications are appearing. Cryptography Chaos theory has been used for many years in cryptography. In the past few decades, chaos and nonlinear dynamics have been used in the design of hundreds of cryptographic primitives. These algorithms include image encryption algorithms, hash functions, secure pseudo-random number generators, stream ciphers, watermarking, and steganography.[119] The majority of these algorithms are based on uni-modal chaotic maps and a big portion of these algorithms use the control parameters and the initial condition of the chaotic maps as their keys.[120] From a wider perspective, without loss of generality, the similarities between the chaotic maps and the cryptographic systems is the main motivation for the design of chaos based cryptographic algorithms.[119] One type of encryption, secret key or symmetric key, relies on diffusion and confusion, which is modeled well by chaos theory.[121] Another type of computing, DNA computing, when paired with chaos theory, offers a way to encrypt images and other information.[122] Many of the DNA-Chaos cryptographic algorithms are proven to be either not secure, or the technique applied is suggested to be not efficient.[123][124][125] Robotics Robotics is another area that has recently benefited from chaos theory. Instead of robots acting in a trial-and-error type of refinement to interact with their environment, chaos theory has been used to build a predictive model.[126] Chaotic dynamics have been exhibited by passive walking biped robots.[127] Biology For over a hundred years, biologists have been keeping track of populations of different species with population models. Most models are continuous, but recently scientists have been able to implement chaotic models in certain populations.[128] For example, a study on models of Canadian lynx showed there was chaotic behavior in the population growth.[129] Chaos can also be found in ecological systems, such as hydrology. While a chaotic model for hydrology has its shortcomings, there is still much to learn from looking at the data through the lens of chaos theory.[130] Another biological application is found in cardiotocography. Fetal surveillance is a delicate balance of obtaining accurate information while being as noninvasive as possible. Better models of warning signs of fetal hypoxia can be obtained through chaotic modeling.[131] As Perry points out, modeling of chaotic time series in ecology is helped by constraint.[132]: 176, 177 There is always potential difficulty in distinguishing real chaos from chaos that is only in the model.[132]: 176, 177 Hence both constraint in the model and or duplicate time series data for comparison will be helpful in constraining the model to something close to the reality, for example Perry & Wall 1984.[132]: 176, 177 Gene-for-gene co-evolution sometimes shows chaotic dynamics in allele frequencies.[133] Adding variables exaggerates this: Chaos is more common in models incorporating additional variables to reflect additional facets of real populations.[133] Robert M. May himself did some of these foundational crop co-evolution studies, and this in turn helped shape the entire field.[133] Even for a steady environment, merely combining one crop and one pathogen may result in quasi-periodic- or chaotic- oscillations in pathogen population.[134]: 169 Economics It is possible that economic models can also be improved through an application of chaos theory, but predicting the health of an economic system and what factors influence it most is an extremely complex task.[135] Economic and financial systems are fundamentally different from those in the classical natural sciences since the former are inherently stochastic in nature, as they result from the interactions of people, and thus pure deterministic models are unlikely to provide accurate representations of the data. The empirical literature that tests for chaos in economics and finance presents very mixed results, in part due to confusion between specific tests for chaos and more general tests for non-linear relationships.[136] Chaos could be found in economics by the means of recurrence quantification analysis. In fact, Orlando et al.[137] by the means of the so-called recurrence quantification correlation index were able detect hidden changes in time series. Then, the same technique was employed to detect transitions from laminar (regular) to turbulent (chaotic) phases as well as differences between macroeconomic variables and highlight hidden features of economic dynamics.[138] Finally, chaos theory could help in modeling how an economy operates as well as in embedding shocks due to external events such as COVID-19.[139] Other areas In chemistry, predicting gas solubility is essential to manufacturing polymers, but models using particle swarm optimization (PSO) tend to converge to the wrong points. An improved version of PSO has been created by introducing chaos, which keeps the simulations from getting stuck.[140] In celestial mechanics, especially when observing asteroids, applying chaos theory leads to better predictions about when these objects will approach Earth and other planets.[141] Four of the five moons of Pluto rotate chaotically. In quantum physics and electrical engineering, the study of large arrays of Josephson junctions benefitted greatly from chaos theory.[142] Closer to home, coal mines have always been dangerous places where frequent natural gas leaks cause many deaths. Until recently, there was no reliable way to predict when they would occur. But these gas leaks have chaotic tendencies that, when properly modeled, can be predicted fairly accurately.[143] Chaos theory can be applied outside of the natural sciences, but historically nearly all such studies have suffered from lack of reproducibility; poor external validity; and/or inattention to cross-validation, resulting in poor predictive accuracy (if out-of-sample prediction has even been attempted). Glass[144] and Mandell and Selz[145] have found that no EEG study has as yet indicated the presence of strange attractors or other signs of chaotic behavior. Researchers have continued to apply chaos theory to psychology. For example, in modeling group behavior in which heterogeneous members may behave as if sharing to different degrees what in Wilfred Bion's theory is a basic assumption, researchers have found that the group dynamic is the result of the individual dynamics of the members: each individual reproduces the group dynamics in a different scale, and the chaotic behavior of the group is reflected in each member.[146] Redington and Reidbord (1992) attempted to demonstrate that the human heart could display chaotic traits. They monitored the changes in between-heartbeat intervals for a single psychotherapy patient as she moved through periods of varying emotional intensity during a therapy session. Results were admittedly inconclusive. Not only were there ambiguities in the various plots the authors produced to purportedly show evidence of chaotic dynamics (spectral analysis, phase trajectory, and autocorrelation plots), but also when they attempted to compute a Lyapunov exponent as more definitive confirmation of chaotic behavior, the authors found they could not reliably do so.[147] In their 1995 paper, Metcalf and Allen[148] maintained that they uncovered in animal behavior a pattern of period doubling leading to chaos. The authors examined a well-known response called schedule-induced polydipsia, by which an animal deprived of food for certain lengths of time will drink unusual amounts of water when the food is at last presented. The control parameter (r) operating here was the length of the interval between feedings, once resumed. The authors were careful to test a large number of animals and to include many replications, and they designed their experiment so as to rule out the likelihood that changes in response patterns were caused by different starting places for r. Time series and first delay plots provide the best support for the claims made, showing a fairly clear march from periodicity to irregularity as the feeding times were increased. The various phase trajectory plots and spectral analyses, on the other hand, do not match up well enough with the other graphs or with the overall theory to lead inexorably to a chaotic diagnosis. For example, the phase trajectories do not show a definite progression towards greater and greater complexity (and away from periodicity); the process seems quite muddied. Also, where Metcalf and Allen saw periods of two and six in their spectral plots, there is room for alternative interpretations. All of this ambiguity necessitate some serpentine, post-hoc explanation to show that results fit a chaotic model. By adapting a model of career counseling to include a chaotic interpretation of the relationship between employees and the job market, Amundson and Bright found that better suggestions can be made to people struggling with career decisions.[149] Modern organizations are increasingly seen as open complex adaptive systems with fundamental natural nonlinear structures, subject to internal and external forces that may contribute chaos. For instance, team building and group development is increasingly being researched as an inherently unpredictable system, as the uncertainty of different individuals meeting for the first time makes the trajectory of the team unknowable.[150] Some say the chaos metaphor—used in verbal theories—grounded on mathematical models and psychological aspects of human behavior provides helpful insights to describing the complexity of small work groups, that go beyond the metaphor itself.[151] Traffic forecasting may benefit from applications of chaos theory. Better predictions of when a congestion will occur would allow measures to be taken to disperse it before it would have occurred. Combining chaos theory principles with a few other methods has led to a more accurate short-term prediction model (see the plot of the BML traffic model at right).[152] Chaos theory has been applied to environmental water cycle data (also hydrological data), such as rainfall and streamflow.[153] These studies have yielded controversial results, because the methods for detecting a chaotic signature are often relatively subjective. Early studies tended to "succeed" in finding chaos, whereas subsequent studies and meta-analyses called those studies into question and provided explanations for why these datasets are not likely to have low-dimension chaotic dynamics.[154] In art (predominately art theory) a possible postpostmodern era has been outlined with emphasis on multiple narratives and the notion that every fictional angle is a possibility. In part this is therefor of a bisociate (trissociative) discourse and can be explained within emphasis on an institutional interchange of subjectivistic agents.[155] See also Examples of chaotic systems • Advected contours • Arnold's cat map • Bifurcation theory • Bouncing ball dynamics • Chua's circuit • Cliodynamics • Coupled map lattice • Double pendulum • Duffing equation • Dynamical billiards • Economic bubble • Gaspard-Rice system • Hénon map • Horseshoe map • List of chaotic maps • Rössler attractor • Standard map • Swinging Atwood's machine • Tilt A Whirl Other related topics • Amplitude death • Anosov diffeomorphism • Catastrophe theory • Causality • Chaos as topological supersymmetry breaking • Chaos machine • Chaotic mixing • Chaotic scattering • Control of chaos • Determinism • Edge of chaos • Emergence • Mandelbrot set • Kolmogorov–Arnold–Moser theorem • Ill-conditioning • Ill-posedness • Nonlinear system • Patterns in nature • Predictability • Quantum chaos • Santa Fe Institute • Shadowing lemma • Synchronization of chaos • Unintended consequence People • Ralph Abraham • Michael Berry • Leon O. 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Bucknell University MS Engineering Thesis – via Bucknell University. Further reading Articles • Sharkovskii, A.N. (1964). "Co-existence of cycles of a continuous mapping of the line into itself". Ukrainian Math. J. 16: 61–71. • Li, T.Y.; Yorke, J.A. (1975). "Period Three Implies Chaos" (PDF). American Mathematical Monthly. 82 (10): 985–92. Bibcode:1975AmMM...82..985L. CiteSeerX 10.1.1.329.5038. doi:10.2307/2318254. JSTOR 2318254. Archived from the original (PDF) on 2009-12-29. Retrieved 2009-08-12. • Alemansour, Hamed; Miandoab, Ehsan Maani; Pishkenari, Hossein Nejat (March 2017). "Effect of size on the chaotic behavior of nano resonators". Communications in Nonlinear Science and Numerical Simulation. 44: 495–505. Bibcode:2017CNSNS..44..495A. doi:10.1016/j.cnsns.2016.09.010. • Crutchfield; Tucker; Morrison; J.D. Farmer; Packard; N.H.; Shaw; R.S (December 1986). "Chaos". Scientific American. 255 (6): 38–49 (bibliography p.136). Bibcode:1986SciAm.255d..38T. doi:10.1038/scientificamerican1286-46. Online version (Note: the volume and page citation cited for the online text differ from that cited here. The citation here is from a photocopy, which is consistent with other citations found online that don't provide article views. The online content is identical to the hardcopy text. Citation variations are related to country of publication). • Kolyada, S.F. (2004). "Li-Yorke sensitivity and other concepts of chaos". Ukrainian Math. J. 56 (8): 1242–57. doi:10.1007/s11253-005-0055-4. S2CID 207251437. • Day, R.H.; Pavlov, O.V. (2004). "Computing Economic Chaos". Computational Economics. 23 (4): 289–301. arXiv:2211.02441. doi:10.1023/B:CSEM.0000026787.81469.1f. S2CID 119972392. SSRN 806124. • Strelioff, C.; Hübler, A. (2006). "Medium-Term Prediction of Chaos" (PDF). Phys. Rev. Lett. 96 (4): 044101. Bibcode:2006PhRvL..96d4101S. doi:10.1103/PhysRevLett.96.044101. PMID 16486826. 044101. Archived from the original (PDF) on 2013-04-26. • Hübler, A.; Foster, G.; Phelps, K. (2007). "Managing Chaos: Thinking out of the Box" (PDF). Complexity. 12 (3): 10–13. Bibcode:2007Cmplx..12c..10H. doi:10.1002/cplx.20159. Archived from the original (PDF) on 2012-10-30. Retrieved 2011-07-17. • Motter, Adilson E.; Campbell, David K. (2013). "Chaos at 50". Physics Today. 66 (5): 27. arXiv:1306.5777. Bibcode:2013PhT....66e..27M. doi:10.1063/PT.3.1977. S2CID 54005470. Textbooks • Alligood, K.T.; Sauer, T.; Yorke, J.A. (1997). Chaos: an introduction to dynamical systems. Springer-Verlag. ISBN 978-0-387-94677-1. • Baker, G. L. (1996). Chaos, Scattering and Statistical Mechanics. Cambridge University Press. ISBN 978-0-521-39511-3. • Badii, R.; Politi A. (1997). Complexity: hierarchical structures and scaling in physics. Cambridge University Press. ISBN 978-0-521-66385-4. • Collet, Pierre; Eckmann, Jean-Pierre (1980). Iterated Maps on the Interval as Dynamical Systems. Birkhauser. ISBN 978-0-8176-4926-5. • Devaney, Robert L. (2003). An Introduction to Chaotic Dynamical Systems (2nd ed.). Westview Press. ISBN 978-0-8133-4085-2. • Robinson, Clark (1995). Dynamical systems: Stability, symbolic dynamics, and chaos. CRC Press. ISBN 0-8493-8493-1. • Feldman, D. P. (2012). Chaos and Fractals: An Elementary Introduction. Oxford University Press. ISBN 978-0-19-956644-0. Archived from the original on 2019-12-31. Retrieved 2016-12-29. • Gollub, J. P.; Baker, G. L. (1996). Chaotic dynamics. Cambridge University Press. ISBN 978-0-521-47685-0. • Guckenheimer, John; Holmes, Philip (1983). Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer-Verlag. ISBN 978-0-387-90819-9. • Gulick, Denny (1992). Encounters with Chaos. McGraw-Hill. ISBN 978-0-07-025203-5. • Gutzwiller, Martin (1990). Chaos in Classical and Quantum Mechanics. Springer-Verlag. ISBN 978-0-387-97173-5. • Hoover, William Graham (2001) [1999]. Time Reversibility, Computer Simulation, and Chaos. World Scientific. ISBN 978-981-02-4073-8. • Kautz, Richard (2011). Chaos: The Science of Predictable Random Motion. Oxford University Press. ISBN 978-0-19-959458-0. • Kiel, L. Douglas; Elliott, Euel W. (1997). Chaos Theory in the Social Sciences. Perseus Publishing. ISBN 978-0-472-08472-2. • Moon, Francis (1990). Chaotic and Fractal Dynamics. Springer-Verlag. ISBN 978-0-471-54571-2. • Orlando, Giuseppe; Pisarchick, Alexander; Stoop, Ruedi (2021). Nonlinearities in Economics. Dynamic Modeling and Econometrics in Economics and Finance. Vol. 29. doi:10.1007/978-3-030-70982-2. ISBN 978-3-030-70981-5. S2CID 239756912. • Ott, Edward (2002). Chaos in Dynamical Systems. Cambridge University Press. ISBN 978-0-521-01084-9. • Strogatz, Steven (2000). Nonlinear Dynamics and Chaos. Perseus Publishing. ISBN 978-0-7382-0453-6. • Sprott, Julien Clinton (2003). Chaos and Time-Series Analysis. Oxford University Press. ISBN 978-0-19-850840-3. • Tél, Tamás; Gruiz, Márton (2006). Chaotic dynamics: An introduction based on classical mechanics. Cambridge University Press. ISBN 978-0-521-83912-9. • Teschl, Gerald (2012). Ordinary Differential Equations and Dynamical Systems. Providence: American Mathematical Society. ISBN 978-0-8218-8328-0. • Thompson JM, Stewart HB (2001). Nonlinear Dynamics And Chaos. John Wiley and Sons Ltd. ISBN 978-0-471-87645-8. • Tufillaro; Reilly (1992). An experimental approach to nonlinear dynamics and chaos. p. 958. Bibcode:1993AmJPh..61..958T. doi:10.1119/1.17380. ISBN 978-0-201-55441-0. {{cite book}}: \|journal= ignored (help) • Wiggins, Stephen (2003). Introduction to Applied Dynamical Systems and Chaos. Springer. ISBN 978-0-387-00177-7. • Zaslavsky, George M. (2005). Hamiltonian Chaos and Fractional Dynamics. Oxford University Press. ISBN 978-0-19-852604-9. Semitechnical and popular works • Christophe Letellier, Chaos in Nature, World Scientific Publishing Company, 2012, ISBN 978-981-4374-42-2. • Abraham, Ralph H.; Ueda, Yoshisuke, eds. (2000). The Chaos Avant-Garde: Memoirs of the Early Days of Chaos Theory. World Scientific Series on Nonlinear Science Series A. Vol. 39. World Scientific. Bibcode:2000cagm.book.....A. doi:10.1142/4510. ISBN 978-981-238-647-2. • Barnsley, Michael F. (2000). Fractals Everywhere. Morgan Kaufmann. ISBN 978-0-12-079069-2. • Bird, Richard J. (2003). Chaos and Life: Complexity and Order in Evolution and Thought. Columbia University Press. ISBN 978-0-231-12662-5. • John Briggs and David Peat, Turbulent Mirror: : An Illustrated Guide to Chaos Theory and the Science of Wholeness, Harper Perennial 1990, 224 pp. • John Briggs and David Peat, Seven Life Lessons of Chaos: Spiritual Wisdom from the Science of Change, Harper Perennial 2000, 224 pp. • Cunningham, Lawrence A. (1994). "From Random Walks to Chaotic Crashes: The Linear Genealogy of the Efficient Capital Market Hypothesis". George Washington Law Review. 62: 546. • Predrag Cvitanović, Universality in Chaos, Adam Hilger 1989, 648 pp. • Leon Glass and Michael C. Mackey, From Clocks to Chaos: The Rhythms of Life, Princeton University Press 1988, 272 pp. • James Gleick, Chaos: Making a New Science, New York: Penguin, 1988. 368 pp. • John Gribbin. Deep Simplicity. Penguin Press Science. Penguin Books. • L Douglas Kiel, Euel W Elliott (ed.), Chaos Theory in the Social Sciences: Foundations and Applications, University of Michigan Press, 1997, 360 pp. • Arvind Kumar, Chaos, Fractals and Self-Organisation; New Perspectives on Complexity in Nature , National Book Trust, 2003. • Hans Lauwerier, Fractals, Princeton University Press, 1991. • Edward Lorenz, The Essence of Chaos, University of Washington Press, 1996. • Marshall, Alan (2002). The Unity of Nature - Wholeness and Disintegration in Ecology and Science. doi:10.1142/9781860949548. ISBN 9781860949548. • David Peak and Michael Frame, Chaos Under Control: The Art and Science of Complexity, Freeman, 1994. • Heinz-Otto Peitgen and Dietmar Saupe (Eds.), The Science of Fractal Images, Springer 1988, 312 pp. • Nuria Perpinya, Caos, virus, calma. La Teoría del Caos aplicada al desórden artístico, social y político, Páginas de Espuma, 2021. • Clifford A. Pickover, Computers, Pattern, Chaos, and Beauty: Graphics from an Unseen World , St Martins Pr 1991. • Clifford A. Pickover, Chaos in Wonderland: Visual Adventures in a Fractal World, St Martins Pr 1994. • Ilya Prigogine and Isabelle Stengers, Order Out of Chaos, Bantam 1984. • Peitgen, Heinz-Otto; Richter, Peter H. (1986). The Beauty of Fractals. doi:10.1007/978-3-642-61717-1. ISBN 978-3-642-61719-5. • David Ruelle, Chance and Chaos, Princeton University Press 1993. • Ivars Peterson, Newton's Clock: Chaos in the Solar System, Freeman, 1993. • Ian Roulstone; John Norbury (2013). Invisible in the Storm: the role of mathematics in understanding weather. Princeton University Press. ISBN 978-0691152721. • Ruelle, D. (1989). Chaotic Evolution and Strange Attractors. doi:10.1017/CBO9780511608773. ISBN 9780521362726. • Manfred Schroeder, Fractals, Chaos, and Power Laws, Freeman, 1991. • Smith, Peter (1998). Explaining Chaos. doi:10.1017/CBO9780511554544. ISBN 9780511554544. • Ian Stewart, Does God Play Dice?: The Mathematics of Chaos , Blackwell Publishers, 1990. • Steven Strogatz, Sync: The emerging science of spontaneous order, Hyperion, 2003. • Yoshisuke Ueda, The Road To Chaos, Aerial Pr, 1993. • M. Mitchell Waldrop, Complexity : The Emerging Science at the Edge of Order and Chaos, Simon & Schuster, 1992. • Antonio Sawaya, Financial Time Series Analysis : Chaos and Neurodynamics Approach, Lambert, 2012. External links Wikimedia Commons has media related to Chaos theory. Library resources about Chaos theory • Resources in your library • Resources in other libraries • "Chaos", Encyclopedia of Mathematics, EMS Press, 2001 [1994] • Nonlinear Dynamics Research Group with Animations in Flash • The Chaos group at the University of Maryland • The Chaos Hypertextbook. An introductory primer on chaos and fractals • ChaosBook.org An advanced graduate textbook on chaos (no fractals) • Society for Chaos Theory in Psychology & Life Sciences • Nonlinear Dynamics Research Group at CSDC, Florence Italy • Nonlinear dynamics: how science comprehends chaos, talk presented by Sunny Auyang, 1998. • Nonlinear Dynamics. Models of bifurcation and chaos by Elmer G. Wiens • Gleick's Chaos (excerpt) Archived 2007-02-02 at the Wayback Machine • Systems Analysis, Modelling and Prediction Group at the University of Oxford • A page about the Mackey-Glass equation • High Anxieties — The Mathematics of Chaos (2008) BBC documentary directed by David Malone • The chaos theory of evolution – article published in Newscientist featuring similarities of evolution and non-linear systems including fractal nature of life and chaos. • Jos Leys, Étienne Ghys et Aurélien Alvarez, Chaos, A Mathematical Adventure. Nine films about dynamical systems, the butterfly effect and chaos theory, intended for a wide audience. • "Chaos Theory", BBC Radio 4 discussion with Susan Greenfield, David Papineau & Neil Johnson (In Our Time, May 16, 2002) • Chaos: The Science of the Butterfly Effect (2019) an explanation presented by Derek Muller Copyright note • This article incorporates text from a free content work. Licensed under CC-BY (license statement/permission). Text taken from Three Kinds of Butterfly Effects within Lorenz Models, Bo-Wen Shen, Roger A. Pielke, Sr., Xubin Zeng, Jialin Cui, Sara Faghih-Naini, Wei Paxson, and Robert Atlas, MDPI. Encyclopedia. To learn how to add open license text to Wikipedia articles, please see this how-to page. For information on reusing text from Wikipedia, please see the terms of use. Chaos theory Concepts Core • Attractor • Bifurcation • Fractal • Limit set • Lyapunov exponent • Orbit • Periodic point • Phase space • Anosov diffeomorphism • Arnold tongue • axiom A dynamical system • Bifurcation diagram • Box-counting dimension • Correlation dimension • Conservative system • Ergodicity • False nearest neighbors • Hausdorff dimension • Invariant measure • Lyapunov stability • Measure-preserving dynamical system • Mixing • Poincaré section • Recurrence plot • SRB measure • Stable manifold • Topological conjugacy Theorems • Ergodic theorem • Liouville's theorem • Krylov–Bogolyubov theorem • Poincaré–Bendixson theorem • Poincaré recurrence theorem • Stable manifold theorem • Takens's theorem Theoretical branches • Bifurcation theory • Control of chaos • Dynamical system • Ergodic theory • Quantum chaos • Stability theory • Synchronization of chaos Chaotic maps (list) Discrete • Arnold's cat map • Baker's map • Complex quadratic map • Coupled map lattice • Duffing map • Dyadic transformation • Dynamical billiards • outer • Exponential map • Gauss map • Gingerbreadman map • Hénon map • Horseshoe map • Ikeda map • Interval exchange map • Irrational rotation • Kaplan–Yorke map • Langton's ant • Logistic map • Standard map • Tent map • Tinkerbell map • Zaslavskii map Continuous • Double scroll attractor • Duffing equation • Lorenz system • Lotka–Volterra equations • Mackey–Glass equations • Rabinovich–Fabrikant equations • Rössler attractor • Three-body problem • Van der Pol oscillator Physical systems • Chua's circuit • Convection • Double pendulum • Elastic pendulum • FPUT problem • Hénon–Heiles system • Kicked rotator • Multiscroll attractor • Population dynamics • Swinging Atwood's machine • Tilt-A-Whirl • Weather Chaos theorists • Michael Berry • Rufus Bowen • Mary Cartwright • Chen Guanrong • Leon O. Chua • Mitchell Feigenbaum • Peter Grassberger • Celso Grebogi • Martin Gutzwiller • Brosl Hasslacher • Michel Hénon • Svetlana Jitomirskaya • Bryna Kra • Edward Norton Lorenz • Aleksandr Lyapunov • Benoît Mandelbrot • Hee Oh • Edward Ott • Henri Poincaré • Mary Rees • Otto Rössler • David Ruelle • Caroline Series • Yakov Sinai • Oleksandr Mykolayovych Sharkovsky • Nina Snaith • Floris Takens • Audrey Terras • Mary Tsingou • Marcelo Viana • Amie Wilkinson • James A. Yorke • Lai-Sang Young Related articles • Butterfly effect • Complexity • Edge of chaos • Predictability • Santa Fe Institute Patterns in nature Patterns • Crack • Dune • Foam • Meander • Phyllotaxis • Soap bubble • Symmetry • in crystals • Quasicrystals • in flowers • in biology • Tessellation • Vortex street • Wave • Widmanstätten pattern Causes • Pattern formation • Biology • Natural selection • Camouflage • Mimicry • Sexual selection • Mathematics • Chaos theory • Fractal • Logarithmic spiral • Physics • Crystal • Fluid dynamics • Plateau's laws • Self-organization People • Plato • Pythagoras • Empedocles • Fibonacci • Liber Abaci • Adolf Zeising • Ernst Haeckel • Joseph Plateau • Wilson Bentley • D'Arcy Wentworth Thompson • On Growth and Form • Alan Turing • The Chemical Basis of Morphogenesis • Aristid Lindenmayer • Benoît Mandelbrot • How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension Related • Pattern recognition • Emergence • Mathematics and art Authority control: National • France • BnF data • Germany • Israel • United States • Czech Republic	Wikipedia

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