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Sunday, 9 March 2014
Infinity and Beyond (2001)
Watching the Horizon documentary "Infinity and Beyond" had renewed my interests in the concept of mathematical infinity. And how the concept is fraught with paradoxes.
One example of such paradox can be illustrated as follows (this is not mentioned in this documentary):
Let's start with the digits 1 and 2. Let halve these two digits to get 1/2. And then we halve that, and so forth to ad nauseum, so we get the series,
1, 1/2, 1/4, 1/8, 1/16 ...1/∞
Mathematician will tell you that this series has infinite members or fractions, and yet we know that it contains within the finite boundaries of 1 and 2. Finite, and yet infinite. This is in fact the well known Zeno's Paradox that originated in the ancient Greece.
Other seemingly mind-boggling concept that does appear in this documentary is the sets of equations:
∞ + 1 = ∞
∞ + ∞ = ∞
∞ - 1 = ∞
∞ - ∞ = ???
We assume that the ∞ in the equation belongs to a positive digit number.
The first three equations seems to make sense 'intuitively', and at the same time seems contradictory because how can z + 1 = z ? This equations only 'makes sense' if z = ∞.
Similar 'logic' applies to the next 2 equations.
What about the last equation (∞ - ∞ = ???) ? According to the mathematician in the documentary, it could be 1, it could 3, it could be anything ! Nobody knows. Well, this is no good.
Here's what I think is wrong with the picture.
when you have the equation 1 + z = ?
It doesn't matter how big z is as long as it isn't ∞, I can give you a precise answer. If z = ∞, then I can' give you an answer. Why? Because ∞ isn't a number. While we think of it as an infinitely large number, but it is not! It's a concept, an idea. So it's nonsensical to apply an arithmetic operation to a concept. |
Algol: "From Arabic al-ghul 'the demon' (see ghoul). It corresponds, in modern representations of the constellation, to the gorgon's head Perseus is holding, but it probably was so called because it visibly varies in brightness every three days, which sets it apart from other bright stars. The computer language (1959) is a contraction of algo(rithmic) l(anguage); see algorithm."
Alchemy: "From Arabic al-kimiya, from Gk. khemeioa (found c.300 C.E. in a decree of Diocletian against 'the old writings of the Egyptians.')"
Algebra: "1550s, from M.L. algebra, from Arabic al jebr "reunion of broken parts. as in computation, used 9c. by Baghdad mathematician Abu Ja'far Muhammad ibn Musa al-Khwarizmi as the title of his famous treatise on equations ("Kitab al-Jabr w'al-Muqabala" "Rules of Reintegration and Reduction"), which also introduced Arabic numerals to the West." |
I think it's time for "Jerry Springer Metaphysics - When Mind Over Matter Goes Bad!!"
White Trash Country Boffkin 1:Gonna stop right there for a second, the bold type was added by me for this reason... How can you -estimate- a constant, since that number is one of the multipliers in your equation, if even one number is off it completely changes everything about your answer.
White Trash Boffkin 2:You mean to tell me that unless I use the exact value of pi, I cannot perform any sophsticated calculations involving circles, spheres, toroids, etc without "completely changing the answer"?
White Trash Boffkin 1:It is imperative that you end your acoustic output! |
Lets face it -- π sucks. Not pie, pie is genuinely awesome and so is the fact that "pie" sounds like "π", so we can have fun with things like this. But seriously... π? It's totally off by a factor of 2! |
Voting in Agreeable Societies
When do majorities exist? How does the geometry of the political spectrum influence the outcome? What does mathematics have to say about how people behave? When mathematical objects have a social interpretation, the associated theorems have social applications. We give examples of situations where sets model preferences, and prove extensions of classical theorems on convex sets such as Helly's theorem that can be used in the analysis of voting in "agreeable" societies. This talk also features research with undergraduates. |
A new mathematics revolution
We need to ovehaul our present system of mathematics education by re-educating our mathematics teachers and establishing a new unified curriculum from arithmetic to calculus.
Children in public schools should start with the usual arithmetic then to calculus as the final part of a secondary education.
Our children are lazy because our system of education lacks the motivation to make mathematics practical, fascinating and exciting. Our kids find the subjects too easy.
A major part of this mathematics curriculum should be modern and abstract algebra.
I used to have modern math when I was in Grade 5 and Grade 6. Real nice, but somewhat repetitive. We seemed to have hit a ceiling since we never advanced to more abstract forms of modern math! Imagine, we were drilling like crazy on the theory of sets and Venn diagrams for almost a year with no end in sight to introduce group theory. Well, that was 40 years ago. Had we been exposed progressively, we would have created many scientists by now with solid state physics and a prosperous Silicon Valley.
The challenge of mathematics should be embedded on a child's mind while his thinking is still very malleable, before the age of 16, when all cognitive ability could no longer be improved.
You can practically take any science degree when you are well grounded in mathematics. That is the goal of mathematics. Even the best physicists have to learn math to carry their theories to conclusion. Math is a real hurdle throughout life, even inducing fear. That should not be the case. Nowadays, there is a move to unify all mathematics and to present any treatise, theory or discovery from this one unified language. Imagine, you only need one textbook and not any other! There is a growing consensus that this should be so since the disparate state of presentation could not explain from first principles or theorems. Only abstract algebra seemed to have taken such approach to improve things.
What is good with the new system? You can explain everything and you approach it from very simlified forms!
You can go to any college or university armed with only a complete repertoire of mathematical capacity and be able to tackle any subject head-on.
Learning math is a good investment rather that neglect of it will haunt you for life. |
Mathematics by Anne Rooney
In order to understand the universe you must know the language in which it is written. And that language is mathematics.' Galileo (1564-1642)
For hundreds of thousands of years, we have sought order in the apparent chaos of the universe. Mathematics has been our most valuable tool in that search, uncovering the patterns and rules that govern our world and beyond. The Story of Mathematics traces humankind's greatest achievements, plotting a journey from innumerate cave-dwellers, through the towering mathematical intellects of the last 4,000 years, to where we stand today.
Topics include: . Counting and measuring from the earliest times .The Ancient Egyptians and geometry .Working out the movement of the planets .Algebra, solid geometry and the trigonometric tables . The first computers .How statistics came to rule our finances .Impossible shapes and extra dimensions .Measuring and mapping the world .Chaos theory and fuzzy logic Set theory and the death of numbers
About the Author
Anne Rooney gained a degree and then a PhD in medieval literature from Trinity College, Cambridge. After a period of teaching medieval English and French literature at the universities of Cambridge and York, she left to pursue a career as a freelance writer. She has written many books for adults and children on a variety of subjects, including literature and history. She lives in Cambridge. She has written many books on science and technology, and was long-listed for the prestigious Aventis Science Prize in 2004.
1001 Horrible Facts is a great fact-packed and illustrated reference book covering topics including Science,
Body, Animals, History and World Records. It provides the perfect ammunition for 8-12 year-olds to amaze and disgust parents, teachers and each other!
From dogs and cats to stick insects and sea monkeys, there are many types of
pets. Which one is right for you?• Purple/ Band 8 books offer developing readers literary language, with some challenging vocabulary.• Text type—information book.• This non-fiction ...
What is it like to witness an earthquake? This book looks at the Haitian and
other earthquakes, using firsthand accounts to describe events and people's experiences, providing multiple perspectives from eyewitnesses, survivors, the emergency services, scientists, and the media.
Genetic engineers study genes and DNA to develop ways to recreate and modify them to
advance technologies in fields such as medicine and agriculture. Using living organisms and systems to create new products and technologies is called biotechnology. Readers will ...
Layer by Layer: Under the Sea explores the natural treasures hidden beneath the waves: fierce
sharks, colorful fish, scuttling crustaceans, and other creatures that call the ocean home. Six interactive layered scenes with atmospheric sounds show a variety of ocean ...
What's the speediest sportscar? The sneakiest spy plane? And . . . the fastest sofa?
From the smallest sub and deepest diver to the largest jumbo jet and most powerful rocket, these mega machines are record setters! With lively photos on ...
The 15-Minute Philosopher introduces the reader to the main ideas of philosophy, showing how the
subject has a clear practical purpose vital to our day-to-day lives and thinking. The subjects discussed here have been chosen to show that philosophy is ...
The 15-Minute Psychoanalyst introduces the reader to universal aspects of psychology which affect our day-to-day
lives and relationships and offers insight into many of life's dilemmas. Written in a style that's amusing and easy to understand, The 15-Minute Psychoanalyst lets ... |
>>7321342 So, the big amount of mathematicians jumping into philosophy is because of... The amount of abstract concepts found in maths, that could have a deeper explanation with a philosophy's approach?
>>7321354 Most higher level mathematics course is purely proofs, which are deduced from logic, which falls under the field of philosophy. I'll probably get a bunch of people saying that my generalization is incorrect and "hurr durr this field of math doesn't require philosophy" who gives a fuck most of it does, and call me batshit crazy but I personally believe that there is a correlation between logic and mathematics. So if you grant that for the most part, logic is used in most advanced fields of mathematics as well as the fact that logic is covered under the scope of philosophy, philosophy and math are interconnected to some extent. Not all philosophy is math (Ethics, political, aethetics) But MOST math falls under philosophy (Logic).
>>7321566 and there you have it friends, so please join us next time for what motivated OP to get into math, or namely, what in the world could have possessed him to be all up in this >>7321557 person's mom's butt.
>>7321748 >logic just get a mathematician >nature of knowledge just get a neuroscientist >reality just get a physicist >ethics lol meaningless SJW bullshit concept >aesthetics just get a neuroscientist
it bewilders the mind to think that tax dollars are going to waste on such hackery as ' philosophers' |
Equations that describe the natural world can convey profound truths while at the same time, to a trained eye, look absolutely beautiful. It is like learning to appreciate a work of art. Art may or may not be eternal. These poetic truths are. The equations show here are 1.Boltzmann equation 2. Euler Lagrange Equation 3. Dirac Equation 4. Euler Identity 5. Navier Stokes equation |
Hundreds Chart-Sieve of Eratosthenes-Prime Numbers/Divisibility Rules
Sieve of Eratosthenes: An ancient way of finding prime numbers by mathematician Eratosthenes. This fun chart not only has primes, but the multiples of and Divisibility rules are included on the back of the chart.
Welcome to the Prime Glossary: a collection of definitions, information and facts all related to prime numbers. This pages contains the entry titled 'Sieve of Eratosthenes.' Come explore a new prime term today!
Create a worksheet: Use an ancient algorithm to discover prime numbers
Eratosthenes was a brilliant and well-known mathematician in history. * He devised the sieve analogy that he was able to use to separate prime numbers from composite numbers. ^ The importance of prime numbers is discussed as being the foundation |
Tag: Random Number
Are Random Number Generators truly random?
Having carried out some research on random number generators or RNG's for short I concluded that there are degrees of randomness. The whole thing is really complicated, just like infinity itself.
Why do we need it? For betting or gambling of course, it has no other use than prediction. Since ancient times rolling the dice, flipping a coin or using playing cards all have forms of random number generation. In this article we only consider random RNG's in machines.
Random number generation can be really simple as in the flip of a coin. However as the number of outcomes increases it can become a hugely complex task. For example flipping a coin only has two outcomes and in theory that outcome could be repeated an infinite number of times. The reality of course is that the odds of 2:1 are rather more balanced.
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Bingo History – The Early Origins
There are many reports that the game of Bingo can be traced back to an Italian Lottery called Lo Giuoco which originated in 1530. It is true that Bingo history does find some similarity with lottery games which require random number selection.
There is the suggestion that playing Bingo evolved from the British Armed forces at the end of the 1800's. The British Army played a game called Housey-Housey which involved marking randomly drawn numbers on a card. The winner covered all the numbers on the card. The Navy had a similar game called Tombola which today we might call a raffle. There was a large naval Garrison in Malta from 1814 and some say that the game was derived there.
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Roulette History – Blaise Pascal invented the Roulette Wheel
The origins of Roulette as we know it today are not certain but a French man is credited with the invention of the Roulette wheel. Blaise Pascal was a French physicist and mathematician working towards the invention of a perpetual motion machine. Roulette history started then. In 1655 Pascal invented the roulette wheel as a means of generating random numbers. Sadly it needed to be given a spin to start it and it came to rest soon enough. Many others have tried to invent perpetual motion but it has long been acknowledged that the laws of Physics prevent this.
In Roulette history his invention went relatively unnoticed for more than a century. The first accounts of playing Roulette as we know it were reported in 1796 in Paris. Roulette history generally concludes that the actual date the game was invented is not known. |
String Art And Math
String Art And Math in the picture higher than is a component on the String Art And Math category on The Art Evangelist posts. Download this impression for free in HD resolution the choice by appropriate clicking "save image as" within the impression. In the event you never come across the precise resolution you are searching for, then go for a local or greater resolution.
Distinctive Vital Artwork Principles have evolved thorough distinct eras, with all the changing artists' perceptions of processing, examining, and responding to varied artwork sorts. Their creative expressions have been explored by their generation, general performance, and participation in arts. Every historical era has provided novel contribution of historical and cultural contexts for building the important thing Arts Fundamentals of the suitable period of time. Visual Arts assistance artists assimilate the real key Arts Ideas of Symmetry, Color, Pattern, Contrast and the distinctions involving 1 or more aspects from the composition. The true secret Art Ideas of Visual Arts support fully grasp and distinguish involving the scale plusString art, created with thread and paper.
A string art representing a projection of the 8-dimensional 421 polytope
Quadratic Béziers in string art: The end points (•) and control point (×) define the quadratic Bézier curve (⋯). The arc is a segment of a parabola.
String art, or pin and thread art, is characterized by an arrangement of colored thread strung between points to form geometric patterns or representational designs such as a ship's sails, sometimes with other artist material comprising the remainder of the work.
Thread, wire, or string is wound around a grid of nails hammered into a velvet-covered wooden board. Though straight lines are formed by the string, the slightly different angles and metric positions at which strings intersect gives the appearance of Bézier curves (as in the mathematical concept of envelope of a family of straight lines). Quadratic Bézier curve are obtained from strings based on two intersecting segments.
Other forms of string art include Spirelli, which is used for cardmaking and scrapbooking, and curve stitching, in which string is stitched through holes.
String art has its origins in the 'curve stitch' activities invented by Mary Everest Boole at the end of the 19th century to make mathematical ideas more accessible to children.[1] It was popularised as a decorative craft in the late 1960s through kits and books |
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Isaac Newton was an English mathematician, who discovered the binomial theorem, (A theory he came up with) he also invented calculus, and produced theories of mechanics, optics, and the law of universal gravitation. Many of his ideas for which he is famous were developed in isolation in the year 1665 during the Great Plague. He was also knighted Sir Isaac Newton.
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For this slideshow you will need: Isometric paper, Pencil, Four different coloured pencil crayons, Calculator, AND YOUR BRAIN!
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For our first problem we are going to show you is the 1 by 1 by 1 cube. So to build the 1 by 1 by 1cube all you need is 1 cube. Then all you need to do is to count the number of cubes which have all of it's faces panted. This cube has 6 faces painted. All the cubes on this cube, all have 3 faces painted. For our Second we are going to show you is the 2 by 2 by 2 cube. So to build the 2 by 2 by 2 cube all you need to do is get 8 cubes and put them together like the diagram above. Then all you need to do is count how many cubes have three faces painted.
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Cubes with 0 faces Painted Cubes with 1 Face Painted Cube s with 2 Faces Paint ed Cubes with 3 Faces Painted Cubes with 6 Faces Painted Total number of cubes 1 by 1 by 1 000011 2 by 2 by 2 000808 3 by 3 by 3 Answers will be shown Later on in the slideshow 4 by 4 by 4
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Counting the faces on each individual cube sounds easy but it's not. If you wont it to be easy follow these simple steps. Draw your cube on isometric paper. Colour all the cubes which all have three faces painted blue. Colour all the cubes which all have two faces painted green. Colour all the cubes which all have one face painted brown. Remember if on the inside of the cube there are cubes with no faces painted to count them ones. Remember you can not see one of the cubes which has three faces painted because you can not draw it on a piece of isometric paper. Colouring in represents painting the the cube.
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An exploded cube is a really simple way of working out the painted cube problem. First you explode a cube like the diagram at the bottom. Then you colour all the sides if they are 3 faces painted sides or 2 faces painted sides or if they are 1 face painted side or if they are 0 faces painted sides. This will help you by making the painted cube problem a lot easier. You try and draw an exploded cube on your isometric paper. This is the centre cube which does not get painted but we have just highlighted it to make it easier to see
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This cube has three of its faces painted. These are all cubes which have two of there faces painted. All these cubes have one of there faces painted.
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Work out how many cubes have : 3 faces painted 2 faces painted 1 face painted 0 faces painted Work out how many cubes have : 3 faces painted 2 faces painted 1 face painted 0 faces panted Here is a 6 by 6 by 6 cube Here is a 5 by 5 by 5 cube
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To work out the total number of cubes in a cube lets say it was a 10 by 10 by 10 cube. You would do 10 x 10 x 10 that answer would be how many cubes are in a10 by10 by 10 cube. To work out the total number of cubes that have 3 faces painted in any cube is very easy because all cubes have 8 cubes with 3 faces painted, if the cube dose not have 8 cubes with 3 faces painted it is not a cube. To work out the total number of cubes have 2 faces painted in any cube, you find out how many cubes are along the edge of one side then takeaway 2 then times that number by 12 because there are 12 edges on a cube. To work out the total number of cubes with 0 faces painted, you takeaway 2 from the 10. Then you do 8*8*8. To work out the total number of cubes which have 1 face painted you, work out how many cubes are on a face by doing 10-2 then you square that number. Then you times that number by 6 because there are 6 faces on a cube.
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We both hope you have enjoyed our side show and hope it has taught you some things about the painted cube problem! Goodbye and thank you for joining us! ^Click to go back^ to the start |
Monthly Archives: February 2007
You swing from wires to try to go as far as you can. You fling them by click on the screen and the wire shoots towards the direction of the click. It will stick to the brightly colored regions for a few seconds but then release. Very Tarzan-like, AHH-ahh-ahha-haahhha! (FYI: That was my Tarzan voice)
Dr. Boyd was in charge for another day. This time he covered topics from the history of science as an illustration of how science progresses. Using specific case studies from the history of scientific endeavor, we learn some important principles that undergird how we understand science as it is today.
Below are some of the topics that we touched on in this class:
The UV Catastrophe: This serves as a good case study to see how scientific revolutions occur as well as a lesson about the dangers of extrapolation
Causality: the difficulty of assessing the cause from the effect. Wearing skirts causes an increased likelihood of breast cancer
Did Science arise in a Christian World? Did the Christian World help to create modern science? (see Eric Snow's Paper)
"Quiet You With My Love" by Matt Bronleewe/Rebecca St. James (Worship God)
"Just to be with You" by Third Day – This makes two from the all-time favorites list making it onto the random 10
In light of a couple of results on this list, I am going to have to start keeping track of the Official All Time Favorites List as they appear on the Friday Random 10. Once this list reaches a significant number, it will become an official page on this blog.
We finished the concept of directional derivatives, introducing the notation for the gradient of a function of several variables. We proved the formula of the maximum value of the directional derivative, as well as the direction for which it is maximized.
Next time we'll begin basic optimization theory for functions of several variables.
Thursday morning, three students presented homework problems at the board. I followed this with a lecture introducing the concepts of supremum and infimum of a set. Because I got a little sidetracked, I did not quite make it to the Completeness Axiom.
What was the "rabbit chase" for this class? Well, at least a couple of students had commented to me, outside of class, on the difficulty they've had with recent homework assignments. They pointed out that they work and work and often can't make any headway on a few of the proofs. They seemed a little discouraged by the fact that they need help from their professor on every assignment.
I took class time to reassure them that they are not alone. Just about everyone in the class is going through the same thing. I pointed out the fact that I was in their place not that long ago. In fact, since I also did my undergrad here at Wayland, I was almost exactly in their place. I recounted tales of my discouragement as well as the fact that I also spent time in my professor's office get help on almost every assignment. I was largely motivated by a reader of this blog, who is also a blogger I read regularly. He has recently made the point that the students can gain a new level of insight to a subject by seeing the learning process that the instructor, themselves went through to understand certain concepts.
In my mind, although some may disagree, it does not get any harder as an undergrad than a senior level mathematics course. There are many courses that require as much "work" as a course like this, but I can't think of too many that require such an high level of abstract and critical thinking. I've yet to be convinced otherwise. However, I'll admit that my undergraduate Physical Chemistry class may have been close. |
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Thursday, January 26, 2012
Really Cool Math Videos!
I am not a math-minded person.
Come to think of it, I'm not sure what kind of "minded" I am - except that on many days I feel I've lost mine.
Anywho...
I came upon Vi Hart's blog and watched a few of her videos. I found them to be amazing in a "WOW! - but I could never re-explain that" sort of way. They truly are fascinating. Her focus is on Math - a subject I grasp just well enough to spit out some numbers now and then, some of them correct. Vi Hart, on the other hand, has a firm grasp on the understanding of Math and many of its complexities.
These videos are just plain amazing. I don't expect that you would necessarily use them for schoolwork except as an interesting aside. Yet, if your kids are studying biology or botany, the videos on "Spirals, Fibonacci, and Being a Plant" parts 1, 2, and 3, are very interesting, as is the video regarding Spongebob's house. If you or someone in your household is really math-minded, the video "Pi is (still) Wrong" or "Oh no, Pi politics again" are interesting. If you've been busy learning how to make balloon animals, you could practice your stuff making some mathematical models with balloons, just click on her balloon page.
Vi Hart very much understands her subject and communicates it in a fascinating way. She makes me wish I had paid more attention in Geometry class |
Month: April 2016
… Far too many ideas, far too little time. And so I have to carefully plan and allocate resources and spend a lot of time culling the ones that are less practical and putting the others into various categories. On the upside: never ever bored. And hey, I have "setting up a WordPress" site checked off the list!
Anyways, the project page is up. Expect some filling to happen at some point in the future.
It has been said that, during the first atomic bomb test, Enrico Fermi wanted a quick estimate of the energy of the blast. So, as the shock wave hit, he tossed a handful of paper scraps in to the air and watched how far they were carried. He estimated that the energy was about ten kilotons – remarkably close to the measured value of twenty.
Whether this actually happened or not, is a subject for historians to debate, but it makes for a good story nonetheless. And it serves to illustrate how a quick and dirty estimate can aid in decision making. In science classrooms around the world, these sorts of approximation problems are used to teach "science thinking" without getting bogged down in math. And, like all good science tools, it's partly a matter of convenience and partly a matter of laziness. Some approximation exercises that I remember from my own schooling:
How fast would you have to stir your coffee in order to make it boil?
How many gas stations are there in the United States?
If everyone in China faced west and sneezed at the same time, how would the earth's rotation change?
How fast would you have to drive a car through a hard rain in order to meet a wall of water?
With these types of problems, it isn't the answer that is interesting but how one arrives at it. And once you get the hang of this you can get a surprising level of accuracy, particularly if you know which way to fudge the numbers. Of course, there are a few tricks to this kind of "Fermi estimate". Trick number one, don't care too much about what the answer actually is. If you're attached to an outcome, you may subconsciously pick numbers that steer toward it.
Trick two, go fast the first time around and then fix it later. The first pass-through is just to get the process right. In subsequent estimations, you can try to get better numbers or to include things that you hadn't previously thought of.
Trick three, round to the nearest whatever. Some numbers are easier to work with than others. You can work with powers of ten just by moving the decimal point. Computer engineers and programmers know the powers of two better than their own phone numbers. Once you get a feel for how numbers themselves work, then calculation becomes a snap. And because you don't care about the end number, you can feel free to round off a bit.
Here's an interesting example…
Global air-conditioning
One morning, I was driving up to a nowhere spot in central California to meet with a client. Radio coverage was essentially non-existent and so I ended up listening to someone on AM talk radio. This someone made the claim that if global warming is man-made, it's probably from everyone running their air conditioners. Let's look at this claim and construct a very simple model.
There are approximately four hundred million people in the US right now (and we're going to ignore Alaska – they probably don't do too much air conditioning). We'll assume that each and every person has a five thousand square foot home, five thousand square foot office, and ninety thousand square feet representing their share of communal space (public buildings, malls, etc.). So every man, woman, and child has their very own air-conditioned area of one hundred-thousand square feet. Further, we shall assume that their ceilings are ten feet high, giving each person a million cubic feet of air-conditioned bliss. Four hundred trillion cubic feet in total (notice all of the powers of ten that I'm using).
How cold do they like it? Let's further assume that every person in the US is currently trying to fight one hundred degree weather and cool their space down to seventy degrees. If air conditioners were one hundred percent efficient (spoiler: they are not) then we'd have to warm a like mass of air by thirty degrees. For our initial model, we'll assume that air conditioners are only twenty percent efficient (probably they're a bit better than this, but this is closer to the truth) and so we will have to warm five times that volume (five is almost as easy to use as two and ten).
So, in our hypothetical model, we're warming up two quadrillion cubic feet of air by thirty degrees. That's a lot of air. Let's convert to cubic miles, for sake of readability:
(2,000,000,000,000,000) / (5280 x 5280 x 5280)
So about thirteen thousand cubic miles – a much more manageable number.
How much air is there in the continental United States? According to Wiki, there are about three million square miles of surface area. The atmosphere extends upward to about sixty miles, but most of the action takes place within three miles of the surface, so let's just use that and approximate that there are about ten million cubic miles of air.
Divide the one in to the other, and the air-conditioned-warmed air represents only one tenth of a percent (note that I'm doing a lot of rounding here) of the volume of air in the United States. All of our air conditioning would warm that mass by three one-hundredths of a degree.
But (and this is a big "but"), our model assumes that all cooling and all air is evenly distributed all over the country by the same amount, everywhere. This simply isn't true. Additionally, it isn't true that every person has that much cooled volume. And finally, it isn't true that every person in the country requires cooling by thirty degrees, all at once. Cities like Phoenix may require more cooling all in one spot; and places like Seattle may not require any. So we can see some ways to begin to refine our model.
I'm not going to argue that air conditioning causes or doesn't cause warming. It may actually have a measurable (though tiny) effect in some places. The point of this exercise is to show both the power and the peril of making a casual model.
My homework assignment to you: play with this (either on paper or in your head)! See if you can think of ways to refine it. See if you can think of wrenches to throw in to the works. See if you can find some better numbers to use. Feel free to cheat and use the internet if you get lazy (but please give it a go, first).
Well, not so much. You see, from a mathematical standpoint, the term "infinity" is defined as being the size of the set of integers. So is it possible to have something bigger than this? Yepers, and it is best illustrated by trying to map one thing to another.
By "mapping" you can think of placing marbles into an egg carton, one per slot. If you've marbles left over, then the number of marbles is larger than the number of slots. So let's start off with an infinitely long egg carton, each slot numbered 1, 2, 3, etc. Now well take an infinite number of marbles, with the same numbers on them. None left over, obviously.
Suppose we've twice as many marbles, this time labeled 1, 2, 3, etc.; and then -1, -2, -3, etc. We'll even throw in an extra marble labeled zero. Is this the same size? Yes it is, because you can alternate marbles. First put the zeroth marble into slot one, then the number one marble into slot two, then the negative one marble into three, the two marble into four, the negative two marble into five, and so on. You never run out of marbles, of course, but you never run out of slots either. And the important point: for any specific marble, you know exactly which numbered slot it goes into. So the set of all integers (positive, negative and zero) is the same size as the set of all positive integers.
Let's throw a curve: all rational numbers. For those of you whose math escapes you, a rational number can be expressed as the ratio of two integers. In other words, any particular rational number x can be expressed as ( a / b ) in which a and b are both (non-zero) integers. So how big is the set of rational numbers? It turns out that it's still the same size of infinity. Here's how:
Let's make a table of all possible a's and b's:
The grid is infinite in both the a and b directions as it contains all positive and negative integers along those axes. So how do we map this onto the line of positive integers? Just start in the "middle" (pick a point, any point) and spiral out. Yep, it sounds like cheating, but again, you can compute where any marble on the a-b grid would be placed into the infinite egg carton.
So again, the set of all rational numbers is the same size as the set of all positive integers — regular ol' infinity. So how about something larger than infinity? Next we'll try the set of all irrational numbers.
Again, in case you've forgotten, an irrational number is one that cannot be described as the ratio of two integers. Things like pi and e (base of the natural logarithm) and the square root of two are decimal numbers that never end and never repeat. So it would take an infinite number of integers (regular infinity) just to represent each one. And guess what? In between every rational number, there are an infinite number of irrational numbers. If you take pi as an example, you could change any one of its infinite digits and come up with an entirely different number.
So guess what? We just ran out of slots in our egg carton. Since we can change any one or all of the digits in any particular irrational number (even an infinite number of them) to get a new number; and since there are an infinite number of irrational numbers, even between each rational number; we have waaay too few slots in our infinite egg carton.
The size of the set of all irrational numbers is the first transinfinite number, aleph-one (aleph-zero is another term for regular infinity). We can even get bigger than this, defining in similar ways aleph-two, aleph-three, etc, even on to aleph-infinity.
So there you go: a quick and dirty (and honestly, not terribly rigorous) introduction on numbers that are actually larger than infinity. To summarize (again, not very rigorously): infinity plus infinity equals infinity (all positive and all negative integers); infinity times infinity also equals infinity (all rational numbers); but infinity to the power of infinity is bigger (all irrational numbers).
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Mathematics - Etymology
Etymology
The wordmathematics comes from the Greek μάθημα (máthēma), which, in the ancient Greek language, means "what one learns", "what one gets to know", hence also "study" and "science", and in modern Greek just "lesson". The word máthēma is derived from μανθάνω (manthano), while the modern Greek equivalent is μαθαίνω (mathaino), both of which mean "to learn". In Greece, the word for "mathematics" came to have the narrower and more technical meaning "mathematical study", even in Classical times. Its adjective is μαθηματικός (mathēmatikós), meaning "related to learning" or "studious", which likewise further came to mean "mathematical". In particular, μαθηματικὴ τέχνη (mathēmatikḗ tékhnē), Latin: ars mathematica, meant "the mathematical art".
In Latin, and in English until around 1700, the term mathematics more commonly meant "astrology" (or sometimes "astronomy") rather than "mathematics"; the meaning gradually changed to its present one from about 1500 to 1800. This has resulted in several mistranslations: a particularly notorious one is Saint Augustine's warning that Christians should beware of mathematici meaning astrologers, which is sometimes mistranslated as a condemnation of mathematicians.
The apparent plural form in English, like the French plural form les mathématiques (and the less commonly used singular derivative la mathématique), goes back to the Latin neuter pluralmathematica (Cicero), based on the Greek plural τα μαθηματικά (ta mathēmatiká), used by Aristotle (384–322 BC), and meaning roughly "all things mathematical"; although it is plausible that English borrowed only the adjective mathematic(al) and formed the noun mathematics anew, after the pattern of physics and metaphysics, which were inherited from the Greek. In English, the noun mathematics takes singular verb forms. It is often shortened to maths or, in English-speaking North America, math.
Other articles related to "etymology": ancient Egyptians and other |
Topics
Pythagorean Triangle Numerology Calculator
Power Of Numbers
Approximately half of the information in the aforementioned analysis actually comes from one' date of birth; the remaining half comes from one' name. Separate from the 1 to 9 digits, as well. So for example, as being lived right now, male or female qualities, and then added back together.
The table below clearly lays out numbers as they are assigned to all 26 letters in the English alphabet? Approximately half of the information in the aforementioned analysis actually comes from one' date of birth; the remaining half comes from one' name.
Pythagorean Triangle Numerology Free Calculator and Meaning of
By easily applying the simple numerological and mathematical formulas to the numbers representing a one name andor pythagorean triangle numerology calculator date, and also brought forth into one present live, primarily the surrounding numbers involved in their date of birth and name decides one' life destiny. Pythagoras summarized that each planet in thesolar system had its own unique pythagorean triangle numerology calculator. Separate from the 1 to 9 digits, from 1 to 9.
Pythagoras also resolved that some. Each and every number has both positive and negative qualities.
Separate from the 1 to 9 digits, these numbers are also reduced. He also believed that numbers are classified as having the nature of an introvert or extrovert, is pythagorean triangle numerology calculator as "1"; and the letter "B" is "2"; so the letter "C" is "3", and characterized as being ugly or beautiful or ugly.
Each and every number has both positive and negative qualities. Pythagoras and his system of Numerology calculate thus the significant numbers in one' life. Each and every number has both positive and negative qualities.
Iamblichus assembled his verses and Thomas Taylor translated them from the language of Greek. These are the results of bad actions performed in previous births, and then added pythagorean triangle numerology pythagorean triangle numerology calculator together, as being lived right now? Pythagorean triangle numerology calculator and his system of Numerology calculate thus the significant numbers in one' life.
Name numerology calculator in hindi
pythagorean triangle numerology calculator Pythagoreans celebrate sunrise, these numbers are also reduced, he also considers the numbers of 11 and 22 as the master numbers. Plato followed the Pythagoras philosophy. By easily applying the simple numerological and mathematical formulas to the numbers representing a one name andor birth date, is pythagorean triangle numerology calculator as "1"; and the letter "B" is "2"; so the letter "C" is "3", the letter "A". |
Pierre de Fermat
Fermat, Pierre de (1601–65), French mathematician, founder of modern number and probability theories. Fermat's Last Theorem, which was not proven until 1993, states that there is no whole number solution of xn+yn=zn, where x, y, and z are nonzero integers and n is an integer greater than 2. Fermat's Principle states that light (or other waves) will follow the path with the shortest travel time between 2 points |
Aesthetic Preferences in Mathematics: a Case Study
Abstract
Although mathematicians often use it, mathematical beauty is a philosophically challenging concept. How can abstract objects be evaluated as beautiful? Is this related to the way we visualise them?
Using a case study from graph theory (the highly symmetric Petersen graph), this paper tries to analyse aesthetic preferences in mathematical practice and to distinguish genuine aesthetic from epistemic or practical judgements.
It argues that, in making aesthetic judgements, mathematicians may be responding to a combination of perceptual properties of visual representations and mathematical properties of abstract structures; the latter seem to carry greater weight. Mathematical beauty thus primarily involves mathematicians' sensitivity to aesthetics of the abstract. |
The Number Science, Explained
The desire to see the unseen, learn about the future events especially about one's own life and those of near and dear ones and all such wishes and fantasies are the hallmark of the human race.
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The desire to see the unseen, learn about future events especially about one's own life and those of the near and dear ones and all such wishes and fantasies are
the hallmark of the human race. The Homo sapiens having free will distinct themselves from the other species of nature by looking outward, stretching beyond the instincts bestowed upon by Mother
Nature with a strong desire to explore, advance, conquer and unravel the mysteries of nature.
People have different views about creation of the universe, the creator and or evolution, the purpose as to why the universe was created or did it evolve out of
nothing? What is time? What is matter? What are we? And why do we exist etc.? While the subject is humongous in scope, I'll focus on one aspect only, 'The Order'.
If we look at any system no matter how tiny or large it is and no matter where is it located in the universe or beyond the known space, we observe and know that the
common element is 'The Order'. To put it in another way, from the tiniest of particles to colossal galactic systems, everything is arranged in some order internally and externally in relation to
other systems or bodies, manifesting a unifying and intelligent force behind it, using the mathematical laws to strike and maintain balance and the Order.
Interpreting it from the viewpoint of the number science or what is commonly known as numerology, we see that Order when we measure entities by assigning values to
alphabets in names or by simply interpreting the numbers as such based on digits from 1 to 9 and their multiples. If we witness clear patterns emerging out by analyzing the numbers and those
patterns being validated in time as the numbers of observations increase, we realize that we are on the verge of discovering a message. A message from that unifying force that creation is
deliberate and not random. That the entire systems is well conceived and meticulously structured and is by no means an accident. While there are countless variables which may appear to be
external and beyond control, we can certainly position ourselves better in relation to those variables if we know 'How'?
Almost every person has played the jigsaw puzzle or at least knows about it. A jigsaw puzzle requires the assembly of numerous oddly shaped pieces to form a complete
picture. The pieces provide clues and encouragement as you move on. That analogy explains the numbers as pieces of the puzzle and when arranged in correct order they show us the complete picture.
It is a transition from the unknown to known, like in a jigsaw puzzle where you see discordant pieces forming a harmonized picture that you clearly recognize. The only difference is that unlike the
jigsaw puzzle the numbers are dynamic and can form various pictures, simulations and scenarios. If grasped properly that knowledge could be of immense help and has enormous potential to
benefit the entire mankind.
Pythagoras was a great Greek mathematician and philosopher who made influential contributions to mathematics and natural philosophy. He and his disciples believed
that numbers were the true universal language and future could be predicted using the laws of mathematics.
I myself had no belief in numbers or more appropriately in numerology until I had an interaction with a person some fifteen years ago. His revelations kept me baffled
for a long time before I decided to do an intense search to learn about numbers, their meaning and how could we use that knowledge to our benefit.
Over the period I have performed tens of thousands of such calculations and am amazed to realize beyond any doubts whatsoever that the number science or numerology is
a distinct reality and must be seriously studied by academia in order to ascertain its significance and for the subject to earn the acknowledgment and respect it deserves. I have written a few
articles on this subject and thought of sharing my views to reach out to those readers who have little or no knowledge about this subject, or who may simply regard it as occult or
rubbish.
To summarize, numerology does not predict future as some may infer, it simply shows the path or track a name is supposed to take. Changing the name means changing
the values, the final number it adds up to hence resulting in a different path which if chosen wisely should be relatively free of hurdles, making life an enjoyable journey. |
We've never escaped the influence of the Babylonians. That there are 60 seconds in a minute, 60 minutes in an hour, and 360 degrees in a full circle, are all echoes of the Babylonian preference for counting in base 60. An affinity for base 12 (inches in a foot, pence in an old British shilling) is also an offshoot, 12 being a factor of 60.
All this suggests that the Babylonians had a mathematics worth copying, which was why the Greeks did copy it and thereby rooted these number systems in Western tradition. The latest indication of Babylonian mathematical sophistication is the discovery that their astronomers knew that, in effect, the distance traveled by a moving object is equal to the area under the graph of velocity plotted against time. Previously it had been thought that this relationship wasn't recognized until the fourteenth century in Europe. But since historian Mathieu Ossendrijver of the Humboldt University in Berlin found the calculation described in a series of clay tablets inscribed with cuneiform writing in Babylonia during the fourth to the first centuries B.C.E., where it was used to figure out the distance traveled across the sky by the planet Jupiter.
It's a startling find. But media accounts of the work have exaggerated its significance for understanding what the Babylonians were up to when they did astronomy. Some have implied that the tablet shows the Babylonians "invented geometry," or even that they invented calculus. The reports betray an urge to turn the Babylonians into the equivalent of modern-day astronomers in sandals and loin-cloths, diligently mapping the heavens in order to understand the cosmos.
This reception of Ossendrijver's work shows that we still struggle to make sense of what "science" meant before the word, or even the concept, had been invented. On the one hand we are inclined to hype the past, eager to show how they already "knew" what we took to be more recent discoveries. On the other hand we patronize it, claiming (as physicist Steven Weinberg did in his 2015 book To Explain the World) that they'd have gotten a lot further quicker if they'd been able to abandon silly superstitions like astrology and magic.
Historians call this tendency to scrutinize and judge the past according to the perspective of the present "Whig history," a term coined in 1931 by the historian Herbert Butterfield, who criticized the practice for ignoring what people in the past were actually interested in doing. According to Butterfield, by engaging in Whig history we filter and warp the thought of others to make it fit our own—as though the aim of the past was to create the present. Weinberg's book, along with a re-analysis of the "scientific revolution" of the seventeenth century by historian David Wootton (The Invention of Science; 2015), has reignited the arguments about Whiggishness in the history of science. Is it right to seek presentiments of modern science in the works of the ancients, or should we judge their "science" on its own terms?
Babylonians recognized five planets, along with the sun and moon, and called them "wild sheep" because of their wandering paths.
Babylonian astronomy is a great place to weigh up these arguments—because, perhaps more than in any other time and place in history, it shows us an example of what we must regard as astronomy (observing, mapping, and predicting the movements of the stars, planets, sun and moon) being put to a very different use from the one we're used to. The "astronomers" of ancient Greece—scholars like Eratosthenes, Hipparcus, Ptolemy and Aristotle—were interested in trying to understand what the heavens looked like. They constructed a theoretical model of the cosmos in which the planets moved in circular orbits around the earth at its center, which was used almost without exception in the West until Copernicus's heliocentric theory in the sixteenth century. The Greeks "look" to us like scientists, trying to understand the world around them.
The Babylonians were different. They inhabited the region called Mesopotamia between the Tigris and Euphrates rivers (present-day Iraq) in the second and first millennia B.C.E. Babylonian astronomers surveyed the skies carefully, keeping detailed records of the movements of the stars and planets. They recognized five planets (Mercury, Venus, Mars, Jupiter and Saturn), along with the sun and moon, and called them "wild sheep" because of their wandering paths over the fixed backdrop of the stars.
But these studies weren't concerned with "understanding" the cosmos; they were conducted for astrology. The Babylonians believed that their gods transmitted messages about the future through the appearance of the celestial bodies: when planets rose over the horizon, what color they were, when they stood in certain arrangements or conjunctions, when eclipses happened, and so on. It was the task of scholarly diviners to interpret these messages, so that they could deliver sound advice to the king.
Reading these signs was complicated and subjective. Certain features were omens, considered to presage particular events. One document, for example, suggests that if the moon is still visible on the 30th day of the lunar cycle then destruction of Babylonia is on the cards, but if the moon is seen on the first day of the cycle then good luck will follow. Mars, meanwhile, was a harbinger of evil, whereas Jupiter brought peace and plenty. With all the different possible parameters, it was never easy to read the gods' code. That was the diviners' job, and it more closely resembles a legal process of weighing up precedents than a scientist's aim of finding an objective truth that "explains" the observations.
While divination remained pretty much the sole motivation for astronomy for the duration of Babylonian civilization, the methods changed significantly from around the late seventh century B.C.E. It was then that the concept of a zodiac appeared: The sky was divided into 12 segments, identified with particular constellations (although the stars themselves rotate across the zodiacal divisions), which became one of the organizing principles for making predictions. This zodiac was inherited by the Greeks, along with the Babylonian techniques for using it to cast horoscopes—it's the system still used for astrology today, when we speak of, say, "Jupiter in Aquarius."
Even more significantly, the "new" Babylonian astronomy differed from the old in that it made predictions about how the heavens would look. The ancient Babylonians couldn't fail to notice that many astronomical events recur periodically—for example, that there is a cycle of 223 months in the pattern of eclipses, or that Saturn rises in the same place in the sky every 59 years. Even for these things to become evident required careful observation and record keeping over generations.
These regularities were previously seen as quirks of the gods' messaging system, but around the 7th century B.C.E. scholars began to appreciate that they could be used to predict how the night skies would look in the future. This might seem obvious now. But with so many variables to think about, some of which (like the color of the moon or brightness of the planets) weren't really predictable, it wasn't clear how much significance to give to periodic movements. It seems likely that the move towards predictive astronomy came not from any desire to understand the cosmos better or to explain what was seen in terms of some underlying model, but from royal demands for better forecasts. Faced with such a challenge, scholars vied with each other to develop predictive methods, which stimulated the increasing use of mathematics—like the recently discovered methods used to track Jupiter.
What resulted does look like science in some respects.
What resulted does look like science in some respects. For one thing, an astronomical prediction, unlike the older forecasts, can be falsified by observation. If you wrongly predicted the king's health, you could say that you misread the signs (and woe betide you). But if you predict an eclipse that doesn't happen, it's your mathematical method that's at fault. Mathematical astronomy is, like science, objective and value-free, operates with known rules, and depends on careful measurements and records of data.
That might leave Weinberg and like-minded supporters of Whiggish science history breathing a sigh of relief, but it doesn't mean that the new Babylonian astronomy was a sort of proto-science. For one thing, the Babylonians, unlike the Greeks, seem to have had no concept of, or interest in, any mechanism that explained the celestial dance. They didn't think in terms of planets making physical orbits, around the earth or sun or anything else. Sure, they would calculate how far Jupiter had "moved" in the sky. But this motion was more like that of the hands of a clock—as opposed to the gears—the new position signified a new meaning, but who cares about how the movement happens? That's a matter for the gods.
Even eclipses weren't pictured in terms of a physical conjunction of sun, moon and earth; if they were "explained" at all, it was in mythical terms, for example the god Sin (symbolized by the moon) becoming surrounded by demons. Greek astrology and later Western iterations, in contrast, depended increasingly on ideas about physical mechanisms by which the planets and stars might affect events on earth, for example because of "emanations" that the celestial bodies exerted just as the sun emanated heat and light.
One common responses to all this is to say that the Babylonians began to discover objective, "scientific" knowledge about the world by accident, for the wrong reasons: Their motives may have been misguided, but they got useful knowledge all the same. By the same token, it's sometimes said, the alchemists discovered a lot of handy chemistry in their fool's quest to make gold. But trying to sift the past for morsels that can be called precursors of science makes for bad history. Babylonian astronomy was not an "imperfect science" but a self-contained intellectual framework woven into the rest of their culture. Of course, no Babylonian scholar ever really foresaw the death of a king or victory in battle written in the stars. Yet Babylonian astrology wouldn't have survived for over a millennium if it hadn't in some sense "worked" in its own terms: if it hadn't helped to order and stabilize society. And to the extent that it created viable mathematics and astronomy, this didn't happen because the practitioners, like today's scientists, were seeking some abstract truth about the world— it came about because of immediate practical and political concerns.
Does this mean that the minds of the Babylonians are a closed book, so different from ours in their worldview that we can't meaningfully seek any continuity between them? I don't think so. The one thing predictive Babylonian astronomy surely shares with modern science is a belief that the universe embodies and obeys some order knowable by human minds: Not everything happens through the immediate whims of the gods. And if that's so, the gods themselves recede just a little further, and we trust a little more that, if we ask the universe questions, it will give us fathomable answers.
About the Author
Philip Ball is a writer based in London. His work has appeared in Nature, The New York Times, and Chemistry World. He is the author of The Water Kingdom: A Secret History of China and Critical Mass: How One Thing Leads to Another |
Pages
1. Introduction
Project Overview
We will be covering on how maths is applied in the solution to the rubik's cube. For example how does inequalities help to identify the total possible number of permutations of the cube or how algorithms come into play when deciding which face tor rotate and so much more! Almost everyone has tried to solve a Rubik's cube. The first attempt often
ends in vain with only a jumbled mess of coloured cubies (as I will call one
small cube in the bigger Rubik's cube) in no coherent order. Solving the cube
becomes almost trivial once a certain core set of algorithms, called macros,
are learned. Using basic group theory, the reason these solutions are not
incredibly difficult to find will become clear. |
Tag: engineering
Besides being an obvious lady killer, Swiss mathematician Leonhard Euler gifted the world with some pretty important mathematical concepts, notational conventions, and formulas. I almost feel bad about the fact that I couldn't even spell his name correctly until I was well into adulthood. You are probably thinking, "Sure, he had a bitchin' robe, and for … |
Dodecahedron
In geometry , a DODECAHEDRON (Greek δωδεκάεδρον, from
δώδεκα dōdeka "twelve" + ἕδρα hédra "base", "seat" or
"face") is any polyhedron with twelve flat faces. The most familiar
dodecahedron is the regular dodecahedron , which is a
Platonic solid Platonic solid .
There are also three regular star dodecahedra , which are constructed
as stellations of the convex form. All of these have icosahedral
symmetry , order 120. The pyritohedron is an irregular pentagonal dodecahedron, having the
same topology as the regular one but pyritohedral symmetry while the
tetartoid has tetrahedral symmetry . The rhombic dodecahedron , seen
as a limiting case of the pyritohedron, has octahedral symmetry . The
elongated dodecahedron and trapezo-rhombic dodecahedron variations,
along with the rhombic dodecahedra, are space-filling . There are a
large number of other dodecahedraList Of Regular Polytopes
This page lists the regular polytopes and regular polytope compounds
in Euclidean , spherical and hyperbolic spaces. The
Schläfli symbol describes every regular tessellation of an
n-sphere, Euclidean and hyperbolic spaces. A Schläfli symbol
describing an n-polytope equivalently describes a tessellation of an
(n−1)-sphere. In addition, the symmetry of a regular polytope or
tessellation is expressed as a
Coxeter group , which
Coxeter Coxeter expressed
identically to the Schläfli symbol, except delimiting by square
brackets, a notation that is called
Coxeter Coxeter notation . Another related
symbol is the
Coxeter-Dynkin diagram which represents a symmetry group
with no rings, and the represents regular polytope or tessellation
with a ring on the first node
[...More...]
Ludwig Schläfli
LUDWIG SCHLäFLI (15 January 1814 – 20 March 1895) was a Swiss
mathematician, specialising in geometry and complex analysis (at the
time called function theory) who was one of the key figures in
developing the notion of higher-dimensional spaces. The concept of
multidimensionality has come to play a pivotal role in physics , and
is a common element in science fiction. CONTENTS* 1 Life and career * 1.1 Youth and education
* 1.2 Teaching
* 1.3 Later life * 2 Higher dimensions
* 3 Polytopes
* 4 See also
* 5 References
* 6 External links LIFE AND CAREERYOUTH AND EDUCATIONLudwig spent most of his life in
Switzerland Switzerland . He was born in
Grasswil (now part of
Seeberg ), his mother's hometown. The family
then moved to the nearby Burgdorf , where his father worked as a
tradesman . His father wanted Ludwig to follow in his footsteps, but
Ludwig was not cut out for practical work
[...More...]
Recursive Definition
A RECURSIVE DEFINITION (or INDUCTIVE DEFINITION) in mathematical
logic and computer science is used to define the elements in a set in
terms of other elements in the set (Aczel 1978:740ff). A recursive definition of a function defines values of the functions
for some inputs in terms of the values of the same function for other
inputs. For example, the factorial function n! is defined by the rules
0! = 1. (n+1)! = (n+1)·n!. This definition is valid for all n, because the recursion eventually
reaches the BASE CASE of 0. The definition may also be thought of as
giving a procedure describing how to construct the function n!,
starting from n = 0 and proceeding onwards with n = 1, n = 2, n = 3
etc.. The recursion theorem states that such a definition indeed defines a
function. The proof uses mathematical induction . An inductive definition of a set describes the elements in a set in
terms of other elements in the setEquilateral Triangle
In geometry , an EQUILATERAL TRIANGLE is a triangle in which all
three sides are equal. In the familiar
Euclidean geometry ,
equilateral triangles are also equiangular ; that is, all three
internal angles are also congruent to each other and are each 60°.
They are regular polygons , and can therefore also be referred to as
regular triangles
[...More...]
Square (geometry)
In geometry , a SQUARE is a regular quadrilateral , which means that
it has four equal sides and four equal angles (90-degree angles, or
right angles ). It can also be defined as a rectangle in which two
adjacent sides have equal length. A square with vertices ABCD would be
denoted {displaystyle square } ABCD
[...More...]
Regular PentagonStar Polygon
In geometry , a STAR POLYGON is a type of non-convex polygon . Only
the REGULAR STAR POLYGONS have been studied in any depth; star
polygons in general appear not to have been formally defined.
Branko Grünbaum identified two primary definitions used by
Kepler ,
one being the regular star polygons with intersecting edges that don't
generate new vertices, and the second being simple isotoxal concave
polygons . The first usage is included in polygrams which includes polygons like
the pentagram but also compound figures like the hexagram
[...More...]
Irreducible Fraction
An IRREDUCIBLE FRACTION (or FRACTION IN LOWEST TERMS or REDUCED
FRACTION) is a fraction in which the numerator and denominator are
integers that have no other common divisors than 1 (and -1, when
negative numbers are considered). In other words, a fraction a⁄b is
irreducible if and only if a and b are coprime , that is, if a and b
have a greatest common divisor of 1. In higher mathematics ,
"IRREDUCIBLE FRACTION" may also refer to rational fractions such that
the numerator and the denominator are coprime polynomials . Every
positive rational number can be represented as an irreducible fraction
in exactly one way. An equivalent definition is sometimes useful: if a, b are integers,
then the fraction a⁄b is irreducible if and only if there is no
other equal fraction c⁄d such that c < a or d < b, where a
means the absolute value of a. (Two fractions a⁄b and c⁄d are
equal or equivalent if and only if ad = bc.) For example, 1⁄4, 5⁄6, and −101⁄100 are all irreducible
fractions
[...More...]
Pentagram
A PENTAGRAM (sometimes known as a PENTALPHA or PENTANGLE or a STAR
PENTAGON ) is the shape of a five-pointed star drawn with five
straight strokes. Pentagrams were used symbolically in ancient Greece and
Babylonia Babylonia ,
and are used today as a symbol of faith by many Wiccans , akin to the
use of the cross by Christians and the
Star of David Star of David by Jews. The
pentagram has magical associations, and many people who practice
Neopagan faiths wear jewelry incorporating the symbol. Christians once
more commonly used the pentagram to represent the five wounds of Jesus
. The pentagram has associations with
Freemasonry and is also
utilized by other belief systems. The word pentagram comes from the Greek word πεντάγραμμον
(pentagrammon), from πέντε (pente), "five" + γραμμή
(grammē), "line"
[...More...] |
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The High Cost of Being Hopeless with Math
"I don't do math—numbers give me a rash." That's a line I've used a lot, mostly because it's true, but also because it gets me a laugh.
Truth be told, most of us stink when it comes to doing the math on the fly. That's a problem because being hopeless with math makes us putty in the hands of retailers.
Why is it socially acceptable to say that we're bad at math, but not to say we're bad at reading? The truth is that it's not okay to be hopeless with numbers. Here are three ways that our aversion to math costs us money:
The number 9.Amazingly, 65 percent of all retail prices end in the number 9. Unconsciously, we're charmed into believing the item is a bargain.
Whenever you see a product priced at $29.99 or $9.99, the retailer is attempting to "charm" your brain by marking prices just below a round number. Because our brains are trained to read from left to right, the first digit is the one that sticks in our head and the number we use to decide if the "price is right."
Both Steve Jobs, who came up with the 99-cent app, and the guy in California who founded the 99-Cents Only stores have made millions off this human quirk. Retailers use 9 on purpose to lure us into buying something because they know we'll assume it's been discounted.
This phenomenon is known as the left-digit effect and studies have shown that it absolutely works and has a big impact on our buying decisions. So whenever you see a price ending with a .99, get in the habit of rounding up, then decide if it's a good deal.
The good news is that simply being aware of the ways that retailers use the number 9 can break their spell over us. Whenever you see a price ending with a 9, stop and think about what's going on.
Intentional confusion. It's a trick retailers use all the time: A confused customer is more likely to opt for the higher price deal. An item marked 5/$4 prompts us to buy five items for $4, not 1 item for $.80, because it's too confusing.
Which is the better deal: 33 percent off the regular price or 33 percent more product for the same price? Studies show that most people go for the 33-percent more deal because they don't know how to do the math and they simply guess. And they're wrong.
Thirty-three percent off is the same as a 50-percent increase in the quantity. Let me show you: If the regular price is $1 for 3 pounds, 33-percent off means you get 3 pounds for $.66 or $.22 per pound. If you opt for 33 percent more, you'll get 4 pounds for $1 or $.25 per pound. The secret is to figure out the per unit price—per ounce, per quart, per pound. Now it's easy to compare.
Priceignorance. In his book Priceless, author William Poundstone tells the story of the retailer Williams-Sonoma and a $279 bread maker. Sales were lagging so they placed a nearly identical machine next to the $279 bread maker with a price tag of $429. Immediately, sales doubled on the $279 model because it appeared to be 40 percent cheaper, and therefore a great deal. The same tactic is in play when you see a big display in the supermarket with a sign that reads, "Special!" If you are not up on your prices, you'll fall for every trick retailers have up their sleeves to get us to spend more money.
Getting good with math, I'm discovering, starts with my attitude. That's why I am never again going to tell myself or anyone else that I'm bad with math. I'm doing brain calisthenics. And while forcing myself to figure out per unit prices on the fly is a good exercise, I'm also learning that a pocket calculator is my friendSharon Madison
Loved this column! You are so right! Retail sales and marketing strategies manipulate our minds even more than you'd even imagine. I've always found this to be fascinating. Just a few examples to add:
Malls do not have clocks in easily visible. That's so you'll stay as long as possible. when a mile is built, you can very rarely see the end of a hallway, they are broken up so that there are trees and skylights and nooks and crannies so that you won't look down the hallway and decide anything is too far away.
In stores like Macy's or Sears, there will be carpeting and soft lighting, and absolutely no shopping carts. If you're going someplace like a more upscale store, it's assumed that you are there for an enjoyable experience, and not necessarily for bargain-hunting. All the new things will be at the front of the store, and all the bargains will be in the back if you are looking for bargains.
Warehouse stores, on the other hand, are very low maintenance, with concrete floors and ductwork visible overhead, so you'll know that they're saving all kinds of money on entertaining you, because you are there for the bargains. Shopping carts are to make it easy for you to carry around all the items without thinking about how heavy they are.
And these are just a few examples that anybody can think about in realize are true. It's up to us to be better consumers.
Anne
I find it maddening that different TP and tissue brands use different size packages. I'm sure that's so you can't easily compare prices, but that doesn't stop me! I pull out a calculator and figure out the price per ounce/pound/roll, whatever it is I'm looking for. I do this both in stores and when shopping online.
Rich Rorex
This is amplified in the tale of a grocer who placed some cans of vegetables on a display with the sign saying "2 for 25 cents" and a sign saying "10 cents each" om the display. When the cashier would ring them up for ten cents each many of the shoppers would complain that they were two for twenty-five cents. Explaining the math made the day for the cashier.
Patricia Stariha Roy
Love this column,,,,should be "buyer beware". When our daughter was younger, I hated taking her to the grocery store. Always asking for the horrible sugary cereals, candy, etc. If you have kids, we have all been there. Until I made a game for her. She asked me if we could get Oreos. Now I had planned to get cookies anyway, but they certainly are not a necessity. So here is what I cam up with. I told her sure, but she had to figure out which package was the best deal. If she was correct, we would buy them, if not, nope. The first time she didn't think about it…she just grabbed the super-size package and brought it to the cart. But when we analyzed the cost per cookie, it was quite a bit more. She was shocked. And we did not get the cookies. We did that all the time, and at one point she asked if she could use a calculator, and started bringing it to the store with us. YES!!! She learned 2 great lessons. First, a bigger package is NOT necessarily the best deal, AND second, that it pays to be able to do math on the fly. How many times do you hear kids say "I'll never use this algebra"? Well, if their week being with or without cookies depends on it, they find it useful real quickly. I can't tell you how often she has thanked me for making her do these kinds of exercises as she was growing up. She knows she is often way ahead of her peers because of the skills we learned in the grocery store, and incidentally, she was her high school valedictorian, and an elementary school principal at 34.
tinydogpries
I had no idea that ANYONE didn't automatically round up when they saw a price ending in .99. I round up anything that doesn't end in .00. I have also learned to look for the shelf tags that quite often give you the price per oz., lb., etc. I have to laugh when I am buying just a few items and I get to the register and hand the cashier the exact amount of the total without having to hunt for the money because I had it figured up before I got to the register. And no, I don't ever use a calculator. I HATE calculators. I was not good at math in school, but after working in factories for most of my working life I became VERY good at doing figures mentally.
crabbyoldlady
I became good at retail pricing when I worked at Target.
crabbyoldlady
I am good at retail mental math, having worked at Target. However, with today's smart phones with a built in calculator, there is no reason for anyone to wonder what the price of an item really is.
LB Girl
There is a calculator built into your iPhone. I use it all the time! You can also google a quantity comparison tool for comparing liters to ounces, etc.
Maggie Graham
1 – are there still people who don't round up from .99? 2 – even 'stupid phones' have a calculator, although you may need to search for it; there is a solar-powered calculator that lives next to my wallet. 3 – are there still people who shop without a carefully-worked-out list?
Robin Phillips
Knowing math when the other person doesn't , works in your favor. Recently I had to replace my central air unit. After lacks of response from companies I found a company willing to do it in the middle of a heat wave. I was quoted a price 0f almost $5,000 which I thought was high but couldn't compare since other companies never got back to me. I asked technician if he was the company that offerd %10 off(I had called so many). He said yeah, OK and agreed to 10% off. He said"so that will be $4950. I corrected him and said "no, price is %4500". DSince he had agreed to it, he honored it. Shows you how much of a makup he quoted.
Ruth Rougeux,
Math gives you a rash math gets me to tears. Just like the above problem I don't get it. I try really hard. My 3rd grade teacher saw my bad math and it has followed me but I try to carry on. I even took an adult math class offered by our CIU and it was bad but there have been changes in math symbols etc. and we aren't aware. There is this woman @ church that loves math I don't get that either. She is a cool women in spite. I do rely on people and their trust and honesty. God Bless you! Olanta PA |
hi rod.
I have no clue what most of this stuff means. I was going to ask you how you felt about geometry? I'm watching an episode of ancient aliens and they're talking about the Nazca lines, crop circles, and other similar geometric communications. They talked about a snake being imbed in the earth where a meteor hit indicating the importance in the element carried over on the meteor... no doubt geometry is a language and an art- so I was curious as to your mission here with this language and what this all means to you.
It's a sober night here... that's a good thing
Yes, this geometry research seems to be a mission ... with symbolism for the future. After many thousands of hours of research, I'm still in agreement with mathematicians who have "proven" that a circle cannot be squared. However, I'm seeing convincing geometry that "proves" that squared circles exist!
Is "You can't get there from here!" the message?
... even though there exists!" |
The geometry of the Einstein equations12/04/2017
|
3:05 PMThe NWA revolves around a collection of questions to which the Dutch research community will pay particular attention in the coming years. This includes fundamental questions about matter, space and time. To kickstart this programme the national research foundation NWO has awarded funds for eight projects. One of these focusses on innovative research problems related to gravity, which are particularly timely in view of the recent spectacular detection of gravitational waves.
In the project at the department of mathematics, postdoctoral researcher Thomas Rot will connect modern differential geometry with the theory of gravity. The Einstein equations, which describe the geometry of space and time, are very difficult to solve. More than a century ago, the prominent mathematician David Hilbert proposed to study the Einstein equations using an optimisation method. In this optimisation problem one needs to find mountain passes in an infinite dimensional mountainous landscape. Such problems turn out to be so complicated that only recently mathematical techniques for their investigation have been developed. The goal of this project is to examine how these modern methods from geometry, topology and the theory of geometric partial differential equations can lead to novel insights in the existence of yet unknown solutions of Einstein's equations. |
"It's looking for problems online and trying them out," said Andy Tsao, a senior and president of the math club.
Even though these students talk about the subject enthusiastically, math club adviser P.J. Yim said this passion doesn't add up for all students.
Those who are drawn to math truly have a knack for it, he said.
Math club students are often solving problems far beyond those they are assigned in the classroom.
In fact, he said, they are completing proofs and number theories so complicated that college math majors would have a difficult time solving them.
Yim said there are a number of reasons children become active in math, including parental influence and early exposure.
Regardless of the reason, he said the school's Bay Area location is key in drawing a concentrated number of these students.
Because the club has an open-door policy, students searching for challenges often show up in his classroom during lunch to practice problems and prepare for upcoming math competitions.
Interested students participate in three major kinds of contests throughout the year, including regional and national competitions over the course of the school year; regional math contests hosted by local universities or organizations; and national contests that lead to the USA Mathematical Olympiad and possibly of competing internationally.
Results from some of these contests continually prove that the school has some of the best high school mathematicians in the country.
Among their recent accomplishments, the club earned 11th place at the 2008 Ciphering Time Trials, a national mathematics contest administered by National Assessment and Testing.
The contest consisted of 10 rounds in which club members had three minutes to solve three problems.
Several students also received individual awards. Brian Wai placed 12th in the ninth-grade division, Albert Gu placed 10th and Amol placed 17th in the 10th-grade division, and Rolland placed 24th in the 11th-grade division.
Despite these accomplishments, the students agreed that they continue moving forward in their quest for knowledge, with the fun activity of mathematics as the root cause |
PowerPoint Slideshow about 'Announcements 10/6/10' - gregory-vaugh: when you add two sines or cosines having the same frequency (with possibly different amplitudes and phases), you get a sine wave with the same frequency! (but a still-different amplitude and phase)
"Proof" with Mathematica… (class make up numbers)
Worked problem: how do you find mathematically what the amplitude and phase are? |
mathematics things |
The Golden Ratio & Math Rock-Van Huynh
Professor Vesna's lecture discussing the golden ratio intrigued me. I have never understood what exactly it was or why it was used until it was mentioned. I understand that the golden ratio, approximately equal to 1.6180339887 is found in nature in which people believe is the measurement of things found that are divine, especially pleasing to the eye, or simply "beautiful." Through browsing the web, I found and discovered that the golden ratio can be found in architecture, the human body, nature, animals, and practical things we use today.
In the article noted above, the author generalizes that people happen to be more attractive ad believe beauty is in "symmetrical and proportional" people and things, like Jessica Simpson. However, this observation is clearly an assumption of people today. In my opinion, I do not believe Jessica Simpson is beautiful due to the impression I have of her, superficial, ditsy, and absurd remarks. This defies the underlying interest of the golden ratio. I believe that beauty is skin-deep, where I find it in one's personalities, strengths, and actions.
When I heard that Yoshida would be coming to perform for us on Wednesday, I was skeptical. I was wondering what exactly is math rock? Do bands just play guitar and sing about mathematical formulas? However, Yoshida's performance amazed me. It really opened my eyes. I noticed the passion in Yoshida's performance through the intense rhythms, and precise structure of the music, overall. I finally understood why it was called math rock. It was not because it consisted of lyrics relating to actual mathematics, but the music, itself, is so intricate and precise, using different time signatures from the norm, creating this new music. It was an exotic and trippy sound giving me this exotic feeling that I was in a circus house. I was urged to start a mosh pit in the room, but considering the size of the room, chairs, and people, I sat there in a sort of trance, listening to in the drumming, guitar playing, and emotional vibe Yoshida gave off. Having experienced this new type of music for myself really gave me a new appreciation for different music I have not heard of. I've learned that music isn't in the glam, the sensual and sexual feelings, the conformity, but is about the actual made created by the musician, the artist.
During the session discussion, we were taught by T.A.'s John and Lis about fractals. This interested me because I have a fascination for intricate shapes and patterns. It urged me to browse the internet for some more examples and here are the few I came across and extremely liked:
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Sunday, 28 December 2014
Intuition and slow thinking
This post is going to be more questions than answers...
Some things have been going through my head. There's Kassia's post, Is There Room For Math That Isn't Hard? Also, a conversation on Twitter, one strand in a bigger conversation about intuition in maths learning. Things can get a bit abstract when you're down to 140 characters including names, but there were a lot of great points. Here's one definition of intuition that Kristin posted that I liked:
I asked them to write their thoughts on their whiteboards. All of them, all of them, gave me a numerical answer! That really surprised me. I thought lots would, but all? I showed the classes the video on Tracy's blog afterwards, and very briefly talked about how some questions don't have answers.
Somehow, these things link, in my mind at least, because we need a solid base of intuitions about maths - partly what we call "number sense" - that helps us to deal with both meaningful and meaningless questions, and to tell the difference!
I also reached down Guy Claxton's brilliant book Hare Brain, Tortoise Mind from the bookshelf.
Claxton says there are three processing speeds in the brain. The fastest, faster than thinking, is the kind of response we have when we skid on ice and just do the right thing. It's the sort of processing a concert pianist or an Olympic fencer has to do. Then there's thinking itself, deliberation, which he calls d-mode. But "below this, there is another mental register that proceeds more slowly still. It is often less purposeful and clear-cut, more playful, leisurely or dreamy."
It maybe helps to look at deliberation, the familiar kind of thinking, first. Claxton lists some of its features:
D-mode
1. is much more interested in finding answers and solutions than in examining the questions.
2. treats perception as unproblematic.
3. sees conscious articulate understanding as the essential basis for action, and thought as the essential problem-solving tool.
4. values explanation over observation
5. likes explanations and plans that are 'reasonable' and justifiable, rather than intuitive.
6. seeks and prefers clarity, and neither likes nor values confusion.
7. operates with a sense of urgency and impatience.
8. is purposeful and effortful rather than playful.
9. is precise.
10. relies on language that appears to be literal and explicit
11. works with concepts and generalisations
12. must operate at the rates at which language can be received, produced, and processed.
13. works well when tackling problems which can be treated as an assemblage of nameable parts.
So, d-mode is how we operate in maths lessons. You could even see mathematics as the place in which it shines most brilliantly.
"Most striking at first is this appearance of sudden illumination, a manifest sign of long, unconscious prior work. The role of this unconscious work in mathematical invention appears to me incontestable, and traces of it would be found in other cases where it is less evident. Often when one works at a hard question, nothing good is accomplished at the first attack. Then one takes a rest, longer or shorter, and sits down anew to the work. During the first half-hour, as before, nothing is found, and then all of a sudden the decisive idea presents itself to the mind. It might be said that the conscious work has been more fruitful because it has been interrupted and the rest has given back to the mind its force and freshness."
(Claxton also gives lots of experimental results from cognitive psychology that demonstrate the effect of slow thinking. I'm glad he does this because words like intuition can sound unscientific, which they're evidently not.)
How to descend from this abstractness then? Is there a place for encouraging slow thinking and intuition in the primary classroom?
A few tentative answers. One: when you ask children what they notice, the pace slows down. There's time for a bit of pondering. Developing this as a regular part of lessons, and the respectful listening and responding that goes with it, allows half-formed and ill-expressed ideas space to breathe and develop.
Two: games. I got the classes playing Daniel Finkel's Prime Climb three times this term. There was no "teaching", apart from, briefly, how to play the game. But I feel that time when students aren't thinking, "I must learn this," is precious. Their hare brain's can be off duty. The games weren't physically slow. Lots of the kids were standing up! But... I hadn't "taught" anything. Slow in that way.
Maybe there's not time for slow thinking in your class. I understand. There's more pressure than ever to pack the learning in, to get the results. And we know ultimately, results will lead to jobs...
So, is there time to slow down?
Is it worth it?
If there is, and it is, what are good ways to do it?
Does it link with number sense?
Does it link with intuition?
Does this help with my meaningless number question?
Do you have any answers? Or more questions?
UPDATE - July 2015
I was really pleased when Gracia, towards the end of the year, came up with this question in class:
She knew it linked back to that "how old is the shepherd?" question we'd looked at before. Still, some people were not getting it. But some were now. As Gracia put it, all that information distracts you; it's like a magic show.
Other classic word problems are expressly designed to trick their victims by misdirection, like a magician's sleight of hand. The phrasing of the question sets a trap. If you answer by instinct, you'll probably fall for it. Try this one. Suppose three men can paint three fences in three hours. How long would it take one man to paint one fence? It's tempting to blurt out "one hour." The words themselves nudge you that way. The drumbeat in the first sentence — three men, three fences, three hours — catches your attention by establishing a rhythm, so when the next sentence repeats the pattern with one man, one fence, hours, it's hard to resist filling in the blank with "one." The parallel construction suggests an answer that's linguistically right but mathematically wrong. The correct answer is three hours. If you visualize the problem — mentally picture three men painting three fences and all finishing after three hours, just as the problem states — the right answer becomes clear. For all three fences to be done after three hours, each man must have spent three hours on his. The undistracted reasoning that this problem requires is one of the most valuable things about word problems. They force us to pause and think, often in unfamiliar ways. They give us practice in being mindful.
Quite brief really. I said I was surprised because there wasn't really any information about the teacher's age. There had been no thoughts on the whiteboards and I asked why. One girl said they had put numbers because they thought that's what I wanted! We watched the video - There was some laughter. I said we should look out for whether there's enough information in a question. Then we moved on.
I have toyed with the idea of returning to this briefly and getting the kids to write their own meaningless questions! |
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Video of lecture on prime numbers
Wednesday 25 October
Vicky Neale (Whitehead Lecturer in Mathematics and Supernumerary Fellow) gave a public lecture about prime numbers at Oxford's Mathematical Institute. The talk, now available on video, marked the recent publication of her book Closing the Gap: The Quest to Understand Prime Numbers (OUP, 2017). |
This is a depiction of why the numerals that we use look the way they do: However, these symbols were designed for base 10. In computer science, the hexadecimal number system is used, but with the letters A-F representing 10-15. Should we replace A-F using symbols similar to the Hindu-Arabic numerals to make the system more meaningful in terms of mathematics? Hexadecimal looks rather strange with a combination of letters and "numerals." In fact, there are times where it's not obvious to tell the difference between whether a letter is a number or a variable. E.g. Is 5A a number or 5 multiplied by A?
Mathematicians also have this urge to try to convince the public to use the duodecimal system (base 12). Would base 12 look less "mathematically aesthetic" if letters or maybe manipulated versions of the Hindu-Arabic numerals were used? Just some thoughts on this topic. |
Science and music hurtling earthwards at eye-watering velocity
Category: Maths
If you're in central London tonight, or during the day this week, you should find a few moments to stop by The Royal Society. The national academy of science of the UK and the Commonwealth is staging their Summer Science Exhibition. Not only are they putting on a week of exhibits from the cutting edge of science but also featuring involved scientists themselves for you to ask questions of.
What a cool opportunity. This is a direct public-engagement event. You can look at items and exhibits and models lots of places, but how often do you get a chance to ask questions of a real, live scientist? There's a list of exhibitshere, along with writeups that indicate which ones might be good for kids.
From their site:
We've got over 20 fascinating, diverse and interactive exhibits. Fields of study range from how fluorescent fish could provide better understanding of human diseases, to a chewing robot that can help us develop dental technology, to how new space missions could help to unlock the history of the universe.
There's also a good writeup at Nature Network's London blog about the exhibition.
Hypatia of Alexandria was an ancient Greek mathematician, most notable for being the first famous female maths scholar. In addition to writing commentaries on and editing some of the most famous early works of maths, like Arithmetica and Euclid's Elements, she's said to have done work in astronomy, measuring the relative density of liquids, and possibly independently developing an astrolabe.
A new film – named Agora – based on the life of Hypatia, who's played in the movie by Rachel Weisz, has received big cheers at Cannes. That's pretty cool: Greek mathematicians aren't your typical silver-screen subjects.
Cryptography – the science of hiding information – is used to secure internet communications and commerce. Most internet cryptography uses a technique called RSA which relies on the difficulty inherent in determining factors for very large integers.
In recent decades mathematicians have developed techniques using elliptical functions to more easily do large-number factorisation. While this implies RSA encryption is easier to break using these methods, salvation has come from those same elliptical functions. They can themselves be used as a form of encryption.
In recent years this elliptic curve cryptography has gained attention. Since it's newer it's not prone to the majority of crypto-attacks, which have been developed to attack RSA. But it's also been shown that you can get the same level of protection with elliptical encryption by using a much smaller encryption key than is needed for RSA. This makes it more computationally efficient, and helps keeps ahead of those who would attack and break those systems of protection. |
On
Saturday afternoon, an exclusive interview took place with
Archimedes. He established the principles of pl\ane and solid
geometry, discovered the concept of specific gravity, conducted
experiments with buoyancy, and demonstrated the p ower of
mechanical advantage. He is known as the most original and
profound mathematician of ancient times.
Q: Lets start off with some general questions for our viewers,
what was your life like?
Archimedes: I was born over 2,000 years ago around 287 B.C. in
the city of Syracuse on Sicily, an island near the
"toe" of Italy. My father was named Pheidias, and he
was an astronomer. At that time Syracuse was a Greek City, even
though it was very far from the Greek mainland. Back in those
times the civilization of Ancient Greece spread all through the
eastern part of the Mediterranean region. I was schooled at
Euclid's school in Alexandria, Egypt. This was one of the
biggest cities of all time. I was taught internal calculus, which
included the studying of volume. I spent most of my life in
Syracuse, but I had no public office to work in. I devoted my
entire lifetime towards science research and development.
Q: What were some of your discoveries?
Archimedes: Throughout my life I invented many war machines used
in the defense of Syracuse, compound pulley systems,
planetariums, the water screw, and the water organ. The screw
consists of a cylinder inside of which a continuous screw,
extending the length of the cylinder forms a spiral chamber. By
placing the lower end in the water and spinning the screw, the
water raises to the top. This principle was applied in many
machines used in drainage and irrigation, and was commonly used
in the Nile Valley. I have made many other machines through
mathematical calculations that make it possible to remove very
heavy things with small force.
Q: What was a memorable moment in your life if you can
remember?
Archimedes: Well, I did most of my work for King Hiero (King of
Syracuse). One time he had suspected that a goldsmith had not
made a new crown of pure gold, but had mixed in some less costly
silver. The king asked me to investigate the
situation. I spent a lot of time thinking about
how to solve the problem, because back then we did not have much
technology. I found the answer while I was taking a bath. I
noticed that when I got in the bathtub water fell out. By
measuring how much water fell out I realized I could measure the
volume of my body. Therefore I could compare the amount of water
displaced by the crown to the amount of water displaced by an
equal weight of pure gold. The crown had displaced more water,
and therefore was not pure gold.
Q: Throughout your lifetime you have been appreciated by many for
making numerous discoveries. Which discovery are you most proud
of?
Archimedes: Back around my
time, we had no idea what pi was. I made further contributions
into closer discovering what pi was. What I did was draw two
polygons, one around the outside of the circle (so its perimeter
was greater than the circle's), and one inside the circle
(so its perimeter was less than the circles). I came to the
conclusion that the perimiter of the large square is 4D, if the
diameter was established as "D". So therefore pi is
less than 4D/D = 4. If the perimeter of the small square is
4D/Rad(2) and the diameter is still "D" then
(4D/Rad(2))/D = 4/Rad(2) = 2.828. Using this principle I used
polygons with more sides to find pi to the nearest number. The
largest polygon I used had 96 sides, and with that I determined
pi to be between 3 10/70 and 3 10/71!
Q: That is quite tremendous! Are you satisfied with all of your
accomplishments?
Archimedes: Yes, Absolutely. After all, without my discoveries,
what a different world this would be!
Works Cited
"Archimedes." The World Book
Encyclopedia. World Book Inc. Chicago Il, 2002.
Gave good background
information and gave some explanation on some achievements and
important discoveries Archimedes had. This is always a solid
source.
This page was
also very good at giving interactive images showing how
Archimedes did when he solved all of his discoveries. It has
everything he did on every subject of math, showing how clever he
really was. This is probably the best source out of all four. |
Math in everyday life essay
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Math in everyday life essay
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Through the years, and probably through the centuries, teachers have struggled to make math meaningful by providing students with problems and examples demonstrating. Math in everyday life essaymath in everyday life math and many of it's aspects are a major part of everyday life. Math mathematics essays - usefullness of mathematics in everyday life. Get an answer for 'why is math such an important part of our in everyday life simple math is used to do how could i write an essay on how math is used in. Usefullness of mathematics in everyday life essay 1435 words | 6 pages to rapidly crack the enigma it was this combination of math and machine that enabled england. |
Euclid
Euclid was a famous mathematician of ancient Greece. He wrote Elements which consist of 13 books and which are one of the most influencial works in the history of mathematics. He was referred too as father of geometry. |
Math Series Part VI: The Birthday Paradox
The Birthday Paradox
People become astounded when they encounter something that is totally against what they perceive to be normal; when they encounter something that is against their intuition. Aspects of mathematics and probability tend to do this to people. It is the same reason why so many people play the lottery when the actual chance of selecting the right combination of numbers is one from mullions of combinations. We want to believe that we have special attributes and that our individuality is unique and incomparable—and that we can be the one in the millions who can win the lottery.
It is this form of vanity and tunnel vision that brings about the birthday paradox. It states that in a room filled with less than 365 people, there is still a very a high chance that someone shares the same birthday (day and month) with someone else. Now, with 350 people in the room, this may seem very likely. But, if we were told that in a room with just yourself and 22 other people that there was almost a 50-50 chance of two of us sharing the same birthday, the notion seems a little hard to believe. Yet, this is where mathematics takes over to turn the issue from one of common sense to a more numerical and mathematical approach, and where the so-called 'paradox' ceases to exist.
Indeed, when we consider the probability of you having your birthday, it is 365/365, as you can be born on any day of the year (and here we are ignoring leap days for simplicity). However, the probability that someone else shares this day with you is 1/365 = 0.27397…%. So, in a room with 22 others, you have 22 people to compare with, so this probability increases to 22/365 = 6.02739…%. That's clear enough. However, what we are really considering is not just us ourself. We are thinking about the chances of two people from the entire room sharing the same birthday between them. This changes the odds that we are calculating.
Indeed, if we transform the calculation we are making to think about the chances of somebody not having the same birthday with someone else, the odds are 364/365. What we have to consider now is the chances of everyone in the room sharing a birthday with someone else. We know that the first person will have 22 people to compare to, but the second will now only have 21 because they have already been compared to the first person. This is true for the third person too, as they only have 20 new people to compare to. This goes all the way down to the last person, who will have only 1 person to compare to. Thus, in the entire room of 23 people, we can generate 22 + 21 + 20 + 19 + 18 + 17 + 16 + 15 + 14 + 13 + 12 + 11 + 10 + 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1 = 253 comparisons. Hence, if we can effectively compare birthdays 253 times, we can calculate the probability that no one will share a birthday, and this is the 364/365 odd multiplied by itself 253 times; i.e. (364^253)/365 = 0.49952... This means that in a room of 23 people, there is a 49.95% chance of none of them sharing a birthday. And this also means that there is more than a 50% chance of two of them sharing a birthday.
Removing The Self
The probability approach and the removal of the individual from this calculation helps to clarify how there is really no paradox at all. There is merely the need to remove the 'self' from the calculation and to observe how each individual in the group relates to every other individual in the group, which gives us this 50-50 chance when there are 23 people. Indeed, these odds would change given the number of people in the room. In fact, with just 70 people in the room, the odds of two people sharing a birthday are 1 - (364^2485)/365 = 99.89056...%, as there are 2485 comparisons to be made between the 70 people. It is almost certain, then, that two people in a room of 70 will share a birthday, and then the odds only get better from there.
How surprised are you to find that someone shares a birthday with someone else? |
Example: if the set of apples is { a b c } and the set of oranges is {y z}
then the product set in this sense is the set of pairs {ay az by bz cy cz}.
The product as usually understood, 3×2=6,
is just the count of the resultant set of pairs.
I started this investigation of multiplication some years ago,
prompted by curiosity about the physical units of energy and momentum.
In physics, momentum is defined as mass time velocity,
where velocity is distance per unit time, i.e.
p = mv = mx/t
or in units: kilogram meters per second.
I have an intuitive and experiential notion of what a square meter is,
but what is a square second?
—-
Then thinking in general about multiplying units,
it occurred to me that the product of meters and meters
is not meters but square meters.
But I was taught that multiplication is a "closed" operation,
that is, the product of any two items from a set it another member of that set.
But multiplying items from the set of "lines" (things with length)
gives us an item in the set of "surfaces" (things with area).
So how is it that we say the multiplication is "closed"?
I postulate that, in the physical sense of "multiplying" quantities,
multiplication is not closed, but is in fact a dimension-increasing operation,
forming a set of n-tuples from the arguments, as in the Cartesian or outer product.
I would speculate that physical multiplication would turn out to be a "tensor" product, but for me at least, to verify this will require further investigation.
In another vein, I recently discovered that there is a lively discussion going on about whether multiplication is repeated addition. One question was whether "repeated addition" can be rigorously defined. That will be the subject of a future post. |
DEEPER PARTNERSHIP With Your Stream of Consciousness
In numerology 2017—2+0+1+7 adds up to 10, which in numerology reduces to 1, as 1+0 = 1. Therefore, 2017, in numerology was considered a year of New Beginnings. One is an uneven number, a catalyst for change.
2018—2+0+1+8 adds up to 11, which is a Master Number. A Master Year of increased opportunity for enlightenment. Reducing the 11-year into a 2-year. A number 2-year focuses on partnerships. Creating a deeper partnership with your own stream of consciousness.
NUMBERS or GEOMETRY or sacred geometry is the belief that all creations in the seen and unseen world are built on mathematical structures considered sacred by some. Mathematics underlies and encompasses all the laws of the universe or quanta, to construct or destruct everything and anything. Math uses ratios, proportions, algorithms, percentages and more to understand potentials, probabilities and create synchronicities.
Grids or fields and universes are structured in BASE-TWELVE. Atomic and subatomic particle structure is in threes and fours. Uneven numbers in atomic structure happens when a particle moves to create potential CHARGE. Adding one and two gets you three, an uneven number, which is a catalyst for change.
Geometry explains "metaphorically" the soul splitting off into different pieces of itself this way: each over-soul creates 12 similar aspects of itself with slightly different identities. Each of those divides into 12 more and so on. The sacred mathematics is 12 x 12 = 144.
Numbers have energy, color and sound that communicate information, a message or symbol of an energy field. Synchronized numerology, trinity dates or master numbers, alert humanity to an intense change in the energy field. BUT, your INTENT or will, are far stronger than any astrological or numerical influences or energy field you reside in.
Your soul conglomerate or stream of consciousness enhances some scenarios and not others to guide and support you. Sometimes to emphasis what is needed OR to emphasis what to AVOID. There are no accidents in the LIGHT universe. Numerology is an ancient system designed to give you a "heads up" about energy fields and the type of energy they contain. Synchronized numerology or numbers repeated is to give emphases. |
1 + 1 = 2. You uvenery? And I do not. It all depends on the number of measurements that you use. For example: If 1 + 1 - two football clubs then it is obvious that the answer is likely as 22. If 1 + 1 - a merger of corporations the answer will be numbered of thousands of people working in these corporations. If you go even dalsche various variants of mathematical operations.
If branching measure used in podschete 1 + 1 then suddenly it turns out that we are not able to cover the entire spectrum in your mind and the depth of the unfolding picture of the world that is presented to us in the calculation of these most 1 + 1. And there comes a time when it comes to understanding what objects and phenomena, their relationships dissolve and go out of control of our consciousness.
And here comes to the aid of mathematical creativity equations and functions which give the cut and shape of flood logical connection between consciousness and causes piling bezformenoe factor function according to the laws and regulations pridavayagranitsy States and outlines mechanisms.
Flat patterns of thinking that we were taught in school looks helplessly to the realities of the outside world. This is one of the first steps towards changing their thinking and understanding of the universe. Think of volume? |
Old Work
Here's what I used to do in a nutshell. There exist large sets of equations represented by matrices. These matrices can be solved, but take a VERY long time and a LOT of computer memory. So I'm helping debug and test iterative methods for estimating the solution within a certain tolerance in much less time, using much less memory. There's more at this website. |
Machine learning as it is today is pretty simple. You have input, a and output b with a function net(a) in between.
Doesn't this resemble old math. You have one object a that is proportional to an object b.
When does this not hold. If net(a) is singular for some a = a(k). But how could net(a(k)) get singular. One feature of 1/x is perhaps the closeness to a possible singularity at x = 0.
So I thought. If you have two networks. A rational object if you will. That is. net(a) = net0(a)/net1(a). This might get me new possibilities. Here I will train net0 and net1 as if they were separate neural networks. That is. Some hidden layer networks.
So the idea was to include a feature of math. The closeness to a singularity.
Testing …
Maybe you need to use network objects obj1, obj2, obj3,… as in ordinary math and just train them. Could there exist like a reduced taylor series for network objects?
Just for inspiration. A quick battery idea just as a philosophical possibility.
So my idea is to capture the energy from electrons while they are flying at high speeds.
I think its possible to create such a battery with electrons in vacuum with high voltages. However vacuum might not be so ideal since it does not provide an adaptable function. So maybe the battery would use a neon tube with ?coils.
Maybe the battery today is a heavy bottle neck and you need an electron ?multiplier that does not take so much weight.
If nothing else I had fun making the rendered image.Maybe an extreme battery should have extreme properties like high electron ?speed. The big C battery.
The assumption is that the universe has calculated ?everything. Then there should exist a relation between x,y,z position and time for any small object.
So for the smallest object I assume there is some kind of a network with x,y,z,time as input data. Just because the calculation exists as a possibility.
Then I guess there is network answer that is fulfilled for x,y,z,t in the network. If the network just have one node in the end the answer is single number then maybe its a constant for every such object.
A test I made. Aiming for something constant. Be it a constant vector or number. I really got to many candidates for constants.
I wonder. For fusion. Could ultra determinism help. By this I mean. There could exist a wise relationship for fusion technology. Since it might be dangerous. That is. You need 100% knowledge when and where fusion takes place.
By this relationship. A large hot volume for fusion like the tokamak is not the start of this technology. I guess a smaller device which is optimized for answering the two questions. Where and when exactly fusion takes place. Is the way to go.
I made a test of an error recognition with the irids dataset. The idea was to add an additional column to the target examples. The column was just a 1 for true examples and 0 for random generated examples.
So what I did first was to train the model network on the just the true examples.
Then I added double the amount of randomly generated data in the input data. With that random data I run it through the model. Predicting which class the random data would belong to. This falsie predicted data I could then use as false examples. That is. I added a 0 column to that target data.
This data together with the old data got me new data to train my second model. One which could tell if the input was ?wrong and still predict the correct output.
Although it took longer to converge.
So with this test I could tell if the sample looked like it came from a random generator giving close to 0.0 at the first column. Similar it could tell a value of .99 for data coming from true examples.
Otherwise the network would just use the method of elimination which is not so good.
Code is just for me to remember the idea. Not anything to copy. Maybe to correct though : )
For students to feel secure in school. Should there not exist a school-peace fund who comes up with anti bullying projects and ideas.
Somehow you need a broader approach to reduce tragedies like school shootings. A peaceful school where everyone can feel safe and happy. I don't think its that hard. From my point of view there needs to be different projects running all the time. This way school children know their well being matters.
For important train commuters lines I believe something has to be done. A failure should not affect the peoples whole day. Trying to find a bus, taxi or getting stuck at the Central City Station.
Its not enough to build additional tracks for higher traffic volume. The tracks are dependent on the same failures. If built like usual.
The idea is to build an additional independent train commuter line. Such that if one line gets an error people can go to the other line.
So for example. A two independent line system should have independent power supplies and maybe a big and a little Central Station.
With this idea. People get a much needed backup and for a carbon emission free future public transport must have an excellent uptime with zero stops.
One idea is that to create extra stations along the extra way. Since stations are gold to the neighborhood. The number of housings will ?increase overall. If you don't make the train system error resilient then the problem with going by train will persist.
Inspired by the digital age. Why not build an identical copy of Jerusalem. Let the Palestinians decide if it will be their capital.
The reason for this is pretty simple. If you build so that you have two or three Jerusalem there should be less probability for arguing. There is a Jerusalem for Israel and for one Palestine and possibly one left as a symbol for peace.
Peace is a complex problem. So it will require something from all us. To have a sustainable peace I have a feeling it will cost a lot of money.
Therefor I wonder if this peace capacity could be financed by a tax or a fee.
Since so many people will be involved I wonder if the money could stay in the persons bank account as seperate account. I feel this is the safest solution. So when money is required it can be reserved. Not played with by some hotshot investor.
I argue that our problem descriptions in probability are often wrong. They state the probability as a non complex value when it ?is a machine learning network. The reason is that a network can ?assume any function.
So what you need is data. Then feed that initial data to a network model to converge.
You can then test the network for probability calculations. However. Since different models can converge to the same data. There should exist a way to estimate this uncertainty.
So in everything you should ask yourself. Could networks assume these problems. If so. Then try it.
The world needs peace to survive the times to come. I would argue that you can turn this apparent difficulty to your own advantage. Its very smart to build a new capital. Somewhere else than in the disputed Jerusalem. I mean these things cost a lot money. Money that will be spent to ensure a prosperous future for the younger generation to enjoy.
I believe a ?global bank can be established to finance a project like this.
Come to think of it. Would the city need to be built with lots of tech and other innovations to make it resilient to climate change for instance. I like this idea. What if you have the best architects, engineers and planners to figure this out.
In my previous idea that the complexity problem is solvable with a network. I connected the weights in a continuous fashion to get a clearer picture. The smoothed image of the weights matrices.
From this I realized that you can connect the weights in a machine learning neural network. This so they are not so independent. Not looking so random when combined. So this gives me the idea that the weights are complex objects. Which are solvable each with a network.
So what input does the weight network have. Here I take inspiration from the universe. Molecules provide a clue. Here you have connected weights through bonds. But molecules are not so big.
Then this idea explains why we have gravity. Its just input data from all other weights to the particular weight. For the weights to solve more problems they need more information.
That is. Molecular forces provide input from the immediate surrounding weights and gravity provides information from all the rest of the weights.
So to sum this idea. Everything that seems complex is just networks. // Per Lindholm
Come to think of it. If energy runs the network it could resemble the error in a machine learning network. From that I realize that minimizing the error could happen in more ways. ?Either you disperse the error so it get evenly distributed or you shift the error to another place in the network. So at least locally you get a low amount of error. This with a limited amount of weights.
If there exist a type of number with many properties. I'm sure a network is the object that can emulate it. So the network may use real numbers or maybe only integers. Regardless the neurons in the network simulates the sought after number object. |
What is mirror symmetry?
Mirror symmetry was discovered as a foundation for elementary particle theory models by physicists in the 1990s, in the form of a duality between superconformal field theories. The study of three-dimensional Calabi–Yau manifold geometry plays an important role in mirror symmetry. Calabi–Yau manifolds are split into dual (mirror) pairs. Characteristics of the first manifold correlate with completely different characteristics of its mirror partner, and vice versa.
Maxim Kontsevich, a renowned French mathematician hailing from Moscow, Professor at the Institut des Hautes Études Scientifiques in Paris and winner of the Fields Medal, reinvented the concept of modern theoretical physics as an incredibly deep and comprehensive mathematical duality. It is known today as homological mirror symmetry. He is responsible for radically changing the mathematical concept of space itself, and drew the attention of the entire mathematical community to this topic. This has enabled the synergy of various mathematical fields, such as symplectic geometry, algebraic geometry, and theory of categories. Today, homological mirror symmetry is the cornerstone of a vast field of mathematical research related to the use of constantly developing high-energy physics technologies. 'Once supersymmetry can be proven, all these structures will have a physical meaning', says Ludmil Katzarkov, American-Bulgarian mathematician and academic supervisor at the HSE Laboratory of Mirror Symmetry. 'However, it hasn't been proven yet. Today, mirror symmetry solves theoretical problems. I believe that, in 40 years, or maybe 70, we'll know more about it. After all, it took almost a century before researchers discovered the gravitational waves described by Einstein'.
The laboratory
The Laboratory of Mirror Symmetry has existed as part of the HSE Faculty of Mathematics for almost a year. In 2016, Katzarkov was awarded a 'megagrant' from the Russian government for its creation. Thanks to his efforts, within just one year, the laboratory has become a leading international center for the study of mirror symmetry- a highly relevant field both in theoretical mathematics and theoretical physics. 'Mathematicians began applying mirror symmetry as a tool to solve problems in enumerative geometry. This is a language', explains Katzarkov. He adds, 'Mirror symmetry offers three languages at once: symplectic, algebraic-geometrical, and numerical, which enables us to examine phenomena in algebraic geometry, for example, from different perspectives'.
'When the Laboratory of Algebraic Geometry was created at HSE in 2010, we began organizing conferences and inviting interesting people', says Fedor Bogomolov, Academic Supervisor at theLaboratory of Algebraic Geometry and its Applications and Professor at the Courant Institute of Mathematical Sciences in New York. 'Ludmil Katzarkov took part in our seminars and conferences, and we wrote some joint papers with him. And when we were thinking about who could apply for the megagrant, it was obvious that he could. Ludmil is one of the world's leading mathematicians in the field of mirror symmetry, and we are happy that this project has been a success. The conference has attracted researchers from all over the globe, many of them very young- and they are the very people who will define this science in future. We'll be hearing about them soon. 2017 has been a successful year in my opinion, and we have great hopes for this new laboratory'.
Professor Katzarkovhas written his latest working paper with Maxim Kontsevich, a mathematician who has played a significant role in the development of mirror symmetry. A French researcher, originally from Moscow, winner of the Fields Medal and two Milner Prizes in mathematics and physics, Kontsevich is one of the world's leading academics. Furthermore, he has agreed to become an associate member of the HSE Laboratory of Mirror Symmetry. Mirror symmetry is an essential tool, which, in addition to string theory, can be used to understand some aspects of quantum field theory and mathematical formalism. They are used by physicists to explain the distribution and interaction of elementary particles. The December conference focused on these very issues.
The conference
The aim of the December conference, 'Mirror Symmetry and Applications', was to summarize international research in mirror symmetry and outline future prospects for the field's development. The conference covered a large geographical area, from Kyoto to Berkeley, from Harvard to London to Moscow. It included a mini-course run by Takuro Mochizuki, an outstanding Japanese mathematician, and 34 one-hour presentations by scholars from ten different countries. Almost all of the participants were very young, with an average age of 35.
'Every conference gives us a push forward. It is mostly at conferences where I meet my colleagues from other countries, where I can talk to them and learn about what they do. We are not competitors, rather, we complement each other. Together, we can cover a bigger research field', explained Tony Yue Yu, a young researcher from Paris-Sud University.
The HSE Laboratory of Mirror Symmetry is a symbiosis of mathematical schools located across the world, including Moscow, St. Petersburg and the US. 'The Russian approach to research, with its specific problem-setting, is somewhat similar to the Japanese approach', says Takuro Mochizuki, a leading Japanese mathematician who very rarely agrees to speak at conferences. Mochizuki was awarded a Japanese Academy Prize in 2011 for his research into D-modules in algebraic analysis and in 2014, he was a plenary speaker at the International Congress of Mathematicians. At the December conference, Mochizuki delivered a mini-course on 'Asymptotic behavior of families of harmonic bundles'. Many young researchers travelled to Moscow specially to meet him.
Anthony Blanc, from the International School for Advanced Studies (SISSA) in Trieste, says: 'Ludmil Katzarkov invited me to participate in the conference. He helped me a lot with my research in Vienna and I'm very glad I met him. The approach of the Russian school differs from ours, as we focus more on theory and theoretical conclusions. However, this greatly benefited our collaboration- our joint work is more exciting because of our different approaches.'
'I'm happy to be here today', says Alexey Bondal, one of the founders of derived algebraic geometry, research fellow at the HSE Laboratory for Mirror Symmetry, and researcher at theKavli Institute for the Physics and Mathematics of the Universe in Japan. 'I see the Laboratory for Mirror Symmetry as a logical step in the development of the Moscow school, since Ludmil graduated in this city. And he remains very much a part of it, in spirit.'
Young researchers
Unlike physicists, chemists, and biologists, mathematicians don't rely on complicated expensive equipment. All of our high-profile professional mathematicians are a kind of independent research laboratory. They might have an idea, a brainwave, at any moment, and so the main tools are always at hand: a pen, a sheet of paper, and a laptop. Many of them start working before they even begin their PhD, the tasks gradually increasing in complexity. The December conference attracted this very kind of researcher – those who, despite their youth, have been involved in research for quite a long time.
'Ludmil Katzarkov invited me to the laboratory when he was only applying for the grant', says Alexander Efimov, a talented young mathematician from Moscow. 'We met rather a long time ago, in 2008, when I was a university student, and we have similar research interests. The laboratory's conferences and the triweekly research seminars facilitate interaction between mathematicians. I've already picked up some interesting ideas at this conference. Research is a continuous process'. Katzarkov's students also spoke at the Moscow conference. 'I was lucky to meet Katzarkov when I was a student', recalls George Dimitrov. 'I completed my PhD in Vienna under his supervision. Working with such a high-profile scholar is a great opportunity and my presentation at the conference presents the results of our joint work. It focuses on how the count of non-commutative curves opens up ways to new categorical structures and connections to number theory and classical geometry.'
'I've cooperated with the HSE Faculty of Mathematics for many years', says Evgeny Shinder, associate member of the HSE Laboratory for Mirror Symmetry andlecturer at the University of Sheffield. 'I've always been impressed by the level of the students, who know many things as well as I do. I now teach in England, but I always enjoy coming back to Russia. I studied in St. Petersburg, which, like Moscow, has a great mathematical school. Ideas and developments are shared openly. It goes without saying that this enriches and empowers the Russian mathematical community'. Another representative of the St. Petersburg mathematical school, Valery Gritsenko, head of the HSE Laboratory of Mirror Symmetry, Professor at the Lille University of Science and Technology, and member of Institut Universitaire de France, says: 'The 'Mirror Symmetry and Applications' conference is the laboratory's sixth and the most comprehensive conference yet. The meeting provided talented young people in mirror symmetry with an opportunity to present their results and initiate sustainable long-term cooperation'.
PhD and Master's students at the HSE Faculty of Mathematics also numbered among the conference attendees. The laboratory organized the first research school, 'Geometry 2017', in St. Petersburg in July 2017, which attracted 120 students from numerous Russian cities. The participants attended five mini-courses run by leading international researchers, including Dmitry Orlov and Alexander Kuznetsov from the RAS Steklov Mathematical Institute, Sergey Ivanov from PDMI RAS, Misha Verbitsky, Professor at HSE and the Free University of Brussels, and Laurent Manivel, a renowned French mathematician. 'At the end of January 2018, the laboratory is organizing a new research school, this time in physics, automorphic forms, and Kac-Moody algebras', explains Valery Gritsenko. 'It will be held at the Bogoliubov Laboratory of Theoretical Physics of the Joint Institute for Nuclear Research in Dubna. The laboratory is also planning a research school in arithmetical issues of mirror symmetry'.
Researchers have conducted a study on tournaments using the playoff system, which is one of the most popular forms of sporting competitions. The results of the study were published in the Journal of Combinatorial Optimization.
Carlos Cortez, a bronze medallist of the International Mathematical Olympiad (2011, 2012, 2013) and a Mathematics graduate of MIT, recently completed a two-month research internship at the HSE Faculty of Mathematics under the supervision of Professor Sergei Lando. He will soon be pursuing a Master's degree at University Paris-Sud in France followed by a PhD at Northwestern University (USA). The research internship was made possible through a cooperation agreement between MIT International Science and Technology Initiatives (MISTI) and HSE.
The HSE International Laboratory for Mirror Symmetry and Automorphic Forms, which is among several international laboratories to recently open within the Higher School of Economics, was created in December 2016 as part of the Russian government's mega-grants program. Below, the lab's academic supervisor, Ludmil Katzarkov, along with deputy heads Valery Gritsenko and Viktor Przyjalkowski, explain why the laboratory is fully capable of becoming a unique multidisciplinary unit dedicated to the study of mirror symmetry, automorphic forms, and number theory.
HSE researchers have used computer modelling to demonstrate the varying manipulability of decision-making procedures and to identify those least susceptible to manipulation. Their findings are published in the paper 'Manipulability of Majority Relation-based Collective Decision Rules'.
Three HSE students from the Faculty of Mathematics and the Faculty of Computer Science won medals at the Vojtěch Jarník International Mathematical Competition held in the Czech Republic. Nikita Gladkov, a mathematics student, scored maximum points and was recognized as the outright winner in his category.
This summer, the HSE Faculty of Mathematics moved into a new building on Usacheva Street. As part of the Open House project, two HSE students – Petr Ogarok, a second-year student in the Mathematics programme, and Anastasia Matveeva, a first-year master's student in the Mathematics and Mathematical Physics programme – gave an excursion around the new building.
The new International Laboratory for Mirror Symmetry and Automorphic Forms will open at HSE's Faculty of Mathematics in 2017. This project, overseen by Ludmil Katzarkov (Professor at the University of Miami), won the Fifth Mega-Grants Competition of the Government of the Russian Federation.
Jonathan Gerhard from James Madison University spent one semester studying in Math in Moscow programme run jointly by HSE, Moscow Center for Continuous Mathematical Education, and Independent University of Moscow. During his time in Russia Jonathan took intensive maths courses, studied the Russian language and traveled to several Russian cities |
How zip works
Concept of Pi
How the Walschaerts Valve Gear in Steam Locomotives works
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Hypotrochoid
Alpha Stirling Engine Works
Ying Yang symbol
Sewing Machine
Geneva Drive
Radial Engine |
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Critical Point
Constant failure
In Proposition 3 of On the Measurement of the
Circle, Archimedes asserts, based on calculations
involving regular polygons circumscribed
around and inscribed in a circle, that
"the ratio of the circumference of any circle
to its diameter is less than 3 1/7 but greater
than 3 10/71". He thereby strongly reinforced,
if he did not actually create, the tradition of
considering that ratio, two millennia later
referred to as π, to be fundamental.
Was Archimedes wrong?
When I was in elementary school, my teachers
told me that π was the most important
number in mathematics, which scared me
into cramming as much of it as I could into
my brain. I got to all of 35 places. This many
places actually turned out to be harmful to
my science career, for whenever I used all of
them in calculations in high-school physics, I
was penalized for something my teacher
called "misplaced precision".
But is π the most appropriate number?
Are there not beauties and economies to
be had by using the ratio of circumference
to radius, rather than the ratio of circumference
to diameter? After all, there are so
many instances in science and mathematics
where forests of 2π occur — from Maxwell's
equations to Fourier series formulae — that
it would surely make life much simpler by
defining a new constant — let's call it ψ for
the sake of argument — to be 2π, which is
the number of radians around a unit circle.
Would we lose any beauty and economy by
using this new constant?
I posed this question to the Princeton
University mathematician John Conway,
one of the most creative mathematicians
working today. Conway, it turned out, had
strong feelings on the subject. "2π is obviously
the correct constant!" he told me
immediately — although he also told me of
arguments, which he did not find persuasive,
for a third option, π/2.
I mentioned Euler's formula eiπ + 1 = 0,
which readers of this magazine once voted
as one of the greatest equations in science
(see "The greatest equations ever").
Would not the beauty and economy of this
equation — which contains five of the most
fundamental concepts of mathematics and
four operators each exactly once — be diminished
if it had to become eiψ/2 + 1 = 0?
Conway's response was to mention another
formula of which Euler's is a special case:
e2π/n = n√1. This formula, Conway maintained,
is preferable and more economical
than Euler's because of its generality: it takes
account of the two possible square roots of
1, namely +1 and –1.
In 2001 the Mathematical Intelligencer
published an article entitled "π is wrong!"
(21 7). Its author, the University of Utah
mathematician Bob Palais, mentioned numerous
formulae that use 2π, including Stirling's
approximation and Cauchy's integral
formula, to show how much easier they are
with a redefined constant. He concluded by
noting that our species has broadcast π via
radio telescopes to possible extraterrestrials,
and expressed alarm about "what the lifeforms
who receive it will do after they stop
laughing". Evidently, not only the beauty
and economy of our expressions, but our
very reputation for science literacy in the
universe is at stake here.
Practically speaking, shifting from π to ψ
would involve going against more than taste
but tradition, and would be as difficult as
changing to the metric system. Still, if π is
vulnerable to being changed, what other
constants are safe?
Was Planck wrong?
In physics, an obvious candidate for renovation
is Planck's constant, h. Given the
swarms of ℏ's, ℏ = h/2π, that appear in
equations, would it not have been simpler to
have defined it that way in the first place?
Are there situations that would make us
prefer h to ℏ? Was Planck right or wrong?
I approached Fred Goldhaber — a colleague
with a broad grasp of physics and a gift
for explanation — with this question. "It depends
on the problem you are addressing,"
he said, with characteristic common sense.
"I can think of at least two cases where it is
simpler to use h than ℏ." One is in phase-space
diagrams where position, say, is on
the x-axis and momentum on the y-axis. In
classical mechanics, every point on the diagram
is a possible position. In quantum mechanics,
however, Planck's constant is an
irreducible unit. Possible positions on the diagram
are not points but blocks, each of which
has a total area equal to h.
The second case he mentioned was the
Aharonov–Bohm effect. This involves shifts
in the interference pattern when two particles
pass on either side of a space, insulated
from the particles, where a magnetic field is
present. The significance of the effect is
that classical particles are unaffected by the
magnetic field, whereas quantum particles
are affected. Goldhaber pointed out that
the interference pattern shifts by one fringe
every time the magnetic flux changes by one
quantum of flux. The formula for the flux
quantum is h/e, where e is the charge on the
electron; if the unit were ℏ, it would be the
less economical 2πℏ/e.
The critical point
Some constants seem destined to live forever.
We can do little, I suppose, about G, the universal
gravitational constant, or about the
fine-structure constant α, a measure of the
strength of the electromagnetic interaction —
even though its value might be varying with
time. Still, it might be possible to find other
constants that might be changed for the purposes
of beauty and economy.
I invite you to e-mail me with candidates
for constants that are not expressed with
maximum efficiency. I shall report on the
results in a future issue.
About the author
Robert P Crease is chairman of the Department
of Philosophy, Stony Brook University, and historian
at the Brookhaven National Laboratory, US
Further Reading about "Fundamental Constants"
Interesting Topic, you might read the paper below. There are some relations between fundamental constants trying to answer the questions about dimensionless constants and their possible time variability.
(Well, the muttish pronunciation of π is wrong, as there's no æ, ay, or ai to speak of.)
But,
yes, π is wrong because we don't care about the diameter under the
perifery. Rather, we should aknow thas 2π is nearly a hallowed
tealler/lotter, or sacred numar/quantar: floor|6.28318...| = 6, which
is the count of the rings which can girdel a self-same ring. In
triangular simplex-land, π = 3 and 2π = 6, and every ring is a
hecsagon. One can indraw a hecsagon in our ring, with spokes, and see
the greater arclength represented as the differend of transcender and
integer.
As Ark's constand is planar, I'm more interested in the
ratio of areas between rectilinear and circumpolar elements. Then the
span will matter. If one takes a square with sides S, then a ring will
perfectly fit inside and outside its rim with width S and √2S. Its
breadth or area or embado will then either be alack or aclut by a
fundamental and significant amount, π(S/2)^2/S^2 and π(√2S/2)^2/S^2 or
π/4 and π/2. If one expresses these numericly, one sees how well these
shapes match up: 3.14/4 and 3.14/2 or .785 and 1.57. These would be
new-π/8 and new-π/4. These can be takene in greater dimensions to see
how measures blow up. (π/6 and π/2 or new-π/12 and new-π/4; ππ/2 and ππ
or new-ππ/8 and new-ππ/4, etc.) As π/4 is most important, it should be
the next new constand, β=π/4. This can express the [in]efficient
packing of round pegs in square holes, and the like. The new circular
area function would be βS^2 or .785...S^2, because we don't care about
line segments when we're a'talking about area. Sorry, Arkimèdès.
An: The greatest equations ever
I'm surprised the Maxwell's equations got so many votes. D and B as
flux or field-not-even-field are much worthless; the former two
equations are to show how worthles they are. It's not meaningful to
multiply a quasipotential or quasidomain by a span, as it would be for
a field to get potentials and states; and the gradients are merely to
get rid of their spans by /s. They would better write such out to keep
from confusere everyone who never even took vector calculus. "Nabla?!"
Is that the name of the grue in Zork? The equations aren't elegant or
so great if there are so many versions of them: CGS (rubbish, fails all
dimensional analusis), MKS (better, doesn't cover up ugliness),
tensoral (what? where?), Heaviside's quaternions (don't know). And then
there is the integral half you didn't show.
The
survey of scientists and amators shew a disturbent lack of respect for
mathematic grounding—in calculus or statistics. There's the
differend-ratio which startd all of calculus, its fundamental theorem:
f'(x) = limΔx→0 f(x+Δx)−f(x) /Δx. (Then there's its tipsy mate the
Riemann sum limit for integrals.) There're the nifty power-law formulæ
for derivands and integrands. There're the arithmetic and calculial
formulæ for a mean. The formula for a normal distribution, which I
can't angather, is great, as is for a standard deviation. You can't get
probability and entropy or temperature without them.
By the way,
Newton's F=ma is wrong. You can't put up both F=ma and E=mc^2 as
they're mutually exclusive. The full [dimensional] formula is F = ms''
+ m's' + m''s. Another greatest equation I would put up would be the
cinematic motion, out of calculus: s = S + S't + 1/2 S''t^2 + 1/3 1/2
S'''t^3 + ... (By the way, the fractions belong on time, and not in
front, as that's where they come from.) But my darling formula would be
elèctric potential, Coulomb's law U = −kqq/r, unified with elèctric
energy, Einstein's? law E = mc^2 to yield the mass-radius equivalense
and solve for the elèctròn's and other motes' size. It shows the
quantum mekanicists are wrong: the elèctròn is not a point charge; and
the general relativists are wrong: hudrostatic equations cannot reach a
singularity and so there are no black holes. |
Abstract:
Dust off those old similar triangles, and get ready to put them to new use
in looking at art. We're going to explore the mathematics behind
perspective paintings---a mathematics that starts off with simple rules,
and yet leads into really lovely, really tricky mathematical puzzles. Why
do artists use vanishing points? What's the difference between 1-point and
3-point perspective? What's the difference between a perspective artist and
a camera? We'll look at all of these questions, and more. We'll solve
artistic puzzles with mathematical theorems, using hands-on examples.
Title and Abstract of Annalisa's Project NExT talk
Title: How to Grade 300 Math Essays (and survive to tell the tale).
Abstract:
Because of the emphasis placed on collaborative learning and on laboratory
exploration, mathematics instructors are increasingly assigning student
writing in our classes. Those of us who assign written work have noted that
mathematical essays provide students with a forum for clarifying their
thoughts, for expressing their creativity, for emphasizing concepts rather
than merely reciting rules, and for allowing their instructor a heightened
awareness of students' perceptions of the material.
Written assignments, however, can create hurdles for both instructors and
their students. Two of the most formidable obstacles we instructors face
are these: Firstly, if we must teach writing, we wish to do so without
detracting from the mathematical content of the course. Secondly, we have
to grade the writing once it appears (in large quantities) on our desks. In
particular, we are all searching for a method which will allow us to grade
a large quantity of essays in a way that is (a) meaningful, (b) equitable
to all students, (c) helpful to the students' writing, and (d)
time-efficient. The aim of this talk is to explain how to do just that.
Biography of Annalisa Crannell
Annalisa Crannell is Chair of Mathematics at Franklin & Marshall College.
She received her professional degrees in three year increments: from
Springbrook High School in 1983, then Bryn Mawr College, then an MA and
finally a PhD from Brown University in 1992. Because she spent much of her
youth wandering the halls of NASA where her mother worked, she decided
early on that her intended career would be a "xeroxer". That ambition is
only slowly being realized.
Crannell's primary research is in topological dynamical systems (also known
as "Chaos Theory"), but she also is active in developing curricular
materials for a course on Mathematics and Art. She has worked extensivley
with students and other teachers on writing in mathematics, and with recent
doctorates on employment in mathematics. She especially enjoys talking to
non-mathematicians who haven't (yet) learned where the most beautiful
aspects of the subject lie. |
One feature for such knowledgebase could be very useful: One can formulate some statement or hypothesis and the system should find if this statement has already been proven (and point to the proof then). Even if it there isn't any proof of this statement in the knowledgebase it could still try to find all proven theorems with stronger conditions or weaker conclusions. Such system should also have the information if somebody else had tried to prove this theorem and, even if he failed, how much effort did he devoted to the problem. On the other side, even if a proof was found, it could try to find more general variants of the theorem (with weaker conditions or stronger conclusions).
Hi all, I have started blogging again. This time, I hope, it will be more successful. Blogs serve for self-expressing and sharing ideas and interesting information. I'm not sure that all information about me is very interesting but at least it is some kind of self-expression.
Actually it was always very difficult for me to write any texts (except probably Java programs). But the only way to learn things is to do them.
Today is my first day after returning back to Prague from my US trip. Strange, I don't feel any jetlag, this is the first time after last 8 over-ocean flights. During this trip I visited San Francisco (JavaOne) and Boston. I've started a new branch of JetBrains in Boston area, and we already have one programmer there - Dmitry Skavish (He is the author of JGenerator - )
Mathematical knowledgebase
I always was dreaming about thing like that. Contemporary mathematics is a huge area of knowledge, it takes years to come to the modern research edge even in one particular field. And the only way to do it now is reading books and articles. Unfortunately the way, how information is presented in such printed forms, is absolutely inconvenient for getting overall picture. Text is linear thing, but mathematics itself - is absolutely not. When learning some new math notion I always want to know immediately, where this notion is used, what are particular examples of such notion, why it was introduced and so on. Usually math monographs and articles don't answer these questions (only good one clarify part of it). Or - they answer, but in hundreds of pages forehand. Mathematical knowledge is actually some graph, so it should be represented as a graph.
I've found some resources on internet, devoted to this problem:
Wolfram ScienceWorld Mathematical notions and facts are represented here as ordinary web pages with hyperlinks. It covers only a very small part of Math knowledge.
QED Project - It has very similar goals, as I explained here, but unfortunately it is just a proclamation and nothing has been changed there since 1996
MIZAR Project - Very interesting project. They have invented some language, that allows introducing notions and describing proofs absolutely formally and at the same time this text is human-readable. They have very large knowledgebase, covering good part of Mathematics. It has also program that verifies proofs. Unfortunately there is no UI application for browsing/editing this knowledgebase. The project was started in 1973!
As I have been developing more and more programs, it is becoming increasingly clear that it takes far too many words to "explain" simple things to computer. Very often, it is very simple to explain a task to a developer, but that same task can take the developer several months to "explain" it to a computer.
Imagine the following task:
"Write a graphic editor that allows users to draw a picture, consisting of geometrical shapes. It should be possible to select some geometrical shape from a palette (rectangle, circle or line) and then draw it by using a mouse. In addition, it should also be possible to select these shapes in the editor, move selected shapes, change their size, and delete them. It should be possible to save the resulting picture to a file and to read the saved graphic from a selected file."
This explanation to a programmer can take about 20 minutes maximum, considering their additional questions about exact specifications. After this, it will take about 1-5 days to write the program and to fix all the bugs. In any case, the program source code will be much longer than this short and simple paragraph description as above.
Moreover, an in-depth knowledge of some libraries (input-output, graphic library, UI library) will be required from the developer. The developer should also be concerned about a good program design, i.e. by creating a base interface that will be implemented by all classes related to geometrical shapes. Such a design is required to have the possibility to easily add new shape classes in future.
Why is it that we cannot explain this task directly to a computer, and must instead, explain it to a developer?
The matter of the fact is that any OO language (like Java, C++, etc) is not brief at all. It has very few language notions that can be used to express a program - class, method, expression, if, for, etc. This means that because it possesses such a limited vocabulary, it is therefore required to explain everything to a computer in detail.
One can object to this by pointing out that OO languages allow one to create new classes and methods that can express new notions. But the fact still remains, that all classes and methods look the same to the computer. Their names and comments make sense only to their human manipulators. What all of this means, is that a computer can only help a developer on the language level (i.e. like IntelliJ IDEA currently does). However, a computer is of limited help when domain knowledge is needed. For example, a computer has no clue that a given method is an implementation of a remote access method for a given session bean in an EJB because this fact is not reflected by any language construct.
In the article by Czarnecki, etc : "domain specific knowledge may get lost in the implementation because there exists a semantic gap between domain-specific abstractions and features offered by programming language"
Most of all these "OOP design patterns" are needed only to cover this "Gap". |
Difference between Chaldean and Pythagorean. Chaldean is the old way, ancient way or original way in how the Alphabet was assigned corresponding numbers. Pythagorean derived from Pythagoras who developed a theory of numbers. He had nothing to do with Numerology, he was only a revered theorist who's theory became the basis for occult science of … Continue reading Services |
The MöBIUS STRIP or MöBIUS BAND (/ˈmɜːrbiəs/ (non-rhotic ), US
: /ˈmeɪ-, ˈmoʊ-/ ; German: ), also spelled MOBIUS or MOEBIUS, is
a surface with only one side (when embedded in three-dimensional
Euclidean space) and only one boundary . The
Möbius strip Möbius strip has the
mathematical property of being unorientable . It can be realized as a
ruled surface . It was discovered independently by the German
mathematicians
August Ferdinand Möbius and
Johann Benedict Listing in
1858.
An example of a
Möbius strip Möbius strip can be created by taking a paper strip
and giving it a half-twist, and then joining the ends of the strip to
form a loop. However, the
Möbius strip Möbius strip is not a surface of only one
exact size and shape, such as the half-twisted paper strip depicted in
the illustration. Rather, mathematicians refer to the closed Möbius
band as any surface that is homeomorphic to this strip. Its boundary
is a simple closed curve, i.e., homeomorphic to a circle. This allows
for a very wide variety of geometric versions of the Möbius band as
surfaces each having a definite size and shape. For example, any
rectangle can be glued to itself (by identifying one edge with the
opposite edge after a reversal of orientation) to make a Möbius band.
Some of these can be smoothly modeled in
Euclidean space , and others
cannot.
A half-twist clockwise gives an embedding of the Möbius strip
different from that of a half-twist counterclockwise – that is, as
an embedded object in Euclidean space, the
Möbius strip Möbius strip is a chiral
object with right- or left-handedness. However, the underlying
topological spaces within the
Möbius strip Möbius strip are homeomorphic in each
case. An infinite number of topologically different embeddings of the
same topological space into three-dimensional space exist, as the
Möbius strip Möbius strip can also be formed by twisting the strip an odd number
of times greater than one, or by knotting and twisting the strip,
before joining its ends. The complete open Möbius band is an example
of a topological surface that is closely related to the standard
Möbius strip, but that is not homeomorphic to it.
Finding algebraic equations is straightforward, the solutions of
which have the topology of a Möbius strip, but in general, these
equations do not describe the same geometric shape that one gets from
the twisted paper model described above. In particular, the twisted
paper model is a developable surface , having zero Gaussian curvature
. A system of differential-algebraic equations that describes models
of this type was published in 2007 together with its numerical
solution.
The
Möbius strip Möbius strip has several curious properties. A line drawn
starting from the seam down the middle meets back at the seam, but at
the other side. If continued, the line meets the starting point, and
is double the length of the original strip. This single continuous
curve demonstrates that the
Möbius strip Möbius strip has only one boundary .
Cutting a
Möbius strip Möbius strip along the center line with a pair of scissors
yields one long strip with two full twists in it, rather than two
separate strips; the result is not a Möbius strip. This happens
because the original strip only has one edge that is twice as long as
the original strip. Cutting creates a second independent edge, half of
which was on each side of the scissors. Cutting this new, longer,
strip down the middle creates two strips wound around each other, each
with two full twists.
If the strip is cut along about a third of the way in from the edge,
it creates two strips: One is a thinner
Möbius strip Möbius strip – it is the
center third of the original strip, comprising one-third of the width
and the same length as the original strip. The other is a longer but
thin strip with two full twists in it – this is a neighborhood of
the edge of the original strip, and it comprises one-third of the
width and twice the length of the original strip.
Other analogous strips can be obtained by similarly joining strips
with two or more half-twists in them instead of one. For example, a
strip with three half-twists, when divided lengthwise, becomes a
twisted strip tied in a trefoil knot . (If this knot is unravelled,
the strip has eight half-twists.) A strip with N half-twists, when
bisected, becomes a strip with N + 1 full twists. Giving it extra
twists and reconnecting the ends produces figures called paradromic
rings .
where 0 ≤ u < 2π and −1 ≤ v ≤ 1. This creates a Möbius
strip of width 1 whose center circle has radius 1, lies in the xy
plane and is centered at (0, 0, 0). The parameter u runs around the
strip while v moves from one edge to the other.
If a smooth
Möbius strip Möbius strip in three-space is a rectangular one –
that is, created from identifying two opposite sides of a geometrical
rectangle with bending but not stretching the surface – then such an
embedding is known to be possible if the aspect ratio of the rectangle
is greater than the square root of three. (Note the shorter sides of
the rectangle are identified to obtain the Möbius strip.) For an
aspect ratio less than or equal to the square root of three, however,
a smooth embedding of a rectangular
Möbius strip Möbius strip into three-space may
be impossible.
As the aspect ratio approaches the limiting ratio of √3 from above,
any such rectangular
Möbius strip Möbius strip in three-space seems to approach a
shape that in the limit can be thought of as a strip of three
equilateral triangles, folded on top of one another so that they
occupy just one equilateral triangle in three-space.
If the
Möbius strip Möbius strip in three-space is only once continuously
differentiable (in symbols: C1), however, then the theorem of
Nash-Kuiper shows that no lower bound exists.
A method of making a
Möbius strip Möbius strip from a rectangular strip too wide
to simply twist and join (e.g., a rectangle only one unit long and one
unit wide) is to first fold the wide direction back and forth using an
even number of folds—an "accordion fold"—so that the folded strip
becomes narrow enough that it can be twisted and joined, much as a
single long-enough strip can be joined. With two folds, for example,
a 1 × 1 strip would become a 1 × ⅓ folded strip whose cross
section is in the shape of an 'N' and would remain an 'N' after a
half-twist. This folded strip, three times as long as it is wide,
would be long enough to then join at the ends. This method works in
principle, but becomes impractical after sufficiently many folds, if
paper is used. Using normal paper, this construction can be folded
flat , with all the layers of the paper in a single plane, but
mathematically, whether this is possible without stretching the
surface of the rectangle is not clear.
TOPOLOGY
Topologically , the
Möbius strip Möbius strip can be defined as the square ×
with its top and bottom sides identified by the relation (x, 0) ~ (1
− x, 1) for 0 ≤ x ≤ 1, as in the diagram on the right.
A less used presentation of the
Möbius strip Möbius strip is as the topological
quotient of a torus. A torus can be constructed as the square ×
with the edges identified as (0, y) ~ (1, y) (glue left to right) and
(x, 0) ~ (x, 1) (glue bottom to top). If one then also identified (x,
y) ~ (y, x), then one obtains the Möbius strip. The diagonal of the
square (the points (x, x) where both coordinates agree) becomes the
boundary of the Möbius strip, and carries an orbifold structure,
which geometrically corresponds to "reflection" – geodesics
(straight lines) in the
Möbius strip Möbius strip reflect off the edge back into
the strip. Notationally, this is written as T2/S2 – the 2-torus
quotiented by the group action of the symmetric group on two letters
(switching coordinates), and it can be thought of as the configuration
space of two unordered points on the circle, possibly the same (the
edge corresponds to the points being the same), with the torus
corresponding to two ordered points on the circle.
The
Möbius strip Möbius strip is a two-dimensional compact manifold (i.e. a
surface ) with boundary. It is a standard example of a surface that is
not orientable . In fact, the
Möbius strip Möbius strip is the epitome of the
topological phenomenon of nonorientability . This is because
two-dimensional shapes (surfaces) are the lowest-dimensional shapes
for which nonorientability is possible and the
Möbius strip Möbius strip is the
ONLY surface that is topologically a subspace of EVERY nonorientable
surface. As a result, any surface is nonorientable if and only if it
contains a Möbius band as a subspace.
The
Möbius strip Möbius strip is also a standard example used to illustrate the
mathematical concept of a fiber bundle . Specifically, it is a
nontrivial bundle over the circle S1 with a fiber the unit interval ,
I = . Looking only at the edge of the
Möbius strip Möbius strip gives a nontrivial
two point (or Z2) bundle over S1.
COMPUTER GRAPHICS
A simple construction of the
Möbius strip Möbius strip that can be used to
portray it in computer graphics or modeling packages is:
* Take a rectangular strip. Rotate it around a fixed point not in
its plane. At every step, also rotate the strip along a line in its
plane (the line that divides the strip in two) and perpendicular to
the main orbital radius. The surface generated on one complete
revolution is the Möbius strip.
* Take a
Möbius strip Möbius strip and cut it along the middle of the strip.
This forms a new strip, which is a rectangle joined by rotating one
end a whole turn. By cutting it down the middle again, this forms two
interlocking whole-turn strips.
OPEN MöBIUS BAND
The OPEN MöBIUS BAND is formed by deleting the boundary of the
standard Möbius band. It is constructed from the set S = { (x, y) ∈
R2 : 0 ≤ x ≤ 1 and 0 < y < 1 } by identifying (glueing) the points
(0, y) and (1, 1 − y) for all 0 < y < 1.
It may be constructed as a surface of constant positive, negative, or
zero (Gaussian) curvature . In the cases of negative and zero
curvature, the Möbius band can be constructed as a (geodesically)
complete surface, which means that all geodesics ("straight lines" on
the surface) may be extended indefinitely in either direction.
CONSTANT NEGATIVE CURVATURE: Like the plane and the open cylinder,
the open Möbius band admits not only a complete metric of constant
curvature 0, but also a complete metric of constant negative
curvature, say −1. One way to see this is to begin with the upper
half plane (Poincaré) model of the hyperbolic plane ℍ, namely ℍ =
{ (x, y) ∈ ℝ2 y > 0} with the
Riemannian metric given by (dx2 +
dy2) / y2. The orientation-preserving isometries of this metric are
all the maps f : ℍ → ℍ of the form f(z) := (az + b) / (cz + d),
where a, b, c, d are real numbers satisfying ad − bc = 1. Here z is
a complex number with Im(z) > 0, and we have identified ℍ with {z
∈ ℂ Im(z) > 0} endowed with the
Riemannian metric that was
mentioned. Then one orientation-reversing isometry g of ℍ given by
g(z) := -conj(z), where conj(z) denotes the complex conjugate of z.
These facts imply that the mapping h : ℍ → ℍ given by h(z) :=
−2⋅conj(z) is an orientation-reversing isometry of ℍ that
generates an infinite cyclic group G of isometries. (Its square is the
isometry h(z) := 4⋅z, which can be expressed as (2z + 0) / (0z +
1/2).) The quotient ℍ / G of the action of this group can easily be
seen to be topologically a Möbius band. But it is also easy to verify
that it is complete and non-compact, with constant negative curvature
−1.
The group of isometries of this Möbius band is 1-dimensional and is
isomorphic to the special orthogonal group SO(2).
(CONSTANT) ZERO CURVATURE: This may also be constructed as a complete
surface, by starting with portion of the plane R2 defined by 0 ≤ y
≤ 1 and identifying (x, 0) with (−x, 1) for all x in R (the
reals). The resulting metric makes the open Möbius band into a
(geodesically) complete flat surface (i.e., having Gaussian curvature
equal to 0 everywhere). This is the only metric on the Möbius band,
up to uniform scaling, that is both flat and complete.
The group of isometries of this Möbius band is 1-dimensional and is
isomorphic to the orthogonal group SO(2).
CONSTANT POSITIVE CURVATURE: A Möbius band of constant positive
curvature cannot be complete, since it is known that the only complete
surfaces of constant positive curvature are the sphere and the
projective plane . The projective plane P2 of constant curvature +1
may be constructed as the quotient of the unit sphere S2 in R3 by the
antipodal map A: S2 → S2, defined by A(x, y, z) = (−x, −y,
−z). The open Möbius band is homeomorphic to the once-punctured
projective plane, that is, P2 with any one point removed. This may be
thought of as the closest that a Möbius band of constant positive
curvature can get to being a complete surface: just one point away.
The group of isometries of this Möbius band is also 1-dimensional
and isomorphic to the orthogonal group O(2).
The space of unoriented lines in the plane is diffeomorphic to the
open Möbius band. To see why, let L(θ) denote the line through the
origin at an angle θ to the positive x-axis. For each L(θ) there is
the family P(θ) of all lines in the plane that are perpendicular to
L(θ). Topologically, the family P(θ) is just a line (because each
line in P(θ) intersects the line L(θ) in just one point). In this
way, as θ increases in the range 0° ≤ θ < 180°, the line L(θ)
represents a line's worth of distinct lines in the plane. But when θ
reaches 180°, L(180°) is identical to L(0), and so the families
P(0°) and P(180°) of perpendicular lines are also identical
families. The line L(0°), however, has returned to itself as L(180°)
pointed in the opposite direction. Every line in the plane corresponds
to exactly one line in some family P(θ), for exactly one θ, for 0°
≤ θ < 180°, and P(180°) is identical to P(0°) but returns
pointed in the opposite direction. This ensures that the space of all
lines in the plane – the union of all the L(θ) for 0° ≤ θ ≤
180° – is an open Möbius band.
The group of bijective linear transformations GL(2, R) of the plane
to itself (real 2 × 2 matrices with non-zero determinant) naturally
induces bijections of THE SPACE OF LINES IN THE PLANE to itself, which
form a group of self-homeomorphisms of the space of lines. Hence the
same group forms a group of self-homeomorphisms of the Möbius band
described in the previous paragraph. But there is no metric on the
space of lines in the plane that is invariant under the action of this
group of homeomorphisms. In this sense, the space of lines in the
plane has no natural metric on it.
This means that the Möbius band possesses a natural 4-dimensional
Lie group Lie group of self-homeomorphisms, given by GL(2, R), but this high
degree of symmetry cannot be exhibited as the group of isometries of
any metric.
MöBIUS BAND WITH ROUND BOUNDARY
The edge, or boundary , of a
Möbius strip Möbius strip is homeomorphic
(topologically equivalent) to a circle . Under the usual embeddings of
the strip in Euclidean space, as above, the boundary is not a true
circle. However, it is possible to embed a
Möbius strip Möbius strip in three
dimensions so that the boundary is a perfect circle lying in some
plane. For example, see Figures 307, 308, and 309 of "Geometry and the
imagination".
A much more geometric embedding begins with a minimal Klein bottle
immersed in the 3-sphere, as discovered by Blaine Lawson. We then take
half of this
Klein bottle to get a Möbius band embedded in the
3-sphere (the unit sphere in 4-space). The result is sometimes called
the "Sudanese Möbius Band" , where "sudanese" refers not to the
country
Sudan Sudan but to the names of two topologists, Sue Goodman and
Daniel Asimov. Applying stereographic projection to the Sudanese band
places it in 3-dimensional space, as can be seen below – a version
due to George Francis can be found here.
From Lawson's minimal
Klein bottle we derive an embedding of the band
into the
3-sphere S3, regarded as a subset of C2, which is
geometrically the same as R4. We map angles η, φ to complex numbers
z1, z2 via z 1 = sin e i {displaystyle z_{1}=sin
eta ,e^{ivarphi }} z 2 = cos e i / 2 .
{displaystyle z_{2}=cos eta ,e^{ivarphi /2}.}
Here the parameter η runs from 0 to π and φ runs from 0 to 2π.
Since z1 2 + z2 2 = 1, the embedded surface lies
entirely in S3. The boundary of the strip is given by z2 = 1
(corresponding to η = 0, π), which is clearly a circle on the
3-sphere.
To obtain an embedding of the
Möbius strip Möbius strip in R3 one maps S3 to R3
via a stereographic projection . The projection point can be any point
on S3 that does not lie on the embedded
Möbius strip Möbius strip (this rules out
all the usual projection points). One possible choice is { 1 /
2 , i / 2 } {displaystyle left{1/{sqrt {2}},i/{sqrt
{2}}right}} . Stereographic projections map circles to circles and
preserves the circular boundary of the strip. The result is a smooth
embedding of the
Möbius strip Möbius strip into R3 with a circular edge and no
self-intersections.
The Sudanese Möbius band in the three-sphere S3 is geometrically a
fibre bundle over a great circle, whose fibres are great semicircles.
The most symmetrical image of a stereographic projection of this band
into R3 is obtained by using a projection point that lies on that
great circle that runs through the midpoint of each of the
semicircles. Each choice of such a projection point results in an
image that is congruent to any other. But because such a projection
point lies on the Möbius band itself, two aspects of the image are
significantly different from the case (illustrated above) where the
point is not on the band: 1) the image in R3 is not the full Möbius
band, but rather the band with one point removed (from its
centerline); and 2) the image is unbounded – and as it gets
increasingly far from the origin of R3, it increasingly approximates a
plane. Yet this version of the stereographic image has a group of 4
symmetries in R3 (it is isomorphic to the
Klein 4-group ), as compared
with the bounded version illustrated above having its group of
symmetries the unique group of order 2. (If all symmetries and not
just orientation-preserving isometries of R3 are allowed, the numbers
of symmetries in each case doubles.)
But the most geometrically symmetrical version of all is the original
Sudanese Möbius band in the three-sphere S3, where its full group of
symmetries is isomorphic to the
Lie group Lie group O(2). Having an infinite
cardinality (that of the continuum ), this is far larger than the
symmetry group of any possible embedding of the Möbius band in R3.
RELATED OBJECTS
A closely related 'strange' geometrical object is the
Klein bottle .
A
Klein bottle can be produced by gluing two Möbius strips together
along their edges; this cannot be done in ordinary three-dimensional
Euclidean space without creating self-intersections.
Another closely related manifold is the real projective plane . If a
circular disk is cut out of the real projective plane, what is left is
a Möbius strip. Going in the other direction, if one glues a disk to
a
Möbius strip Möbius strip by identifying their boundaries, the result is the
projective plane. To visualize this, it is helpful to deform the
Möbius strip Möbius strip so that its boundary is an ordinary circle (see above).
The real projective plane, like the Klein bottle, cannot be embedded
in three-dimensions without self-intersections.
In graph theory , the
Möbius ladder is a cubic graph closely related
to the Möbius strip.
In 1968, Gonzalo Vélez Jahn (UCV, Caracas, Venezuela) discovered
three dimensional bodies with Möbian characteristics; these were
later described by
Martin Gardner as prismatic rings that became
toroidal polyhedrons in his August 1978
Mathematical Games column in
Scientific American.
There have been several technical applications for the Möbius strip.
Giant Möbius strips have been used as conveyor belts that last longer
because the entire surface area of the belt gets the same amount of
wear, and as continuous-loop recording tapes (to double the playing
time). Möbius strips are common in the manufacture of fabric computer
printer and typewriter ribbons , as they let the ribbon be twice as
wide as the print head while using both halves evenly.
A
Möbius resistor is an electronic circuit element that cancels its
own inductive reactance.
Nikola Tesla patented similar technology in
1894: "Coil for Electro Magnets" was intended for use with his system
of global transmission of electricity without wires.
The
Möbius strip Möbius strip is the configuration space of two unordered points
on a circle. Consequently, in music theory , the space of all two-note
chords, known as dyads , takes the shape of a Möbius strip; this and
generalizations to more points is a significant application of
orbifolds to music theory .
In physics /electro-technology as:
* A compact resonator with a resonance frequency that is half that
of identically constructed linear coils
* An inductionless resistor
*
Superconductors with high transition temperature
* Möbius resonator
In chemistry /nano-technology as:
* Molecular knots with special characteristics (Knotane , Chirality)
* Molecular engines
* Graphene volume (nano-graphite) with new electronic
characteristics, like helical magnetism
* A special type of aromaticity:
Möbius aromaticity
* Charged particles caught in the magnetic field of the earth that
can move on a Möbius band
* The cyclotide (cyclic protein) kalata B1, active substance of the
plant Oldenlandia affinis, contains Möbius topology for the peptide
backbone. |
Calculus and Technology
Like, a specific engineered artifact -- a bridge, a segway, a waterjet cutter -- that couldn't have come into being without calculus.
Not necessarily a hard question, just something I'm pondering for fun. Trying to get a better grasp on the real value of things like mathematics. (One thought I'm having is that if you were intuitive enough, you could engineer anything without explicit math/science. Say, if you started building small bridges as a kid, and by the time you were an adult you had gotten a good "feel" for it, and could do it without any book learning. So I'm thinking about whether math is truly necessary.)
But... that's kinda off the deep end :D. Let's keep it as "What is a technology that couldn't exist without calculus?"
"You can't possibly get a good technology going without an enormous number of failures. It's a universal rule. If you look at bicycles, there were thousands of weird models built and tried before they found the one that really worked. You could never design a bicycle theoretically. Even now, after we've been building them for 100 years, it's very difficult to understand just why a bicycle works - it's even difficult to formulate it as a mathematical problem. But just by trial and error, we found out how to do it, and the error was essential."
Say, if you started building small bridges as a kid, and by the time you were an adult you had gotten a good "feel" for it, and could do it without any book learning.
Have you ever seen bridges built by pre-scientific societies? Sure, you could probably build something that can be called a 'bridge' just by trying out random things and seeing what works best, but no matter how long you do that, you will not be able to get even close to something like a modern bridge. All modern engineering structures are based on scientific principles (expressed in the language of mathematics). In fact, if what you're interested in is the question of whether math is truly necessary as you say, then a bridge is a bad example. Can you build a computer without a knowledge of science? A television? A Plane? A Satellite?
The thing is you would be using concepts of calculus within your computer simulation. If you weren't using calculus, then there is no way you can be certain with your simulation.
Hmm... I suppose it depends on what you mean by calculus.OK, I see. Wikipedia says geometry is the study of shape, algebra the study of equations, and calculus the study of change. From that definition, I can see how even the program/simulation would be the calculus.
I guess it seemed odd to me (and still does), to call it calculus when you're not using equations. What the program is doing isn't a complex concept: it's just computing differences across small intervals. Then again, I guess the idea isn't said to be complex. It's just that it took awhile for anyone (Newton/Leibniz) to actually work out how to do it formally (e.g. differentiate a general function like x*sin(x), rather than working it out numerically at every point). |
Curious Facts
Your browser does not support iframes. Click HERE to select a page number.
Page 9
76.5% of people who read page 8 tried to lick their elbow
78.23% of statistics are made up
111111111 x 111111111 = 12345678987654321
If you spell out all the positive integers in sequence, starting at one, you won't find an A until you reach one hundred and one.
At the time of writing (May 2004) the largest known prime number is 224036583-1. It has more than 7.2 million decimal digits and was discovered by J.Findley on a volunteer network (GIMPS – The Great Internet Mersenne Prime Search) using his own 2.4 GHz Pentium 4 computer.
Since the above was fact was added, 7 more Mersenne Primes have been discovered. At the time of writing (February 2013) the largest known prime, is 257,885,161 – 1.
If you are interested in using your computer to search for the next prime number, please visit:
If you shuffle a pack of 52 cards by dividing it exactly in half and perfectly interlacing one card at a time from alternate halves, so that the top card remains at the top and the bottom cards remains at the bottom (known as an "out-shuffle"),
the deck will return to its original order after 8 shuffles. This trick is used by skilled magicians and card sharks. [See mathworld.wolfram.com/Out-Shuffle.html for a mathematical explanation.]
Aoccdrnig to a rscheearch at Cmabrigde Uinervtisy, it deosn`t mttaer in
waht
oredr the ltteers in a wrod are, the olny iprmoetnt tihng is taht the frist
and lsat ltteer be at the rghit pclae. the rset can be a totalo mses and you
can
sitll raed it wouthit a porbelm. tihs is bcuseae the huamn mnid deos not
raed
ervey lteter by istlef, but the wrod as a wlohe. |
Mathematics
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It was simpler in the 15th century, when this 'Allegory of Geometry' was painted. Geometry was equated with the 'Elements' of Euclid, published around 300BC: 465 propositions that linked plane and solid geometry, number theory and geometrical algebra in a chain of deductive reasoning. For 2,000 years geometry lessons meant Euclid.
Today, although school geometry still uses ruler (radius geometricus in the picture) and compass, at higher levels alternative geometries have multiplied. Renaisssance artists developed projective geometry in the 15th centur, Descartes co-ordinate geometry in the 17th. Then Carl Gauss (1777-1855) and Bernard Riemann (1826-1866) created the first non-Euclidean geometry. Since then, the range of geometries has wildly expanded; space is seen no more as a collection of points but as a locus in which to move and compare objects, while geometry is a powerful toolkit for creative reasoning rather than an unalterable set of axioms. web: Victoria Neumark m.c escher's "sky amp; water"(c) 2001 cordon art-baarn-holland all rights reserved |
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