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One of the many things we love about knitting and crochet - and all the fiber arts - is how it satisfies on many levels: It's fun, it's great to look at, it keeps us warm, no creature needs to suffer in order for us to obtain the materials and it's soothing to the spirit. As many of us know, it's also an excellent way to practice our math skills. Some schools use it to teach the basics and beyond. So, when we discovered this website, we were smitten:
Pat Ashforth and Steve Plummer are self-descirbed "mathekniticians". It's the perfect description for two educators who use textile art as one of the vehicles for making mathematics easier to comprehend. Now retired from formal teaching, their knitting and crochet work has expanded well beyond their beginnings and brought them a new kind of notice and a career in design. It's a great story and here's a brief rendition. You can read a more complete version by visiting their website which is filled with fascinating facts and great pictures.
A married British couple, both math teachers were active on an online knitting forum when they were asked to produce an afghan pattern based upon a mathematical formula. They wound up producing 4 designs and it began a journey that led them on a new and exciting path. Recognizing that the afghan was the perfect template for expressing math formulas, they employed it as as a canvas and, simultaneously created some visually exciting works of art. At last count, that was 90 afghans ago. Take a look at some stunning examples:
Square Deal: the smallest possible example of a square divided into smaller squares, where the sides of each of the squares are all whole numbers, and where no two squares are the same size. Photograph: Pat Ashforth
Counting Pane: a grid of the numbers from 1 to 100. Each number cell contains the colors of the numbers from 1 to 10 that divide it, with 1 being blue, 2 being yellow, 3 red, and so on. So 12, which is divisible by 1, 2, 3, 4 and 6 has the colours of blue, yellow, red, green and black. A copy of this was sold to the Science Museum. Photograph: Pat Ashforth
This one is one of my favorites. It also makes me want to crochet.
Psesudoku: A crochet version of three superimposed Sudoku patterns. Photograph: Pat Ashforth
Like the math they taught (how I wish one of them had taught me), the patterns are easy to understand and require basic skills. If one wishes to understand theory, it's explained on their website, but a person can undertake it for the sheer enjoyment of sailing along on a work of art, secure in the knowledge that science is in accord.
Ashforth and Plummer have become celebrities in the world of mathematical crafts. Some of their afghans have been bought by the Science Museum in London. That said, they don't sell their finished products, but they do sell the patterns via their website, which is also filled with information about the mathematical truths behind the "proof" afghans. There are also other product patterns offered for sale, including "toys" which demonstrate theory in a playful way.
This is a knit version of a popular toy. It is made up from eight cubes, joined in a special way. you can fold and unfold the large cube continuously to reveal several different faces. There is also a crochet version available.
When traversing their website, it quickly becomes apparent that each of the couple have strong right and left brain capabilities. On one hand, they have the ability to visualize a creative way to showcase a mathematical fact, while, on the other, each possesses the ability to explain how to produce it. Here's a quote from Pat, which says it well:
"We enjoy the challenge of seeing an idea then working out how it can be made into an afghan in a way that would be easy enough for anyone else to recreate. It is like trying to solve a puzzle and refining it to give the best possible solution."
Both Ashforth and Plummer are accomplished knitters and, when conducting workshops, they are equal partners in knitting and/or crochet skills. A recent article in "The Guardian", a British publication, quotes Pat: "We always try to make sure that knitting is not seen as a female activity and Steve always knits at any event to emphasize the point," says Ashforth. "We find more reluctance from women who say they can't do math than from men who say they can't knit."
Here are a few more "afghan-a-matics" (my word).
Pythagoras tree: an image based on Pythagoras's theorem. For each black triangle you see the square on the hypotenuse and the squares on the other two sides. The Science Museum have the original. Photograph: Pat Ashforth.;My comment: Given the colors used, this one is evocative of african tribal art.
Ashforth says that another part of the enjoyment of making the afghans is seeing "... the effect we have had on children, either directly by them seeing our big colorful blankets and suddenly understanding something they had previously struggled with, or because other teachers have used our ideas (not always in knitted form) to help teach math in an unconventional way. And influencing the lives of so many (most often women) maths-phobics who would not dream of becoming involved with anything mathematical in other circumstances."
in the next post, we will follow up with a look at their newest fascination: Illusion Knitting. In the meantime, should you wish to visit their site, which contains lots more pictures and information, just click on the header on top. Pat can be found on Ravelry here. Steve can be found here, and their group on Ravelry is here. In addition, you will find them on twitter: Pat Ashforth is@matheknitician and Steve Plummer @IllusiveSteve |
never gets mentioned in the various mathematical studies is that the figure is an unfolded tesseract (4 dimensional hypercube), also called the net of the tesseract. A folded hypercube would have four physical dimensions and not be visualizable. Just as the six faces of a cube could be unfolded to form a flat cross, the eight 3D faces of a tesseract could theoretically be unfolded into this shape. |
Archives
Is Math Fiction? (part 2)
The last post asked, "Is mathematics fiction or nonfiction?" Here's a recap of the discussion so far:
One definition for math is "the science of patterns."
Science and technology depend on math. Disciplines like chemistry, physics, and computer science are made possible by the quantitative study of real-world patterns. This makes it seem like math is nonfiction.
Some mathematical scenarios, like nine-dimensional space, do not correspond to the real world in any obvious way. Since it is possible to study patterns that do not exist, it would seem that math is fiction.
If you went to a typical American high school, you almost certainly learned Euclidean geometry. This geometry was formalized* by the ancient Greek mathematician Euclid. The principles of Euclid's system seem straightforward enough. Any two points can be connected by a straight line. All right angles are equal. Given a line and a point not on that line, there is exactly one line that can be drawn through the point that is parallel to the original line.
*This was important for a lot of reasons and will hopefully be discussed in a future post.
Our real-world experience tells us that making things bigger won't change their shape. A triangle with sides of 3 feet, 4 feet and 5 feet should be the same shape (and have the same angles) as a triangle with sides 30, 40 and 50 feet. Fortunately, Euclidean geometry meshes with our idea of reality. It also does a fantastic job describing things like tables, beach balls, and skyscrapers. There is just one problem: apparently, the universe doesn't work that way.
Space is curved. You may have heard this before—the idea became famous when Einstein developed his theory of relativity. Einstein did not use Euclidean geometry in his model. He used a curved Riemannian geometry. Riemannian geometry is just as mathematical as Euclidean geometry is, but it clashes with concrete intuition. Small changes in its basic structure shove curved geometry outside the realm of experience. But curved geometry, not Euclidean geometry, fits relativity. In other words, physics says that curved geometry describes our universe, and Euclidean geometry does not. Now, which of these ideas is a fiction? Is Euclidean geometry "nonfiction" because it is true to human experience? Is Riemannian geometry "nonfiction" because it is used in physics? Are both nonfiction, or maybe neither?
In the end, it depends what you mean by "fiction" and "nonfiction." Math is not information we gather about the natural world. You can't pick up a piece of math or look at it under a microscope. No scientific experiment can prove a theorem. No historical record contains math that "happened." Every bit of math in the history of civilization occurred in someone's mind. If "nonfiction" is defined as "a description of something physical," then math is fiction. However, math gives us insight into the natural world. We use math to organize the information we gather with our senses. Math can be used to explain and predict physical patterns. It makes the difference between drinking "coffee, then coffee, then coffee" and drinking "three cups of coffee." It's what takes you from "This coin sometimes comes up heads" to "What are the odds it will happen?" If nonfiction includes "that which humans use to understand reality,"then math is nonfiction.
So, that is an answer. It depends on how you define nonfiction. But the debate is far from over. For one thing, these posts have not discussed the controversial idea of "mathematical truth." The tremendous enjoyment of seeing the order and harmony in math is often accompanied by a sense that math is "real" in some way: that it is something humans perceive, not imagine. This adds another facet to the fiction vs. nonfiction debate. Maybe it will get its own post later on.
Thank you for reading!
There were many, many people whose ideas went into the formulation of this series. I tried to present the points smoothly, without pointing out whose idea was whose. But to everyone who discussed this with me: thanks for your input!
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3 thoughts on "Is Math Fiction? (part 2)"
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Completely amazed to find the logic behind the mathematical operations just using 1 and 0
just studied half-adder and full-adder circuit's and figured out the logic behind the mathematical operations being performed by electronic circuits. Designing digital circuits is much like programming, using deductive analysis we have to find the appropriate elements and construct a circuit with boolean algebra. It may look a bit like reverse engineering on the start, but eventually you will feel that it is THE LOGIC SCIENCE |
Revue d'histoire des mathématiques - Titles - 13
Éric Vandendriessche String figures: a mathematical activity in some traditional societies
Abstract and full text
Marco Panza What is new and what is old in Viète's analysis restituta and algebra nova, and
where do they come from? Some reflections on the relations between algebra and
analysis before Viète
Abstract and full text
Frédéric Brechenmacher The 1874 Controversy between Camille Jordan and Leopold Kronecker
Abstract and full text
Cinzia Cerroni The contributions of Hilbert and Dehn to non-Archimedean geometries and their
impact on the Italian school
Abstract and full text
Sabine Rommevaux The Similitude of Equimultiples in the Definition of Noncontinuous Proportion in
Campanus' Edition of Euclid's Elements : An Obstacle in the Reception of the
Theory of Proportions in the Middle Ages
Abstract and full text |
Links
DO the math, DON'T overpay. We make high quality, low-cost math resources a reality.
Thursday, June 4, 2015
Throwback Fact: Emmy Noether's Math Journey
Emmy Noether
The way we name proofs in mathematics, most often after their founder, immortalizes mathematicians who could otherwise slip into history unnoticed. But 96 years ago today, an event occured that helped to keep one particular female mathematician from obscurity.
Emmy Noether (1882 - 1935) almost missed her chance at mathematical fame. Born to a Jewish family in the German town of Erlangen, Noether showed few signs of her mathematical talent until she reached a college age. Noether originally planned to learn to teach English and French, but she attended math courses at the University of Erlangen where her father lectured. There, she earned her doctorate in mathematics in 1907, and worked at the same university for 7 years, but didn't earn a single payment for her research.
David Hilbert
Noether developed a professional relationship with Austrian-born mathematician David Hilbert (who we featured in our Math Madness bracket!). Hilbert invited Noether to the University of Göttingen, but the university would not grant a professorship to a woman. Hilbert and fellow mathematician Felix Klein convinced Noether to stay at the university, unpaid, while they fought a political battle for her professorship. During this period, Noether gave lectures unofficially. To get around the university, Hilbert would advertise a lecture as his own "with the assistance of Dr. E. Noether," and she would then give the lecture.
But on this day in 1919, Noether was finally granted official permission to teach at the University of Göttingen in a Privatdozent position. She began to gain recognition for her work under her own name and flourished. See this more complete biography for a discussion on what mathematics she worked on.
One of the halls of Bryn Mawn college
Noether moved to the United States in 1933 after accepting a position at Bryn Mawr college. She worked there until her sudden death in 1935. Today, Emmy Noether is recognized worldwide for her contributions to mathematics. She was even made into a recent Google Doodle on the anniversary of her birth! Without the hard work of Hilbert and Klein, we may not have remembered Noether like we do today. |
Everything and Anything Anytime
Amazing Fractal Images
A fractal is a mathematical set that exhibits a repeating pattern displayed at every scale. It is also known as expanding symmetry or evolving symmetry. If the replication is exactly the same at every scale, it is called a self-similar pattern. Fractals can also be nearly the same at different levels. This latter pattern is illustrated in small magnifications of the Mandelbrot set. Fractals also include the idea of a detailed pattern that repeats itself.
Fractals are different from other geometric figures because of the way in which they scale. Doubling the edge lengths of a polygon multiplies its area by four, which is two (the ratio of the new to the old side length) raised to the power of two (the dimension of the space the polygon resides in). Likewise, if the radius of a sphere is doubled, its volume scales by eight, which is two (the ratio of the new to the old radius) to the power of three (the dimension that the sphere resides in). But if a fractal's one-dimensional lengths are all doubled, the spatial content of the fractal scales by a power that is not necessarily an integer. This power is called the fractal dimension of the fractal, and it usually exceeds the fractal's topological dimension.
I'm not exactly sure what the paragraph above means. But they sure are nice to look at. |
Links
Numbers [en
inglés] - When you have a bit of time to fill, check out
this page and it's links! It's about how our numbers came to be and
how our numbers and those of other cultures diverged through time.
The images explain why some people cross their sevens, the bizarre
history of how the two and the three got flipped around in arabic,
and how the five, which originally was a circle (it represented a fist),
got the fancy poise. Enjoy! It's interesting. Unfortunately,
since the site is on Geocities, it's probably blocked for viewing by
the LBUSD from school, so check it out at home. |
Q We return to the two number systems that you have defined i.e.
horizontal and vertical. Can you show now how these lead to both a linear
and circular interpretation of number respectively?
PC It is important to remember that even though the same symbols are
used in both cases they actually relate to very distinctive interpretations
of number.
In the first system though the number (as quantity) can vary the dimension
(as quality) remains fixed as 1,
i.e. 11, 21 ,3 1, 41,
…
In the second system though the number (as dimension) can vary the number
(as quantity) remains fixed as 1,
i.e. 11, 12, 13, 14, …
So in the first system, the natural number symbols 1, 2, 3, 4, … refer
to quantities that are defined with respect to a fixed (qualitative) dimension
of 1.
In the second system, the same natural number symbols 1, 2, 3, 4 … refer
to qualities (i.e. dimensions) to which the fixed quantity of 1 is raised.
However as they stand these two systems are not comparable in terms
of each other.
What is requires therefore is a means to express the vertical (qualitative)
in terms of the horizontal (quantitative) system and likewise - in reverse
fashion - the horizontal (quantitative) in terms of the vertical (qualitative)
system.
Q So how do you indirectly express the (qualitative) vertical in
terms of the (quantitative) horizontal system? In other words how can you
indirectly express the notion of a dimension in terms of an object phenomenon?
PC We have already dealt with basic mathematical symbols from both a
holistic arithmetic and geometric standpoint. We now do so from a more
powerful holistic algebraic perspective (which includes the other two interpretations).
As always mathematical operations have both analytic and holistic aspects.
Firstly we will show how to express the vertical (qualitative) number
- indirectly - in horizontal (quantitative) terms.
To do this let us introduce a very basic bit of algebra.
Once again in the horizontal system the number as quantity can vary
while the number as quality (i.e. dimension) remains fixed as 1.
i.e. 11, 21 ,3 1, 41,
…
So if x1 is a number in this system then x can take on any
of the values 1, 2, 3, 4, … (as quantities).
Now as the first (horizontal) system is expressed with respect to an
invariant fixed dimension of 1, to express the second (vertical) system
in this fashion requires taking the corresponding root of each number (where
x now represents the root number).
Thus in the case where x = 1 and 1x = 11, we obtain
the 1st root of 11 which remains the same as 1 (i.e. strictly
11).
Now when we plot these points, we can represent them at opposite ends
of the line diameter, that bisects the circle (of unit radius) into two
halves.
What is truly remarkable is that we obtain subsequent roots in this
fashion - defined in the complex plane - they will always lie as equidistant
points on the same circle of unit radius.
Thus through attempting to express the vertical (qualitative) system
- indirectly - in horizontal (quantitative) terms with respect to the invariant
dimension of 1, we create a coherent circular number system i.e. where
all numbers can be expressed as equidistant points on the circle of unit
radius.
Of special interest here in the context of my approach are the algebraic
(and subsequent geometrical) representations for 2, 4 and 8 roots.
As we have seen for x = 2, when 1x = 12, the 2nd
(i.e. square) root of 12 has two possible solutions + 1 and
- 1 (defined with respect to a fixed dimension of 1).
These two results (i.e. + 1 and - 1) can then be plotted as opposite
ends of the horizontal line (diameter) - drawn in the complex plane - that
bisects the circle of unit radius.
Then for x = 4, when 1x = 14, the 4th root of
14 now has four possible solutions two of which are real i.e.
+ 1 and - 1 and two of which are imaginary i.e. + i and - i
(where i is the square root of - 1).
Once again the two real roots + 1 and - 1 can be plotted as opposite
end points of the horizontal line diameter (through the circle of unit
radius).
The two imaginary roots + i and - i can then be plotted
as opposite end points of the corresponding vertical line diameter (through
the same circle).
Furthermore these two line diameters literally divide the circle into
four quadrants (now with a precise mathematical interpretation).
So here we have the first vital clue as to the all important mathematical
relationship as between quantitative and qualitative (which plays such
a vital role in the corresponding holistic interpretation of the four quadrants).
Thus if we represent the horizontal line diameter (or axis) in real
terms as representing quantities, then the vertical line diameter (or axis)
as representing dimensions is thereby imaginary (in precise mathematical
terms).
Finally for x = 8, when 1x = 18, the 8th root
of 18 now has eight possible solutions.
We have already encountered four of these with out two real (i.e. +
1 and - 1) and two imaginary roots (+ i and - i).
As we have seen the two real (+ 1 and - 1) and the two imaginary (+
i and - i) roots can be represented as opposite end points
of the horizontal and vertical line diameters (axes) respectively of the
circle (of unit radius).
The four complex roots can be represented as end points of the two corresponding
diagonal line diameters (drawn at an equal distance from both horizontal
and vertical).
Thus k(1 + i) and k(- 1 - i) lie at opposite ends of the
diagonal that bisects the upper right and lower left quadrants.
k(- 1 + i) and k(1 - i) lie at opposite ends of the other
diagonal that bisects the upper left and lower right quadrants.
Now remarkably the diagonal lines (in each quadrant) have an alternative
interpretation as null lines = 0. This can be easily shown through using
the Pythagorean theorem.
Again the holistic explanation of this dual interpretation in terms
of equal complex co-ordinates and null lines plays a vital role when we
attempt to give a precise scientific explanation of the nature of diagonal
poles.
Q So your purpose here is to show how all numbers can be given a
coherent linear and circular interpretation in analytic terms. You then
will presumably move on to showing how we derive the corresponding dynamic
holistic interpretation of the circular system?
PC Yes! That is correct.
Though the circular system is of course recognised in mathematical terms,
its great potential significance has been greatly overlooked. A reduced
interpretation of this system inevitably results for - as we have seen
- ultimately all interpretation is with respect to the linear system (with
an invariant dimension of 1).
The true significance however of the circular system comes from holistic
interpretation where - when properly understood - it provides the appropriate
basis for an integral TOE (indeed a series of integral TOEs paradoxically
of both increasing complexity and yet increasing simplicity).
It is certainly true that an implicit holistic understanding of this
circular system of number exists in many esoteric traditions.
If modern writers Jung had an especially strong sense of its importance.
Thus when Marie Louis von Franz said that
"Jung devoted practically all of his life to the importance of the number
4" she implied the circular interpretation of 4 (in a dynamic holistic
sense).
Indeed Jung pointed to the integral significance of mandalas the most
important of which entail ornately designed patterns that are fundamentally
similar to the geometrical representations above).
However the deep mathematical significance of these representations
- serving as the basis for a truly scientific approach to development -
now needs to be made fully explicit.
Q Before moving on to holistic interpretation can you briefly show
how - in reverse fashion - the horizontal (quantitative) can be indirectly
expressed in terms of the vertical (qualitative) system? In other words
how can one indirectly express a phenomenal object in terms of a general
dimension?
Whereas in the previous case, numbers as dimensions were expressed in
terms of a fixed dimension of 1 (through obtaining successive roots defined
by these same numbers), in this instance, numbers as quantities are expressed
in terms of a fixed quantity of 1 (though obtaining successive powers defined
by these same numbers).
Now this leads directly back to the vertical system.
So when x (as quantity) = 1, 1x = 11.
When x (as quantity) = 2, 1x = 12.
Now the deeper significance of this dimension of 2 is that it really
is defined in terms of a both/and logic that combines two directions i.e.
combining both positive (+) and negative (-) real directions of unity.
This again can be represented in circular terms as end points of the horizontal
line diameter that bisects the circle of unit radius (in the complex plane).
When x (as quantity) = 4, 1x = 14.
Again the deeper significance of this dimension of 4 is that it is defined
in terms of a both/and logic that combines four directions i.e. both positive
(+) and negative (-) directions of unity in real and imaginary terms. This
again can be represented in circular terms as end points of both horizontal
and vertical line diameters that bisect the circle of unit radius.
Likewise when x (as quantity) = 8, 1x = 18.
Once more the deeper significance of this dimension of 8 is that it
is defined in terms of a both/and logic that combines eight directions
i.e. both the positive (+) and negative (-) directions of unity in real
and imaginary terms and four complex directions (that are simultaneously
of equal magnitude in both real and imaginary terms).
This again can be represented in circular terms as end points of both
horizontal, vertical and diagonal line diameters that bisect the circle
of unit radius.
Two Logical Approaches
Q I think I am beginning to see what you are getting at and I will
try and summarise.
The true significance of your linear and circular type number systems
is that number can be defined in terms of two utterly distinctive logical
systems.
The conventional understanding - which is directly suited to linear
interpretation - is based on an unambiguous either/or logic.
However there is an alternative system - directly suited to circular
interpretation - based on a paradoxical both/and logic. (However because
conventional mathematics does not recognise this logic it can only deal
with this system in a distorted reduced fashion).
Each system can be expressed indirectly in terms of the other.
Thus the linear (vertical) system has an indirect circular interpretation
in horizontal terms.
Likewise the linear (horizontal) system likewise has an indirect
circular interpretation in vertical terms.
Equally - starting from the circular viewpoint - we can say that
the circular (vertical) system has an indirect linear interpretation (in
horizontal terms).
Likewise the circular (horizontal) system has an indirect linear
interpretation in vertical terms.
Therefore each number system has both a linear or circular interpretation
(depending on the logical system through which it is viewed).
However the great problem with conventional mathematics is that it
operates solely in terms of one logical system. Therefore notions - which
are properly circular - can only be understood in a reduced linear (i.e.
one-dimensional) fashion.
Thus the very notion of dimension (as greater than 1) is inherently
circular.
This has immense consequences as it shows that the real meaning of
dimension conforms to a circular notion of direction (that directly conforms
to the mathematical notion of a root).
Therefore - as we have already seen in the previous discussion -
the fundamental structure of the physical (and psychological) dimensions
of reality is inherently mathematical in this circular sense.
Finally, when related to each other - which is always necessarily
the case - number (as quantity) and number (as dimension) are always -
properly - linear and circular (or alternatively circular or linear).
Thus if we interpret number (as quantity) in linear terms, number
(as dimension) is thereby circular. Alternatively, if we interpret number
(as quantity) in circular terms, number as quality (or dimension) is thereby
linear.
Thus by exploring this hidden (circular) aspect of number interpretation
- initially in analytic terms - we are led to the brink of the truly dynamic
holistic understanding.
I think that does it. However can you briefly clarify again the two
ways in which the circular can be interpreted (and the two ways likewise
of the linear).
PC When we obtain the various roots of the number 1 (i.e. 2nd
and higher), the results conform - even in analytic terms - to a circular
pattern (as equidistant points on the circle of unit radius).
However when we interpret these results in terms of the linear (either/or)
logic - conforming properly to the horizontal quantitative system - they
are separated. So for example the square root of 1 is either + 1 or - 1.
Now - in reverse fashion - when we raise the number 1 to various powers
or dimensions (2 or higher), the results this time conform - in direct
fashion - to a circular pattern (again as equal points on the circle of
unit radius).
This time the interpretation is in terms of the circular (both/and)
logic. Therefore the square of 1 (i.e. 12) properly refers to
the dimension of 2 (combining the two directions + 1 or - 1 as complementary).
However because conventional mathematics does not recognise this alternative
paradoxical both/and logic, this interpretation of dimension is completely
lost and all that remains is the indirect linear interpretation (where
such numbers now represent dimensions on a vertical linear axis).
Once again however we are on the verge here of the dynamic holistic
interpretation of these symbols to which we should now perhaps return.
Imaginary Number Interpretation
Q This is all of course very subtle and I can see absolutely fundamental.
I am beginning to glimpse here a key implication for the interpretation
of an imaginary number. Am I correct in assuming that the true meaning
of such an imaginary number requires circular understanding (and the corresponding
use of paradoxical both/and logic)?
However because conventional mathematics only recognises the either/or
logical system can we thereby only attempt to deal with imaginary numbers
in an indirect linear fashion which hides their true nature!
PC Yes! You are fully correct.
I must confess that I have long had a great fascination with imaginary
numbers. Even from an early age I felt that there was an important hidden
meaning that conventional mathematics somehow had failed to demonstrate.
Then gradually after several decades the secret at last revealed itself
when it dawned on me that whereas a real number corresponds directly to
linear logic, an imaginary number corresponds directly to circular interpretation.
Thus for example - in linear terms - a number can be either + 1 or -
1. However it cannot be + 1 or - 1 simultaneously. So either/or logic inherently
governs real number interpretation.
However an imaginary number inherently operates according to an alternative
logic where opposite polarities are simultaneously combined. Thus i
(i.e. the square root of - 1) has a direct interpretation as a number which
is simultaneously + 1 and - 1. However this has no meaning in conventional
either/or terms and therefore its secret remains locked within. So operations
can be only carried out indirectly on imaginary numbers, as if they were
real numbers (according to an either/or logic).
This also means that imaginary numbers truly come into their own within
a dynamic holistic interpretation (which properly incorporates circular
both/and logic).
Q I can also see once again the importance of the binary approach.
Not alone are the binary numbers so fundamental but every number (whether
as quantity or dimension) has a binary interpretation as linear (1) and
circular (0) and - in reverse terms - circular (0) and linear (1) respectively
.
In this context can you explain how for example with respect to 12
the dimension - which properly conforms to circular interpretation - is
equal to 0. How can 2 = 0?
Thus the number representing the dimension in this case (i.e. 2) properly
points to the complementarity of opposite positive (+) and negative (-)
poles which - in terms of a both/and logic cancel out as 0.
However when we try to represent this in the language of separation
we concentrate on the fact that there are two distinct poles. So the number
2 - in linear terms - represents these two distinct poles.
So again 14 (as dimension) in proper circular terms is again
= 0. This time the two real poles (+ 1 and - 1) and the two imaginary poles
(+ i and - i) using complementary both/and logic in both
cases cancel out = 0.
However in linear (separate) terms we again concentrate on the fact
that there are now four separate poles. So the dimension is thereby represented
as 4.
Q This relates to what you were saying earlier where the very nature
of the linear approach is to see opposite poles from a merely positive
perspective (i.e. where each pole is posited separately). Equally it also
entails seeing such poles in solely real terms. This logically leads to
the four-quadrant approach being equally a four-dimensional approach in
a reduced manner (where understanding takes place in terms of four separated
poles).
PC It also of course ties in with the notion that the linear approach
is suited directly for differentiation whereas the circular approach is
directly suited for integration.
So strictly speaking whereas a differentiated interpretation may be
four-quadrant (and thereby four-dimensional), from an integral perspective
it is strictly non-quadrant (and thereby non-dimensional).
In other words from an integral perspective the notion of separate quadrants
dissolves in the realisation of nondual understanding.
Holistic Integration: Type 0
Q Can we now move on to the holistic interpretation of these number
relationships.
Let is start with the simplest case where there is no difference
as between the horizontal and vertical interpretation of number.
PC When we look at both systems we can see that 11 belongs
to both systems.
We will examine here more carefully first what happens when we try to
convert the vertical interpretation (with respect to dimension) indirectly
in terms of the horizontal system. This requires - in this instance - obtaining
the 1st root of 11 which gives an unchanged answer
i.e. 11.
The implication is this is that the transformation is strictly linear
in this case (and does not lie on a circle).
In holistic terms this means that in the linear transformation circular
(paradoxical) notions are not formally recognised and thereby reduced in
linear terms. In other words the distinction of qualitative and quantitative
is effectively ignored (with the qualitative effectively reduced in quantitative
terms).
Now alternatively we can attempt to express the horizontal in terms
of the vertical number system (i.e. by indirectly expressing quantitative
notions in qualitative terms).
This requires - in this instance obtaining the 1st power
of 1 (11) which again gives an unchanged answer i.e. 11.
Again the implication is that the transformation is strictly linear
(again not lying on a circle).
So once more in holistic terms through linear transformation (in this
case from quantitative to qualitative) the distinction as between linear
and circular notions is not formally recognised (with the quantitative
now reduced in qualitative terms).
Put another way with linear interpretation relationships are formally
interpreted solely through either/or logic. This renders it impossible
to thereby preserve the key distinction as between quantitative and qualitative
notions which are respectively based on linear (either/or) and circular
(both/and) logic respectively.
So in precise holistic mathematical terms linear understanding is one-directional
with respect to both (quantitative) objects and (qualitative) dimensions.
Therefore with linear understanding formal understanding with respect
to both the quantitative and qualitative aspects of phenomena is merely
posited (in conscious terms).
This in turn means that linear understanding is based on the separation
of opposite poles (leading to dualistic interpretation).
Such dualistic understanding is associated with - in any given context
- unambiguous asymmetrical connections between variables.
By its very nature linear understanding - which is based on unitary
form (1) - is not geared to deal with notions of emptiness (0). Such notions
require the complementary appreciation of opposite poles (both positive
and negative) using circular both/and logic.
Linear logic is thereby geared for differentiated understanding of phenomena.
However it is quite unsuited for corresponding integral understanding.
Thus it can only deal with integration through effectively reducing
it to differentiation.
So in terms of complementarity, I refer to linear understanding as Type
0 complementarity (i.e. the absence of complementarity).
This is likewise associated with Type 0 differentiation (i.e. unambiguous
one-directional differentiation associated with a corresponding lack of
complementary understanding).
Such understanding is associated with the middle level of the Spectrum
(L0, H0) where the specialisation of rational understanding occurs.
This is the understanding that typifies conventional scientific understanding
and indeed the great bulk of intellectual discourse of all kinds.
Q Can you now briefly illustrate the nature of this understanding
with respect to conventional scientific understanding?
Though quantitative and qualitative aspects of understanding dynamically
interact it may be helpful to initially approach the issue where we identify
- in any context - perception with the horizontal (quantitative) and the
corresponding concept with the vertical (qualitative) aspect.
Now the every nature of such scientific i.e. analytic understanding
is that is has no appropriate means to preserve the distinction of quantitative
and qualitative (and qualitative and quantitative) and therefore must necessarily
reduce one to another.
So if we for example take the perception of an actual (i.e. real) molecule",
the corresponding concept of "molecule" will be viewed as applying to all
actual molecules. Therefore a direct correspondence is assumed to exist
as between perceptions and their corresponding concepts.
Likewise if we start from the concept of an actual molecule again it
will be viewed as relating to any particular molecule perception (within
its class) so that again a direct correspondence is assumed to exist -
in reverse manner - as between concepts and their corresponding perceptions.
In this way therefore in science a (conceptual) theory is interpreted
as directly applying to all corresponding empirical (perceptual) facts
within its class. Likewise empirical facts are interpreted as applying
to corresponding (conceptual) theories.
So in effect this double correspondence as between theory and facts
(and facts and theory) implies equally a double form of reductionism whereby
the qualitative is reduced to the quantitative (and the quantitative to
the qualitative).
Putting it another way - because analytic scientific understanding is
so heavily based on linear (either/or) interpretation - it cannot properly
distinguish as between the differentiation and integration of phenomena.
As we have seen such integration requires an alternative circular (both/and)
logical interpretation.
Q I know it is your contention that this form of linear reductionism
pervades most forms of intellectual discourse include those that that specifically
aim at a comprehensive integral view of development. Can you attempt to
explain this firther.
PC Yes it is possible to have an extremely comprehensive model of development
(based on a genuine spiritual integral vision) which is yet heavily linear
in intellectual interpretation (conforming to both Type 0 complementarity
and Type 0 separation).
Now perhaps it is unfair to single out Ken Wilber but he is undoubtedly
the best known proponent of the integral approach. However, I would find
it that his characteristic style of interpreting development is in fact
strongly linear and thereby not properly suited as an integral approach.
In other words - again in terms of intellectual interpretation - to
my mind he consistently confuses integration with multi-differentiation.
For example it is certainly true that a more comprehensive approach
to development requires the various types of differentiation with accompanying
sophisticated analysis that Ken has so brilliantly demonstrated throughout
the years. So among his many refinements his model carefully distinguishes
prepersonal, personal and transpersonal stages, levels of self and levels
of reality, structures states and bodies of development, the basic levels
(or waves) and transitional levels, many distinct lines (streams) of development
moving through the various waves, the interpretation of all these waves
and streams in terms of the four quadrants etc.
Now it is also true that he strongly advocates the need for authentic
spiritual practice as a prerequisite for genuine integration. However the
problem that I find is that such practice leading ultimately to nondual
realisation is treated largely separately from his characteristic analytic
understanding of development.
In other words there is a considerable discontinuity evident in terms
of his somewhat dualistic intellectual treatment of development (strongly
based on linear asymmetrical notions) and his admittedly very sincere nondual
approach to spiritual development.
However from a dynamic perspective dual and nondual continually interact
mutually changing the manner in which both aspects are realised and this
is especially true at the "higher" levels of development.
So the very purpose of an integral approach - in dynamic terms - is
to properly show how initial rigid notions of asymmetrical connection (based
on Type 0 interpretation) gradually give way to a much more refined paradoxical
interpretation properly consistent with spiritual nondual awareness. In
the terms that I describe it this therefore requires clarifying the nature
and role of Type 1, Type 2 and Type 3 understanding (that properly defines
the integral understanding of the "higher" spiritual levels of development).
Finally we show how these forms of circular understanding are then fully
incorporated with Type 0 linear interpretation.
Thus the radial approach is concerned with demonstrating both the relative
independence and interdependence of linear (differentiated) and circular
(integral) understanding.
Q With reference to the stages of development already mentioned can
you briefly explain the nature of an Integral 0 approach (i.e. based on
Type 0 complementarity).
PC Firstly prepersonal, personal and transpersonal are treated in a
somewhat discrete asymmetrical manner. So prepersonal unambiguously relate
to the lower, personal to the middle and transpersonal to the higher stages
of development.
However such distinctions are properly suited for differentiated rather
than integral understanding. So from a correct integral perspective prepersonal
and transpersonal (and transpersonal and prepersonal) are mutually complementary
and ultimately with nondual awareness identical. In other words balanced
integration occurs in both a top-down (transpersonal to prepersonal) and
bottom-up fashion (Likewise the personal are complementary with both the
(integrated) personal and transpersonal stages. In other words what is
neither prepersonal not transpersonal (i.e. personal) is complementary
with what is both prepersonal and transpersonal.
However as complementary understanding is strictly incompatible with
asymmetrical interpretation and undermines its very assumptions.
Likewise the holarchical approach to stages of development is very much
a linear asymmetrical approach being based on one-sided interpretation
of the dynamic relationship as between whole and part (i.e. where the whole
is part of a higher whole).
Whereas holarchy is suitable as one way of differentiating stages, it
is not suited as a means of integration which requiring the mutual interdependence
of holarchy i.e. where each whole is part of a higher whole with partarchy
i.e. where each part is a whole (in the context) of other parts.
Likewise any approach to interpreting the four quadrants based on unambiguous
notions of Right-Hand or Left-Hand (and Upper and Lower) is again - by
definition - linear as I define it. In effect such unambiguous identification
always entails understanding using isolated reference frames (based on
separation of polar opposites).
So for example to identify the Right-Hand with exterior reality requires
treating the exterior pole as relatively independent of the interior.
However whereas this again is initially suitable as a means of differentiating
quadrants it is not appropriate as a means of integration.
Again an integral approach establishing interdependence - requires the
two way interaction of complementary poles. All fixed notions of location
such as Right or Left (and Upper or Lower) are thereby rendered paradoxical.
Finally the notion of lines of development as relatively independent
(again defined as passing through stages in a holarchical manner) is very
much linear. (The very use of the terms "lines" clearly implies this!)
Again the identification of such "lines" is suited to the task of differentiating
development. However it is not suited for integration.
Q Do you imply that someone using a linear approach - as you define
it - actually believes that development unfolds in a linear manner?
PC Of course not! Linear has a very precise meaning in my approach implying
literally one-dimensional interpretation (i.e. one-directional). So a linear
approach is based - in any context - on interpretation of the relationship
between development variables in unambiguous sequential terms. This in
turn is always associated with asymmetrical understanding.
Thus if one maintains for example that the atom is contained in the
molecule (but the molecule not contained in the atom) then this is a clear
example of linear understanding.
Circular understanding by contrast is always based on the symmetrical
recognition of relationships between variables that simultaneously are
bi-directional (moving in opposite directions from each other).
Though such recognition is directly intuitive and spiritual, indirectly
it dynamically associated with a paradoxical means of intellectual interpretation.
Likewise I do not suggest that those who use linear approach believe
that integration is the same as differentiation (or that the qualitative
aspect is the same as the quantitative) . Rather I am pointing out that
- in effect - asymmetrical interpretation always leads to a reduction of
one aspect in terms of the other. |
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Ten most popular ordinal numbers
1
2
3
4
5
6
7
8
9
10
And that was an astoundingly bad joke. For more of the same sort, except illustrated, why not visit my daily growing jar of doodles, Lemmata?
(And the tragedy of this list is that 1, 2, 3 and so on aren't ordinal numbers in the mathematical sense, or in the linguistic sense: the first are monsters, the second are third, fourth and so on. Just that plain "arithmetic" meaning meant here. Also, is it funny or just weird that I feel compelled to add notes like this?) |
25+ Easter Eggs Kids (& Adults) Can Discover On Wolfram|Alpha
Recently the Wolfram|Alpha team blogged "10 Fun Questions Kids Can Answer with Wolfram|Alpha." It's worth a read, as it lists some great examples of what can be done with this "computational engine," if only because Wolfram|Alpha can seem awkward to use at first. Just some of the suggestions from the blog post are:
Find out the popularity of your name
Find out how many sides a given shape has
Find words that rhyme
Wolfram|Alpha has some cool parental uses too, in that it can answer some typically childlike questions, such as, "When I grow up, will I be as tall as you?" or satisfy a child's curiosity (and the Rainman in all of us) to know exactly how long it will be until their next birthday.
Furthermore, virtual Easter eggs buried within the index could almost be used to alleviate some of the more mind numbing tasks of dealing with children's questions. For example, next time you find yourself resigned to saying, "Just because," to your kids, instead, you could conceivably reply, "Why don't you ask Wolfram|Alpha?"
1. Are we there yet?
2. Whhhhhhhy?
3. Why NOT?
And, if you're not ready for "the birds and the bees" chat yet, you could see what your kid makes of Wolfram|Alpha's answer.
Those were the best I could find. Can you find any more? Drop a link in the comments if you do.
Undoing Conspiracy Theory
On the topic of conspiracy theories and weird science, in contrast to the entire Internet, this computational engine is refreshingly partisan on certain topics. Perhaps, "no-nonsense" would be a better way to put it, typical of an academic scholar. In fact, you almost get the feeling Stephen Wolfram himself may be speaking through the medium.
25. Some of the answers will knock any adult off their chair. Despite a philosophy degree, I was still convinced there was no definitive answer to the question "if a tree falls in the wood, does it make a sound?" Clearly, I should have paid more attention in class.
26. I got a shudder when I discovered the following little gem. It's almost evidence of the type of computational irregularity which might cause a machine to become sentient or perceive a soul. Albeit a dark and morbid soul, like HAL from "2001: A Space Odyssey".
Bear with me, as what follows is a bit obtuse. Ludwig Wittgenstein, a great philosopher of mind and mathematical genius, argued that we could not know the absolute essence of a soul, even if we all agreed and believed in the existence of it. Using the beetle in a box "mind game" he illustrated that the certainty with which human beings discuss the existence of their soul, is actually an agreement about nothing in particular.
He argued that the concept of a soul, is similar to everyone being in possession of a box, in which they were hiding a beetle, but they were forbidden to reveal it to anybody. While everyone was allowed to discuss the beetle they had in the box, they could not compare them with each other. Therefore, they operated on the assumption that everyone's beetle was the same as their own, despite never having any proof to that effect. In short, he threw into question the absolute certainty with which previous philosophers had posited the existence of "meta-physical" phenomena, such as souls and minds, as no one could verify the existence of anyone else's beetle.
In many ways, Wittgenstein's rationale opened the doors to people being allowed to have their own beliefs, about human nature and human experience, rather having those beliefs prescribed by the church. Original sin was one such idea that needed to be re-examined as a result.
So, I repeat, it is with a shudder that Wolfram|Alpha's answer to What is in the box? takes the refutation almost forward and backward simultaneously. Our latter day HAL citing a character from the sci-fi novel "Dune" echoes an undeniable and common experience of life, so close to original sin of the past and yet so modern in it's context. It's genius.
As you can see from all of the queries above, the Wolfram|Alpha team and their troupe of editors and academics gave the engine a bit of personality and playfulness. It's brilliant. I love it. I would go as far to say that you could call Wolfram|Alpha the Stephen Fry (once, the most followed man on Twitter) of search engines. But don't let a robot tell you, or your kids, the meaning of life |
Computers vs. People: Writing Math
Readers of this 'blog know I actively use many forms of technology in my teaching and personal explorations. Yesterday, a thread started on the AP-Calculus community discussion board with some expressing discomfort that most math software accepts sin(x)^2 as an acceptable equivalent to the "traditional" handwritten .
Some AP readers spoke up to declare that sin(x)^2 would always be read as . While I can't speak to the veracity of that last claim, I found it a bit troubling and missing out on some very real difficulties users face when interpreting between paper- and computer-based versions of math expressions. Following is an edited version of my response to the AP Calculus discussion board.
MY THOUGHTS:
I believe there's something at the core of all of this that isn't being explicitly named: The differences between computer-based 1-dimensional input (left-to-right text-based commands) vs. paper-and-pencil 2-dimensional input (handwritten notation moves vertically–exponents, limits, sigma notation–and horizontally). Two-dimensional traditional math writing simply doesn't convert directly to computer syntax. Computers are a brilliant tool for mathematics exploration and calculation, but they require a different type of input formatting. To overlook and not explicitly name this for our students leaves them in the unenviable position of trying to "creatively" translate between two types of writing with occasional interpretation differences.
Our students are unintentionally set up for this confusion when they first learn about the order of operations–typically in middle school in the US. They learn the sequencing: parentheses then exponents, then multiplication & division, and finally addition and subtraction. Notice that functions aren't mentioned here. This thread [on the AP Calculus discussion board] has helped me realize that all or almost all of the sources I routinely reference never explicitly redefine order of operations after the introduction of the function concept and notation. That means our students are left with the insidious and oft-misunderstood PEMDAS (or BIDMAS in the UK) as their sole guide for operation sequencing. When they encounter squaring or reciprocating or any other operations applied to function notation, they're stuck trying to make sense and creating their own interpretation of this new dissonance in their old notation. This is easily evidenced by the struggles many have when inputting computer expressions requiring lots of nested parentheses or when first trying to code in LaTEX.
While the sin(x)^2 notation is admittedly uncomfortable for traditional "by hand" notation, it is 100% logical from a computer's perspective: evaluate the function, then square the result.
We also need to recognize that part of the confusion fault here lies in the by-hand notation. What we traditionalists understand by the notational convenience of sin^2(x) on paper is technically incorrect. We know what we MEAN, but the notation implies an incorrect order of computation. The computer notation of sin(x)^2 is actually closer to the truth.
I particularly like the way the TI-Nspire CAS handles this point. As is often the case with this software, it accepts computer input (next image), while its output converts it to the more commonly understood written WYSIWYG formatting (2nd image below).
Further recent (?) development: Students have long struggled with the by-hand notation of sin^2(x) needing to be converted to (sin(x))^2 for computers. Personally, I've always liked both because the computer notation emphasizes the squaring of the function output while the by-hand version was a notational convenience. My students pointed out to me recently that Desmos now accepts the sin^2(x) notation while TI Calculators still do not.
Desmos:
The enhancement of WYSIWYG computer input formatting means that while some of the differences in 2-dimensional hand writing and computer inputs are narrowing, common classroom technologies no longer accept the same linear formatting — but then that was possibly always the case….
To rail against the fact that many software packages interpret sin(x)^2 as (sin(x))^2 or sin^2(x) misses the point that 1-dimensional computer input is not necessarily the same as 2-dimensional paper writing. We don't complain when two human speakers misunderstand each other when they speak different languages or dialects. Instead, we should focus on what each is trying to say and learn how to communicate clearly and efficiently in both venues.
6 responses to "Computers vs. People: Writing Math"
While I was only an acorn this past year, one thing that seemed abundantly clear to me is that there is no "always". The way we were instructed to score certain problems seemed arbitrary between problems and sometimes inside a problem. For instance, students were given credit for an answer with only two decimal places when the third was a zero on part (b) of BC#2. They lost points though for the same thing on part (d) of the same problem. While I am sure the Question Leader and Chief Reader have their reasons, it was not made clear to everyone else. As for the notation, the very same problem (BC#2) awarded point for an expression for speed. Students with an expression of (x'(3))^2 + (y'(3))^2 inside the radical were given credit immediately. Students with the expression of x'(3)^2 + y'(3)^2 only earned the point for the expression in the presence of the correct answer. The notation was called "ambiguous" and needed the correct answer to show the student used it as required for distance.
Thanks, Dennis. Your note reminds me why I emphasize clear writing and understanding of audience to all students (not just AP).
For the AP, students need to understand that their graders are absolutely not the same as their teachers. A teacher knows what students have learned and probably knows what was mean when by unconventional notations or phrasings. Unfortunately, that assumption can never be made for AP graders. Those who grade AP responses need to be convinced that a student knows what he or she is doing in a very precise way. They don't know the students, and they should not be expected to give any credit for even marginally vague explanations. It is the students' jobs to write clearly. It is our job as teachers to help them learn that skill. It's a good habit for any writing that will be (or might be) read by someone unknown to the writer.
I agree. I was trying to make the point that both notations are entirely logical within their respective environments.
For someone steeped in handwritten math to complain about computer syntax is much like an English speaker complaining that a French speaker puts adjectives after a noun (e.g. chapeau blanc) instead of before the before the noun like they're supposed to be [in English] (e.g. white hat). In this example, the complainer misses the point. Both constructions are correct within their environments. |
My teacher,
Heather Clewett-Jachowski, strongly believes that many (and perhaps
even all) crop circles have Squaring the Circle encoded in them in some
form. We are just failing to see it. She could well be right.
The geometrical basis of for instance the 1.5 Yin Yang formation helps us in a very elegant way to square the circle
and we nearly missed it!
See
Squaring Yin Yang
Connected to this formation is the Furze Knoll crop circle of 20 June
2008.
The diagram above is a
representation of the core of the crop circle that arrived on 20 June
2008. The day Summer and Winter were true at the same time. They were
one. This Oneness is exactly what Yin Yang stands for and what was so
profoundly expressed in the ´Yin Yang´ crop circle of 8 May 2008.
In the article
Möbius Strip Crop Circle
I showed how the 20 June 2008 crop circle is closely related
to the Yin Yang crop circle.
They have the identical core. The core of two circles wrapped inside One
circle.
Because the above diagram
representing the 20 June 2008 crop circle at Furze Knoll, Wiltshire,
England, has the same core (twice) as the Yin Yang does, it also
contains Squaring the Circle twice! See Squaring Yin Yang.
When you look at the diagram below on the left, you will see an amazing
feature. The red arrows show were the Circles of the two Squaring the
Circles are intersecting. We now can construct a square that goes
through these intersections.
Yes, you have guessed
correctly. The blue square and the red circle are squaring the circle.
The perimeter of the blue square is equal to the circumference of the
red circle with a precision of 99.4%. This is really amazing!
The core of the formation is based on two overlapping Yin Yangs. Since
every separate Yin Yang contains Squaring the Circle, the core of the 20
June 2008 formation contains two overlapping 'squaring the circle'. The
overlap in itself defines a new square, that Squares the total
perimeter. Wow, wow, wow.
Again, you don't have to stretch your mind a lot to go from the diagram above to the diagram below.
It is so amazing to watch
how the crop circle phenomenon works. It gives us clue after clue, where
the clues are often different for every different observer. Then, when we
have failed to recognise the given clue, it will give us new clues to
help us understand the first clues. And so on and on. It therefore can
happen that you will 'discover' new things in old crop circles. Like I
did with the 1994 crop circle near Boven-Smilde, the Netherlands.
See:
Squares and Circles
Many crop circles - and perhaps even all of them as my teacher
Heather Clewett
suggests - have come down
with Squaring the Circle encoded in them. Many times we have failed to recognise this feature.
On 8 May 2008 a 1.5 Yin Yang symbol arrives. The symbol of the Unity of Opposites and again we nearly missed the obvious:
the encoding of Squaring the Circle. Also the connection with the Furze Knoll
formation that came down on 20 June 2008 did not immediately lead to recognising
Squaring the Circle. It is shouting in our face and we keep missing it.
Lets look better from now on! |
More Activities:
Mathematicians are not the people who find Maths easy; they are the people who enjoy how mystifying, puzzling and hard it is. Are you a mathematician?One Torch Tunnel
Solve the problem of getting four people through a tunnel with one torch in the minimum amount of time. The solution is not obvious but when you get it you see the elegance and beauty of its cunning |
What are the advantages and disadvantages of Mathematical model - disadvantage: it often cannot account for all possible factors Advantage to a mathematical model is it can be applied to categories of any size and can predict complex … |
A A who uses logic so cleverly that everyone thinks it is magic. Leading them through the complexities of logic, he takes them to a different island, where robots are programmed to make other robots with programmes of their own, and the reader has to work out what the programmes will be. The book ends with a guided tour of infinity, and we hear the remarkable story of how the devil was once outwitted by a student of the mathematician Georg Cantor. Raymond Smullyan is a mathematician and logician. He has also written "The Lady or the Tiger?", "Forever Undecided - a Puzzle Guide to Goedel", "To Mock a Mockingbird", "The Chess Mysteries of Sherlock Holmes" and "The Chess Mysteries of The Arabian Knights". ...Continua Nascondi |
Tuesday, 21 August 2012
HISTORY OF TRIGONOMETRY
Ansua XI C
The history of trigonometry goes back to the earliest recorded mathematics in Egypt and Babylon. The Babylonians established the measurement of angles in degrees, minutes, and seconds. Not until the time of the Greeks, however, did any considerable amount of trigonometry exist. In the 2nd century bc the astronomer Hipparchus compiled a trigonometric table for solving triangles. Starting with 7½° and going up to 180° by steps of 7½°, the table gave for each angle the length of the chord subtending that angle in a circle of a fixed radius r. Such a table is equivalent to a sine table. The value that Hipparchus used for r is not certain, but 300 years later the astronomer Ptolemy used r = 60 because the Hellenistic Greeks had adopted the Babylonian base-60 (sexagesimal) numeration system.
In his great astronomical handbook, The Almagest, Ptolemy provided a table of chords for steps of y°, from 0° to 180°,that is accurate to 1/3600 of a unit. He explained his method for constructing his table of chords, and in the course of the book he gave many examples of how to use the table to find unknown parts of triangles from known parts.
Indian astronomers had developed a trigonometric system based on the sine function rather than the chord function of the Greeks. This sine function, unlike the modern one, was not a ratio but simply the length of the side opposite the angle in a right triangle of fixed hypotenuse. The Indians used various values for the hypotenuse.
Late in the 8th century, Muslim astronomers inherited both the Greek and the Indian traditions, but they seem to have preferred the sine function.Trigonometric calculations were greatly aided by the Scottish mathematician John Napier.
Finally, in the 18th century the Swiss mathematician Leonhard Euler defined the trigonometric functions in terms of complex numbers. This made the whole subject of trigonometry just one of the many applications of complex numbers, and showed that the basic laws of trigonometry were simply consequences of the arithmetic of these numbers.
Monday, 20 August 2012
AryabhatTa
Bijeesh A B - VI C -
Aryabhata was the great Hindu Mathematician. He lived from 475 AD to 550 AD. He wrote on arithmetic including undo this heading arithmetic and geometric progressions, quadratic equations, and indeterminate equations. His work often called Aryabhatiya consists of a collection of astronomical tables and the Aryastasata which include the Ganita, a note on arithmetic, the Kalakriya on time and its measure and the Gola on the sphere. Aryabhata was one of those ancient scholars of India who could stand with pride among the greatest scholars even of the modern age.
Puzzle
1)There is a number which is very peculiar. This number is 3times the sum of its digits. Can you find the number?
2) Write the biggest number that can be written with four one's?
3) Can you write 1789 in roman numbers?
4) What is the sum of first 70 odd numbers?
5) Find the value of:
CCLXV+CXVI+XIII+VI=?
Beginning of algebra
It is said that algebra as a branch of mathematics began about 1550 BC, that means more than 3500 years ago, when people in Egypt started using symbols to denote unknown numbers .Around 300 BC, use of letters to denote unknown and forming expressions from them was quite common in India. Many great Indian mathematicians Aryabhata, Brahmagupta, Mahavira and Bhaskara II and others, contributed a lot to the study of Algebra.
Branches of mathematics
The branches of mathematics in which we study about numbers is called arithmetic. The branch in which we learn about figures in two and three dimensions and their properties is called geometry. The branch in which we use letters along with numbers to write rules and formulas in general is called algebra.
·Different names for the number 0
include zero, nought, naught, nil, zilch and zip.
·The = sign ("equals sign")
was invented by 16th Century Welsh mathematician Robert Recorde, who was fed up
with writing "is equal to" in his equations.
·Googol (meaning & origin of Google brand)
is the term used for a number 1 followed by 100 zeros and that it was used by a
nine-year old, Milton Sirotta, in 1940.
·The name of the popular search engine
'Google' came from a misspelling of the word 'googol'.
·Abacus is considered the origin of the
calculator.
·12,345,678,987,654,321 is the product
of 111,111,111 x 111,111,111. Notice the sequence of the numbers 1 to 9 and
back to 1.
·Plus (+) and Minus (-) sign symbols
were used as early as 1489 A.D.
·An icosagon is a shape with 20 sides.
·Trigonometry is the study of the
relationship between the angles of triangles and their sides.
·If you add up the numbers 1-100
consecutively (1+2+3+4+5...) the total is 5050.
·2 and 5 are the only primes that end in
2 or 5.
·From 0 to 1,000, the letter "A"
only appears in 1,000 ("one thousand").
·'FOUR' is the only number in the
English language that is spelt with the same number of letters as the number
itself
·40 when written "forty" is
the only number with letters in alphabetical order, while "one" is
the only one with letters in reverse order.
·Among all shapes with the same
perimeter a circle has the largest area.
·Among all shapes with the same area
circle has the shortest perimeter .
·In 1995 in Taipei, citizens were
allowed to remove '4' from street numbers because it sounded like 'death' in
Chinese. Many Chinese hospitals do not have a 4th floor.
·In many cultures no 13 is considered
unlucky, well,there are many myths around it .One is that In some ancient
European religions, there were 12 good gods and one evil god; the evil god was
called the 13th god.Other is superstition goes back to the Last Supper. There
were 13 people at the meal, including Jesus Christ, and Judas was thought to be
the 13th guest.
Some
beautiful examples with multiplication
12345679 x 9 = 111111111;
12345679 x 8 = 98765432
An
interesting fact about primes
Mathematicians of XVIIIth century proved
that numbers 31; 331; 3331; 33331; 333331; 3333331; 33333331 are primes. It was a big temptation to think that all numbers of such kind
are primes. But the next number is not a prime. 333333331 = 17x19607843 |
Using random numbers to estimate Pi
02 Jan 2015
I'm a big fan of Monte Carlo methods. There is something neat about performing a complex integration or simulation fueled by an engine running on random numbers.
In this post, I'm going to talk about a canonical example used to demonstrate Monte Carlo integration: estimating the area of the unit circle via hit-or-miss Monte Carlo sampling. Indeed, this toy example is so ubiquitous that it is the focus of the Wikipedia Monte Carlo integration page (I was even asked to describe this problem during my Masters defence).
Here we go:
The unit circle follows the equation $x^2 + y^2 = 1$, and the area of the full circle can be determined from the integral of a quadrent (multiplied by 4 to account for all four quadrents). The area is then expressed as the definite integral,
$A_{\mathrm{circle}} = 4\int_{0}^{1} \sqrt{1-x^2}dx$.
Well, sometimes we just don't feel like doing textbook trigonometric substitutions. In this case we're essentially going to brute force the solution using so-call 'hit-or-miss' Monte Carlo. The procedure is as follows:
generate two random numbers, $x$ and $y$, both in the range [0,1]
if $x^2 + y^2 < 1$ then add $(x,y)$ to the 'hit' bucket, otherwise add $(x,y)$ to the 'miss' bucket
repeat steps 1-2 a total of $n$ times
finally, use the fraction of 'hits' to estimate the area of the circles via the formula $4 \frac{\mathrm{hits}}{\mathrm{hits}+\mathrm{misses}}$
The following function uses Python's standard pseudo-random number generator to complete steps 1-3:
Finally, we can summarize the result, estimate the area, and compare tne result to the known value ($\pi$),
In [18]:
fromtabulateimporttabulateimportmathdefestimate(hits,misses):''' Estimate the area of the unit circle '''return4.*float(len(hits))/float(len(hits)+len(misses))defsummary(hits,misses):printprinttabulate([["Number of points inside the circle",len(hits)],["Number of points outside the circle",len(misses)],["Estimate of circle area (4*h/(h+m))",estimate(hits,misses)],["Percent different from pi",round((estimate(hits,misses)-math.pi)/math.pi*100.,1)]],tablefmt="plain")summary(hits,misses)
Number of points inside the circle 7890
Number of points outside the circle 2110
Estimate of circle area (4*h/(h+m)) 3.156
Percent different from pi 0.5
With 1000 samples, we get an estimate that is within a couple percent.
It should be pointed out that Monte Carlo integration isn't very fast. In fact, this method converges as $1/\sqrt{n}$, which is to say, if you want a 10% increase in the precision, you need to add 100x more samples. This scaling relationship between the precision of the estimate, and the number of samples is shown here: (where the green line is $\sim1/\sqrt{n}$)
In [6]:
importnumpyasnpns=[]stdevs=[]sqrts=[]fornin[10,25,50,100,250,500,1000,2500,5000,10000]:samples=[]forsampleinrange(100):hits,misses=generate_points(n)samples.append((estimate(hits,misses)-math.pi)/math.pi)ns.append(n)stdevs.append(np.std(samples))sqrts.append(0.5/math.sqrt(n))plt.loglog(ns,stdevs,'o',ns,sqrts,'-')plt.xlabel("Number of points, n")plt.ylabel("Precision of estimate")plt.show()
The $1/\sqrt{n}$ scaling isn't that attractive for 1-D integrations like above (the trapezoid rule, for example, goes as $\sim1/n^2$). However the power of the Monte Carlo method is that the scaling relation holds for higher dimensions, where methods like the trapezoid rule fall down. This is great, as we can still fairly efficiently perform integrations for problems with very large dimensionality.
Many fields use Monte Carlo methods to simulate physical processes. In particle physics (which is what I'm most familar with), for example, the trajectories of particles are simulated, along with their interactions with materials and other particles, to estimate likely outcomes for experiments. An simulation package used extensively by particle (and medical) physicists is GEANT4.
While GEANT4 is a large and very powerful software package, I'm going to use it here in a somewhat silly way to perform the same hit-or-miss integration of the area of a circle. Essentially, I'm going to simulate an electron being shot at a circular target, where the origin point of the electron is uniformly distributed in an encompassing square (just like above). Then, in the same way, we can count the number of electrons that hit the target, and estimate just how close to $\pi$ we can get!
A note: GEANT4 can be finicky to install. Here I took advantage of a Docker container put together by Simon Biggs, which removed most of the installation hassle and also took advantage of the Python bindings and runs inside an IPython notebook. How cool is that. I've modified Simon's example to my own devious ends:
First off, we'll create the world volume and a thin cylinder target. That will be it for the simulation geometry -- we don't actually want anything to get in the way of the electrons (not even simulated air), because any scattering will cause perturbations in the electron trajectories or even cause the emission of secondary particles.
In [2]:
classMyDetectorConstruction(G4VUserDetectorConstruction):def__init__(self):G4VUserDetectorConstruction.__init__(self)self.solid={}self.logical={}self.physical={}#root volumnself.create_world()#targetself.create_target()defcreate_world(self):material=gNistManager.FindOrBuildMaterial("G4_Galactic")self.solid['world']=G4Box("world",4000/2.,4000/2.,4000/2.)self.logical['world']=G4LogicalVolume(self.solid['world'],material,"world")self.physical['world']=G4PVPlacement(G4Transform3D(),self.logical['world'],"world",None,False,0)visual=G4VisAttributes()visual.SetVisibility(False)self.logical['world'].SetVisAttributes(visual)defcreate_target(self):translation=G4ThreeVector(0,0,0)material=gNistManager.FindOrBuildMaterial("G4_Si")visual=G4VisAttributes(G4Color(1.,1.,1.,0.1))mother=self.physical["world"]self.solid["target"]=G4Tubs("target",0.,1000,0.0001/2.,0.,2*pi)self.logical["target"]=G4LogicalVolume(self.solid["target"],material,"target")self.physical["target"]=G4PVPlacement(None,translation,"target",self.logical["target"],mother,False,0)self.logical["target"].SetVisAttributes(visual)defConstruct(self):# return the world volumereturnself.physical['world']
Here, we define the primary particle emission, which in our case is the electrons. We'll shoot those electrons in the direction $-1\vec{z}$, with energies of 6MeV (which shouldn't really matter in our case). Each iteration, a random starting position is choosen for the electron in GeneratePrimaries.
Here is the sneaky bit. For every step of the simulation, we'll check if the primary electron is inside of the target volume. If yes, we'll increment a hit counter. Please note that this isn't actually a good GEANT4 practice: usually, if you want to check if a particle has traveled through a volume, you'll define a SensitiveDetector, and it'll take care of figuring out if any particles hit that volume. However, since we're already doing naughty things, we'll take this shortcut.
In [4]:
classMySteppingAction(G4UserSteppingAction):def__init__(self):G4UserSteppingAction.__init__(self)self.cnt=0#hit counterdefUserSteppingAction(self,step):track=step.GetTrack()iftrack.GetVolume().GetName()=='target'andtrack.GetParentID()==0:#check if the primary electron hit the targetself.cnt=self.cnt+1
Here, we register our user-defined functions with the GEANT4 run manager. We'll use a default Physics List -- event though it defines a lot more processes than we'll actually utilize, it is easier to use a pre-defined list than add all of the processes one-by-one.
Just like the first section, we see points both inside and outside of the target circle. We can then access the stepping_action hit counter, which recorded the number of electrons that passed through the target volume:
In [9]:
stepping_action.cnt
Out[9]:
15642
With a couple extra lines, we can reuse the summary code from above to make the circle area estimate and percent error:
Number of points inside the circle 15642
Number of points outside the circle 4358
Estimate of circle area (4*h/(h+m)) 3.1284
Percent different from pi -0.4
Not so bad, for a fairly inappropriate use of the GEANT4 framework. Note that, while we could add more electrons to our samples, using this method we won't end up getting a better estimate of $\pi$. This is mainly because the simulation volume used by GEANT4 isn't actually a circle. You can see in the above figure that it is acutally a regular polygon with n=22 or so. Therefore, adding more samples will actually give us a more precise estimate of the area of that polygon, but a biased estimate of $\pi$ (which is a not-bad example of the difference between accuracy and precision).
I've also hand-waved over a bunch of assumptions about the simulation: like if the electron trajectories are actually expected to be straight lines in this medium (I didn't actually check if G4_Galactic has a mean-free-path), also, I made the target volume very thin to hopefully avoid any scattering or secondary generation within the target itself, but when using a default physics list, you could possible end up with some hadronic interaction or large angle scattering that could double-count some electrons in the target. But ultimately, doing something unusal, even if a bit hand-wavy, is the fun part. |
Marwan S.
Math, it's more than just the numbers. It's all about the methods and logic behind those numbers.
As an engineer from both Intel and Apple |
ASK
Where Can You Find A Math Expert To Answer Your Questions Online? #ask
#ask a math question #
Where Can You Find A Math Expert To Answer Your Questions Online?
A math expert is a person well versed in mathematics and/or the sciences which require advanced computing skills. A few years back consulting with a math expert ay be hard or at least inconvenient since the only way to do this is via mail, telephone or actual visits. Nowadays, if a person needs to a mathematics expert all he or she has to do is go to the internet and type in a search sentence or phrase. The goal of this article is to put a bit of methodology on the question "where can you find a math expert to answer your questions online?"
The first thing that a person searching for a math expert has to do is to specify the query. This is because chances are typing in the words "where can you find a math expert to answer your questions online?" will populate the computer screen with multiple results that number in the hundreds if not thousands. Reading thru all this will be an exercise in futility. Narrowing down the search to a specific topic will definitely make life a lot easier. For example, if an individual is looking for a math expert on physics then type in "math expert on physics" if a person is looking for an expert in geometry then type in "math expert on geometry." Always remember narrow down the subject matter to narrow down results. A very good tip when looking for math experts is to go to university websites and search for the proper department. This should provide the searcher with names of mathematics professors, the next thing to be done is to just email the math professor. The worst case scenario is he or she does not reply, the best case scenario: he or she is the expert the searcher has been looking for and if not, he or she might be able to forward the searcher to a colleague who is.
The second thing that a person needs to know in order to find the proper math expert online is to determine what kind of application the mathematics problem will be used for or is related to. Some math problems may be related to biology such as DNA or it may be related to engineering, aerodynamics or it can be related to a thesis that is being written. Whatever the particular application is, it would be best if the same is entered as part of the keyword. For example, if a person is looking for a math expert on geometry in order to determine or answer questions regarding aerodynamics then type in "math expert on geometry for aerodynamics" if a person is looking for a physics expert for aerodynamics then type in "math expert on physics for automobile aerodynamics." Again, narrowing down the search gives the person more time to read thru the numerous queries that will pop up.
The third thing a person needs to know is to try and localize the search to universities, NGO's and blogs within the locality. Chances are there is a math expert within the vicinity and contacting that person online maybe more convenient at first but in some cases it would be best to actually sit down face to face and discuss the issues involved. If this is the case, it's better that you find a math expert residing as near as possible. Only after a localized query finds no sufficient results should the searcher slowly increase the coverage of the search. For example, at first type in the words "math expert on physics for aerodynamics in Los Angeles" if no sufficient queries pop out then modify the search to "math expert on physics for aerodynamics in California" or "math expert on physics for aerodynamics in the east coast."
Always plan ahead, it would be very inconvenient to find out that the math expert that has been corresponding with the searcher lives too far for a proper consultation. Typing in "where can you find a math expert to answer your questions online?" is a place to start but a good researcher always plans ahead. Remember, a good researcher always thinks a step ahead. |
Tangram, The link between mathematics and art
Leaving aside the controversial forms of art, the artist and academic at the University of Nicosia, Maria Chrisfororou presents the event Tangram: The link between mathematics and art. The event includes an interactive workshop along with interesting visual and digital interventions.
Maria Christoforou deals with the correlation of mathematics and art, inspired by Tangram's philosophy, and focuses mainly on Geometry and the geometric shapes that are essentially between us.
Even though mathematics and art rely on two different themes, they are probably not so different from each other! Mathematics and art are mainly interrelated with their "philosophy," the depiction of the abstract or realistic reality of which we are all part of it.
As the artist says, "We can argue that mathematics and art are two examples of human consciousness, that is, the human perception process, to understand reality. The artist, as well as the mathematician, participate in the human need and understanding of the world. In conclusion, an artistic creation/artwork seems to be realistic or abstract in a mathematical construction. "
Maria Christoforou proposes a delightful artistic and creative experience with Tangram, this ancient and unique Chinese puzzle consisting of seven geometric pieces: One square, one parallelogram and five triangles (two large, two small and a medium sized). The relationship between the shapes is such that, by following some rules, it allows them to fit precisely and create different shapes. With Tangram we create very beautiful artworks in varied and numerous combinations. |
Firstly, I'll take this opportunity to wish James Cranch a marvellous 30th birthday.
This post was inspired by the conception of a new 'origami society' at Trinity. Having never tried it before, it took me a while to decipher the instructions for the construction of a dragon.
There are rather restrictive constraints on what possible patterns of folds can be realised. Firstly, arrangements of folds are represented as plane graphs (known as 'crease patterns') with two distinct types of edges, namely 'mountain folds' and 'valley folds'. There is actually a third type of edge, a 'boundary edge', since we're dealing with a finite square rather than R^2; the boundary edges form a cycle such that all other vertices lie in the closed subset of the plane bounded by this cycle.
Maekawa's theorem: Each non-boundary vertex must have even degree 2n, such that there are either n + 1 valley folds and n − 1 mountain folds, or vice-versa. As a consequence of the even degree property, the dual graph (or, more pedantically, the induced subgraph of the dual graph obtained by removing the vertex at infinity) has a chromatic number of 2.
This is the only combinatorial restriction, and yields graphs such as this one:
There are, however, more geometrical constraints.
Kawasaki's theorem: the sum of the alternate angles around each interior vertex is π. This is a well-defined concept, since Maekawa's theorem forces the degree of each interior vertex to be even. Also, the configuration must be realisable in Euclidean geometry.
Finally, there is the constraint that the three-dimensional embedding of the folded paper does not self-intersect. This is much more difficult to verify from the diagram than the other two properties.
Postulates
Since origami requires precision, it would be ideal to be able to produce exact crease patterns. This can be accomplished with a pencil, compass and straightedge, as in Euclidean constructions. Basically, the operations achievable with compass and straightedge are:
Given two points, A and B, construct the unique line passing through A and B;
Given two points, A and B, construct the unique circle centred on A and passing through B;
Given two curves which intersect in finitely many points, construct these points of intersection.
It's not too difficult to show that these operations are equivalent to solving a quadratic given its coefficients. Three famous classical problems, namely 'doubling the cube', 'trisecting the angle' and 'squaring the circle', are unsolvable with these operations (proof: Galois theory for the first two, and transcendentality of pi for the third).
It can be shown via Galois theory that the nth roots of unity can be constructed if and only if n is a product of a power of two and distinct primes of the form 2^k + 1 (i.e. Fermat primes). For instance, Gauss managed to express exp(2πi/17) in terms of square-roots, thus effectively constructing the regular 17-gon.
However, this seems to be digressing somewhat from the original topic of origami. It is overkill to add three new instruments (a pencil, straightedge and compass) to our arsenal, since origami alone can be used for geometrical constructions. Even though circles cannot be constructed for obvious reasons, it is a remarkable fact that we can still construct all of the same points as with the compass and straightedge. Indeed, as will be revealed shortly, a proper superset of those points are origami-constructible.
The ordinary Euclidean constructions are replaced with seven different postulates, known as Huzita-Hatori axioms (despite not actually being axioms).
Given two points, A and B, we can construct a unique fold (line) passing through both of them;
Given two points, A and B, we can construct a unique fold mapping one of them to the other (the perpendicular bisector);
Given two lines, l and l', we can construct a fold (two if the lines intersect, or one if they're parallel) mapping one of them to the other (angle bisector);
Given a line l and a point P not incident with l, we can construct a unique fold passing through P and perpendicular to l;
Given two points A and B and a line l, we can construct a fold passing through A and mapping B onto l;
Given two points, P_1 and P_2, and two lines, l_1 and l_2, we can construct a fold capable of mapping P_1 onto l_1 and P_2 onto l_2;
Given a point A and two lines, l and l', we can construct a fold perpendicular to l' mapping A onto l.
The sixth postulate is the powerful one, which (in conjunction with the others) enables the general cubic to be solved. This solves both the problems of trisecting the angle and doubling the cube, and allows the construction of nth roots of unity if and only if n is a product of a power of two, a power of three and distinct Pierpont primes.
It is conceivable that someone could write an origami version of Euclid's Elements. Also, this would be more powerful than Euclid: Morley's angle trisector theorem cannot be proved by compass and straightedge, since the equilateral triangle cannot be constructed in general, but is within the power of origami. |
The Mandelbrot Set
Where does the color come from?
We know the points in the set are coded black. The Java applet you will see took the
function out to 100 iterates. After n iterates, the point may leave the set proceeding
on its way to infinity. If n is a small number, then the point goes out to infinity
very quickly. If n is close to 100, then the point's orbit, or iterate path, stays
inside the set for awhile before it leaves.
Color
Let's take our earlier example, z0 = 0.5 + 0.75 i. This appears to have left
the set after the 1st or 2nd iterate. We can check this by finding the
distance from z2 to the origin. How do we compute distances between two complex
numbers? Look it up. If the result is greater than a radius of 3,
then the point has left the set. Since n = 2, it would be assigned the color for 2, say
it is purple.
Go on to the Julia set. |
A short account of the history of mathematics by W. W. Rouse Ball(
Book
) 7
editions published
in
1907
in
French and Undetermined
and held by
25 WorldCat member
libraries
worldwide
"Aristotle, Galileo, Kepler, Newton -- you know the names, here's the book that will tell you what they really did. It covers
in detail the historical development of mathematics and includes the early major theorists and their discoveries. The basis
of arithmetic starts with the ancient Greeks, including Plato and the introduction of geometry in the method of analysis.
The use of Arabic numerals brought math lightyears into the future, and was critical in the development of both commerce and
science. The introduction of calculus helped make space travel possible, while the abacus was the precursor of the calculator.
There is Leonardo da Vinci, whose fame as an artist has overshadowed his achievements as a mathematician; Pascal, who laid
down the principles of the theory of probabilities; and Fermat and his intriguing theorem. Without these pioneers, our lives
would be very different, because the growth of our society is interwoven with that of mathematics. And this is how it began.
Book jacket."--Jacket |
<< No reason why you can't learn times tables as well--and why is maths
considered 'higher' anyway?in my experience, mathematicians are just like
the rest of us--and many of them live in a fantasy world, too. >>
Some of us *are* mathematicians.
Solomon Deems
Well, I feel that the most efficient way of learning something is to be properly motivated. The approach of turning learning into a game is applied to young
Message 2 of 6
, Sep 2, 2000
Well, I feel that the most efficient way of learning something is to be
properly motivated. The approach of turning learning into a game is applied
to young children with flashcards and "game shows" in school. Looking back
on my high school days, I think it worked pretty well. I don't think we
should place more emphasis on the learning of math rather than more
imaginative areas such as fantasy, I think they must be combined. I happen
to be excellent at math (which I do not at all consider to be unimaginative
by the way), only because I like it so much. Geometry came easily to me, I
was a natural at it. Algebra was not quite as easy for me, but I loved it
as well. Why? Because I looked further than the times tables, I was
looking not at math but at the most basic language of reality, because I was
discovering a perfect science that has always been there, because I was
gaining new insight into problem solving and common sense and discovering
the laws that govern the universe.
I think that strong math skills elevate one's ability to comprehend advanced
concepts, as well as creating a more complex general thinking structure, a
sort of unconscious identification between our common knowledge and
unconscious knowledge. By learning complex elements of math, it is almost
as if you are building default thought patterns and plotting paths for
problem solving in a way we could not understand. We already know math, we
do nightmarishly complicated calculus every time we catch a baseball, jog,
get out of bed...
I think the secret to getting our children to focuss on math is not to
force memorization, but to motivate their imaginations to explore the
universe and it's mysteries, to teach them how to look at life like an
adventure, to give them a thirst for the heart's desire, for it is in pusuit
of the heart's desire that one is lead to the beauty in life. A person who
enjoys life's adventure willingly will explore all that s/he finds to be
stimulating and has only to see what makes anything interesting. For this
one must naturally look for what is stimulating in all that surrounds, and
that is why we have fantasy. Among other things, it shows us how to look at
the wold from the heart, and it is from the heart- only- that we might see
what the world trully has to offer.
And then one day I began reading about chaos theory and I was hopelessly
obsessed.... |
This post originally appeared on The Aperiodical. We republish it here with permission.
Being a mathematician, I often get asked if I'm good at calculating tips. I'm not. In fact, mathematicians study lots of other things besides numbers. As most people know, if they stop to think about it, one of the other things mathematicians study is shapes. Some of us are especially interested in the symmetries of those shapes, and a few of us are interested in both numbers and symmetries. The recent announcement of "Pariah Moonshine" has been one of the most exciting developments in the relationship between numbers and symmetries in quite some time. In this blog post I hope to explain the "Pariah" part, which deals mostly with symmetries. The "Moonshine", which connects the symmetries to numbers, will follow in the next post.
What is a symmetry?
First I want to give a little more detail about what I mean by the symmetries of shapes. If you have a square made out of paper, there are basically eight ways you can pick it up, turn it, and put it down in exactly the same place. You can rotate it 90 degrees clockwise or counterclockwise. You can rotate it 180 degrees. You can turn it over, so the front becomes the back and vice versa. You can turn in over and then rotate it 90 degrees either way, or 180 degrees. And you can rotate it 360 degrees, which basically does nothing. We call these the eight symmetries of the square, and they are shown in Figure 1.
Figure 1. The square can be rotated into four different positions, without or without being flipped over, for eight symmetries total.
If you have an equilateral triangle, there are six symmetries. If you have a pentagon, there are ten. If you have a pinwheel with four arms, there are only four symmetries, as shown in Figure 2, because now you can rotate it but if you turn it over it looks different. If you have a pinwheel with six arms, there are six ways. If you have a cube, there are 24 if the cube is solid, as shown in Figure 3. If the cube is just a wire frame and you are allowed to turn it inside out, then you get 24 more, for a total of 48.
Figure 2. The pinwheel can be rotated but not flipped, for four symmetries total.
Figure 3. The cube can be rotated along three different axes, resulting in 24 different symmetries.
These symmetries don't just come with a count, they also come with a structure. If you turn a square over and then rotate it 90 degrees, it's not the same thing as if you rotate it first and then flip it over. (Try it and see.) In this way, symmetries of shapes are like the permutations I discuss in Chapter 3 of my book, The Mathematics of Secrets: you can take products, which obey some of the same rules as products of numbers but not all of them. These sets of symmetries, which their structures, are called groups.
Groups are sets of symmetries with structure
Some sets of symmetries can be placed inside other sets. For example, the symmetries of the four-armed pinwheel are the same as the four rotations in the symmetries of the square. We say the symmetries of the pinwheel are a subgroup of the symmetries of the square. Likewise, the symmetries of the square are a subgroup of the symmetries of the solid cube, if you allow yourself to turn the cube over but not tip it 90 degrees, as shown in Figure 4.
Figure 4. The symmetries of the square are contained inside the symmetries of the cube if you are allowed to rotate and flip the cube but not tip it 90 degrees.
In some cases, ignoring a subgroup of the symmetries of a shape gets us another group, which we call the quotient group. If you ignore the subgroup of how the square is rotated, you get the quotient group where the square is flipped over or not, and that's it. Those are the same as the symmetries of the capital letter A, so the quotient group is really a group. In other cases, for technical reasons, you can't get a quotient group. If you ignore the symmetries of a square inside the symmetries of a cube, what's left turns out not to be the symmetries of any shape.
You can always ignore all the symmetries of a shape and get just the do nothing (or trivial) symmetry, which is the symmetries of the capital letter P, in the quotient group. And you can always ignore none of the nontrivial symmetries, and get all of the original symmetries still in the quotient group. If these are the only two possible quotient groups, we say that the group is simple. The group of symmetries of a pinwheel with a prime number of arms is simple. So is the group of symmetries of a solid icosahedron, like a twenty-sided die in Dungeons and Dragons. The group of symmetries of a square is not simple, because of the subgroup of rotations. The group of symmetries of a solid cube is not simple, not because of the symmetries of the square, but because of the smaller subgroup of symmetries of a square with a line through it, as shown in Figures 5 and 6. The quotient group there is the same as the symmetries of the equilateral triangle created by cutting diagonally through a cube near a corner.
Figure 5. The symmetries of a square with line through it. We can turn the square 180 degrees and/or flip it, but not rotate it 90 degrees, so there are four.
Figure 6. The symmetries of the square with a line through it inside of the symmetries of the cube.
Categorizing the Pariah groups
As early as 1892, Otto Hölder asked if we could categorize all of the finite simple groups. (There are also shapes, like the circle, which have an infinite number of symmetries. We won't worry about them now.) It wasn't until 1972 that Daniel Gorenstein made a concrete proposal for how to make a complete categorization, and the project wasn't finished until 2002, producing along the way thousands of pages of proofs. The end result was that almost all of the finite simple groups fell into a few infinitely large categories: the cyclic groups, which are the groups of symmetries of pinwheels with a prime number of arms, the alternating groups, which are the groups of symmetries of solid hypertetrahedra in 5 or more dimensions, and the "groups of Lie type", which are related to matrix multiplication over finite fields and describe certain symmetries of objects known as finite projective planes and finite projective spaces. (Finite fields are used in the AES cipher and I talk about them in Section 4.5 of The Mathematics of Secrets.)
Even before 1892, a few finite simple groups were discovered that didn't seem to fit into any of these categories. Eventually it was proved that there were 26 "sporadic" groups, which didn't fit into any of the categories and didn't describe the symmetries of anything obvious — basically, you had to construct the shape to fit the group of symmetries that you knew existed, instead of starting with the shape and finding the symmetries. The smallest of the sporadic groups has 7920 symmetries in it, and the largest, known as the Monster, has over 800 sexdecillion symmetries. (That's an 8 with 53 zeros after it!) Nineteen of the other sporadic groups turn out to be subgroups or quotient groups of subgroups of the Monster. These 20 became known as the Happy Family. The other 6 sporadic groups became known as the 'Pariahs'.
The shape that was constructed to fit the Monster lives in 196883-dimensional space. In the late 1970's a mathematician named John McKay noticed the number 196884 turning up in a different area of mathematics. It appeared as part of a function used in number theory, the study of the properties of whole numbers. Was there a connection between the Monster and number theory? Or was the idea of a connection just … moonshine?
Can we live without the idea of purpose? Should we even try to? Kant thought we were stuck with purpose, and even Darwin's theory of natural selection, which profoundly shook the idea, was unable to kill it. Indeed, teleological explanation—what Aristotle called understanding in terms of "final causes"—seems to be making a comeback today, as both religious proponents of intelligent design and some prominent secular philosophers argue that any explanation of life without the idea of purpose is missing something essential. In On Purpose, Michael Ruse explores the history of the idea of purpose in philosophical, religious, scientific, and historical thought, from ancient Greece to the present. Read on to learn more about the idea of "purpose," the long philosophical tradition around it, and how Charles Darwin fits in.
On Purpose? So what's with the smart-alecky title?
It was a friend of Dr. Johnson who said that he had tried to be a philosopher, but cheerfulness always kept breaking in. Actually, that is a little bit unfair to philosophers. Overall, we are quite a cheerful group, especially when we think that we might have been born sociologists or geographers. However, our sense of humor is a bit strained, usually—as in this case—involving weak puns and the like. My book is about a very distinctive form of understanding, when we do things in terms of the future and not the past.
In terms of the future? Why not call your book On Prediction?
I am not talking about prediction, forecasting what you think will happen, although that is involved. I am talking about when the future is brought in to explain things that are happening right now. Purposeful thinking is distinctive and interesting because normally when we try to explain things we do so in terms of the past or present. Why do you have a bandage on your thumb? Because I tried to hang the picture myself, instead of getting a grad student to do it. Purposeful thinking—involving what Aristotle called "final causes" and what since the eighteenth century has often been labeled "teleological" thinking—explains in terms of future events. Why are you studying rather than going to the ball game? Because I want to do well on the GRE exam and go to a good grad school.
Why is this interesting?
In the case of the bandaged thumb, you know that the hammer hit you rather than the nail. In the case of studying, you may decide that five to ten years of poverty and peonage followed by no job is not worth it, and you should decide to do something worthwhile like becoming a stockbroker or university administrator. We call this "the problem of the missing goal object." Going to grad school never occurred, but it still makes sense to say that you are studying now in order to go to grad school.
Is this something that you thought up, or is it something with a history?
Oh my, does it ever have a history. One of the great things about my book, if I might show my usual level of modesty, is that I show the whole problem of purpose is one with deep roots in the history of philosophy, starting with Plato and Aristotle, and coming right up to the modern era, particularly the thinking of Immanuel Kant. In fact, I argue that it is these three very great philosophers who set the terms of the discussion—Plato analyses things in terms of consciousness, Aristotle in terms of principles of ordering whatever that might mean, and Kant opts for some kind of heuristic approach.
If these thinkers have done the spadework, what's left for you?
I argue that the truth about purposeful thinking could not be truly discovered until Charles Darwin in his Origin of Species (1859) had proposed his theory of evolution through natural selection. With that, we could start to understand forward-looking thinking about humans—why is he studying on such a beautiful day? He wants to go to grad school. About plants and animals—why does the stegosaurus have those funny-looking plates down its back? To control its temperature. And why we don't use such thinking about inanimate objects? Why don't we worry about the purpose of the moon? Perhaps we should. It really does exist in order to light the way home for drunken philosophers.
Why is it such a big deal to bring up Darwin and his theory of evolution? Surely, the kind of people who will read your book will have accepted the theory long ago?
Interestingly, no! The main opposition to evolutionary thinking comes from the extreme ends of the spectrum: evangelical Christians known as Creationists—biblical literalists—and from professional philosophers. There are days when it seems that the higher up the greasy pole you have climbed, the more likely you are to deny Darwinism and be a bit iffy about evolution generally. This started just about as soon as the Origin appeared, and the sinister anti-evolutionary effect of Bertrand Russell and G. E. Moore and above all Ludwig Wittgenstein is felt to this day. A major reason for writing my book was to take seriously Thomas Henry Huxley's quip that we are modified monkeys rather than modified mud, and that matters.
Given that you are a recent recipient of the Bertrand Russell Society's "Man of the Year" Award, aren't you being a bit ungracious?
I have huge respect for Russell. He was a god in my family when, in the 1940s and 50s, I was growing up in England. One of my greatest thrills was to have been part of the crowd in 1961 in Trafalgar Square listening to him declaim against nuclear weapons. But I think he was wrong about the significance of Darwin for philosophy and I think I am showing him great respect in arguing against him. I feel the same way about those who argue against me. My proudest boast is that I am now being refuted in journals that would never accept anything by me.
One of the big problems normal people today have about philosophy is that it seems so irrelevant. Initiates arguing about angels on the heads of pins? Why shouldn't we say the same about your book?
Three reasons. First, my style and approach. It is true that most philosophy produced by Anglophone philosophers today is narrow and boring. Reading analytic philosophy is like watching paint dry and proudly so. Against this, on the one hand I am more a historian of ideas using the past to illuminate the present. That is what being an evolutionist is all about. Spending time with mega-minds like Plato and Aristotle and Kant is in itself tremendously exciting. On the other hand, I have over fifty years of teaching experience, at the undergraduate level almost always at the first- and second-year level. I know that if you are not interesting, you are going to lose your audience. The trick is to be interesting and non-trivial.
Second, I don't say that my book is the most important of the past hundred-plus years, but my topic is the most important. Evolution matters, folks, it really does. It is indeed scary to think that we are just the product of a random process of change and not the favored product of a Good God—made in His image. Even atheists get the collywobbles, or at least they should. It is true all the same. Fifty years ago, the geneticist and Nobel laureate Hermann J. Muller said that a hundred years without Darwin is enough. That is still true. Amen.
Third, deliberately, I have made this book very personal. At the end, I talk about purpose in my own life. Why, even though I am a non-believer, I have been able to find meaning in what I think and do. This ranges from my love of my wife Lizzie and how with dedication and humor we share the challenges of having children—not to mention our love of dogs, most recent addition to the family, Nutmeg a whippet—through cooking on Saturday afternoons while listening to radio broadcasts of Metropolitan Opera matinees, to reading Pickwick Papers yet one more time. I suspect that many of my fellow philosophers will find this all rather embarrassing. I mean it to be. Philosophy matters. My first-ever class on the subject started with Descartes' Meditations. Fifteen minutes into the class, I knew that this was what I was going to do for the rest of my life. Nearly sixty years later I am still at it and surely this interview tells you that I love it, every moment.
So, why should we read your book?
Because it really does square the circle. It is cheerful and philosophical. It is on a hugely important topic and there are some good jokes. I am particularly proud of one I make about Darwin Day, the celebration by New Atheists, and their groupies of the birthday of Charles Darwin.
Which is?
Oh, hell no. I am not going to tell you. Go out and buy the book. And while you are at it, buy one for your mum and dad and one each for your siblings and multi-copies for your students and…. I am seventy-seven years old. I need a bestseller so I can retire. You need a bestseller so I can retire.
Michael Ruse is the Lucyle T. Werkmeister Professor of Philosophy and Director of the Program in the History and Philosophy of Science at Florida State University. He has written or edited more than fifty books, including Darwinism as Religion, The Philosophy of Human Evolution, and The Darwinian Revolution.
While writing Unsolved! The History and Mystery of the World's Greatest Ciphers from Ancient Egypt to Online Secret Societies, it soon became clear to me that I'd never finish if I kept stopping to try to solve the ciphers I was covering. It was hard to resist, but I simply couldn't afford to spend months hammering away at each of the ciphers. There were simply too many of them. If I was to have any chance of meeting my deadline, I had to content myself with merely making suggestions as to how attacks could be carried out. My hope was that the book's readers would be inspired to actually make the attacks. However, the situation changed dramatically when the book was done.
I was approached by the production company Karga Seven Pictures to join a team tasked with hunting the still unidentified serial killer who called himself the Zodiac. In the late 1960s and early 70s, the Zodiac killed at least five people and terrorized entire cities in southern California with threatening letters mailed to area newspapers. Some of these letters included unsolved ciphers. I made speculations about these ciphers in my book, but made no serious attempt at cracking them. With the book behind me, and its deadline no longer a problem, would I like to join a code team to see if we could find solutions where all others had failed? The team would be working closely with a pair of crack detectives, Sal LaBarbera and Ken Mains, so that any leads that developed could be investigated immediately. Was I willing to take on the challenge of a very cold case? Whatever the result was, it would be no secret, for our efforts would be aired as a History channel mini-series. Was I up for it? Short answer: Hell yeah!
The final code team included two researchers I had corresponded with when working on my book, Kevin Knight (University of Southern California, Information Sciences Institute) and David Oranchak (software developer and creator of Zodiac Killer Ciphers. The other members were Ryan Garlick (University of North Texas, Computer Science) and Sujith Ravi (Google software engineer).
My lips are sealed as to what happened (why ruin the suspense?), but the show premieres Tuesday November 14, 2017 at 10pm EST. It's titled "The Hunt for the Zodiac Killer." All I'll say for now is that it was a rollercoaster ride. For those of you who would like to see how the story began for me, Princeton University Press is making the chapter of my book on the Zodiac killer freely available for the duration of the mini-series. It provides an excellent background for those who wish to follow the TV team's progress.
If you find yourself inspired by the show, you can turn to other chapters of the book for more unsolved "killer ciphers," as well challenges arising from nonviolent contexts. It was always my hope that readers would resolve some of these mysteries and I'm more confident than ever that it can be done!
Craig P. Bauer is professor of mathematics at York College of Pennsylvania. He is editor in chief of the journal Cryptologia, has served as a scholar in residence at the NSA's Center for Cryptologic History, and is the author of Secret History: The Story of Cryptology. He lives in York, Pennsylvania.
It's tough to argue with the idea that passion is an admirable aspect of the human condition. Passionate people are engaged in life; they really care about their values and causes and being true to them. However, a big minefield of passion is when people use it to excuse or explain away unseemly behavior. We saw this during the summer of 2017 in how the White House press secretary, Sarah Huckabee Sanders, responded to the infamous expletive-laced attack of Anthony Scaramucci on his then fellow members of the Trumpteam, Steve Bannon and Reince Priebus. According to The New York Times, (July 27, 2017), "Ms. Sanders said mildly that Mr. Scaramucci was simply expressing strong feelings, and that his statement made clear that 'he's a passionate guy and sometimes he lets that passion get the better of him.' " Whereas Ms. Sanders acknowledged that Mr. Scaramucci behaved badly (his passion got the better of him), her meta-message is that it was no big deal, as implied by the words "mildly" and "simply" in the quote above.
The passion plea is by no means limited to the world of politics. Executives who are seen as emotionally rough around the edges by their co-workers often defend their behavior with statements like, "I'm just being passionate," or "I am not afraid to tell it like it is," or, "My problem is that I care too much."
The passion plea distorts reality by glossing over the distinction between what is said and how it is said. Executives who deliver negative feedback in a harsh tone are not just being passionate. Even when the content of the negative feedback is factual, harsh tones convey additional messages – notably a lack of dignity and respect. Almost always, there are ways to send the same strong messages or deliver the same powerful feedback in ways that do not convey a lack of dignity and respect. For instance, Mr. Scaramucci could have said something like, "Let me be as clear as possible: I have strong disagreements with Steve Bannon and Reince Priebus." It may have been less newsworthy, but it could have gotten the same message across. Arguably, Mr. Scaramucci's 11-day tenure as White House director of communications would have been longer had he not been so "passionate" and instead used more diplomatic language.
Similarly, executives that I coach rarely disagree when it is made evident that they could have sent the same strong negative feedback in ways that would have been easier for their co-workers to digest. Indeed, this is the essence of constructive criticism, which typically seeks to change the behavior of the person on the receiving end. Rarely are managers accused of coming on "too strong" if they deliver negative feedback in the right ways. For example, instead of saying something about people's traits or characters (e.g., "You aren't reliable") it would be far better to provide feedback with reference to specific behavior (e.g., "You do not turn in your work on time"). People usually are more willing and able to respond to negative feedback about what they do rather than who they are. Adding a problem-solving approach is helpful as well, such as, "Some weeks you can be counted on to do a good job whereas other weeks not nearly as much. Why do you think that is happening, and what can we do together to ensure greater consistency in your performance?" Moreover, the feedback has to be imparted in a reasonable tone of voice, and in a context in which people on the receiving end are willing and able to take it in. For instance, one of my rules in discussing with students why they didn't do well on an assignment is that we not talk immediately after they received the unwanted news. It is far better to have a cooling-off period in which defensiveness goes down and open-mindedness goes up.
If our goal is to alienate people or draw negative attention to ourselves then we should be strong and hard-driving, even passionate, in what we say as well as crude and inappropriate in how we say it. However, if we want to be a force for meaningful change or a positive role model, it is well within our grasp to be just as strong and hard-driving in what we say while being respectful and dignified in how we say it.
Joel Brockner is the Phillip Hettleman Professor of Business at Columbia Business School.
The Prizes celebrate outstanding science writing and illustration for children and young adults and are meant to encourage the writing and publishing of high-quality science books for all ages. AAAS believes that, through good science books, this generation, and the next, will have a better understanding and appreciation of science.
As of late 2015, we have a new way of probing the cosmos: gravitational radiation. Thanks to LIGO (the Laser Interferometer Gravitational-wave Observatory) and its new sibling Virgo (a similar interferometer in Italy), we can now "hear" the thumps and chirps of colliding massive objects in the universe. Not for nothing has this soundtrack been described by LIGO scientists as "the music of the cosmos." This music is at a frequency easily discerned by human hearing, from somewhat under a hundred hertz to several hundred hertz. Moreover, gravitational radiation, like sound, is wholly different from light. It is possible for heavy dark objects like black holes to produce mighty gravitational thumps without at the same time emitting any significant amount of light. Indeed, the first observations of gravitational waves came from black hole merger events whose total power briefly exceeded the light from all stars in the known universe. But we didn't observe any light from these events at all, because almost all their power went into gravitational radiation.
In August 2017, LIGO and Virgo observed a collision of neutron stars which did produce observable light, notably in the form of gamma rays. Think of it as cosmic thunder and lightning, where the thunder is the gravitational waves and the lightning is the gamma rays. When we see a flash of ordinary lightning, we can count a few seconds until we hear the thunder. Knowing that sound travels one mile in about five seconds, we can reckon how distant the event is. The reason this method works is that light travels much faster than sound, so we can think of the transmission of light as instantaneous for purposes of our estimate.
Things are very different for the neutron star collision, in that the event took place about 130 million light years away, but the thunder and lightning arrived on earth pretty much simultaneously. To be precise, the thunder was first: LIGO and Virgo heard a basso rumble rising to a characteristic "whoop," and just 1.7 seconds later, the Fermi and INTEGRAL experiments observed gamma ray bursts from a source whose location was consistent with the LIGO and Virgo observations. The production of gamma rays from merging neutron stars is not a simple process, so it's not clear to me whether we can pin that 1.7 seconds down as a delay precisely due to the astrophysical production mechanisms; but at least we can say with some confidence that the propagation time of light and gravity waves are the same to within a few seconds over 130 million light years. From a certain point of view, that amounts to one of the most precise measurements in physics: the ratio of the speed of light to the speed of gravity equals 1, correct to about 14 decimal places or better.
The whole story adds up much more easily when we remember that gravitational waves are not sound at all. In fact, they're nothing like ordinary sound, which is a longitudinal wave in air, where individual air molecules are swept forward and backward just a little as the sound waves pass them by. Gravitational waves instead involve transverse disturbances of spacetime, where space is stretched in one direction and squeezed in another—but both of those stretch-squeeze directions are at right angles to the direction of the wave. Light has a similar transverse quality: It is made up of electric and magnetic fields, again in directions that are at right angles to the direction in which the light travels. It turns out that a deep principle underlying both Maxwell's electromagnetism and Einstein's general relativity forces light and gravitational waves to be transverse. This principle is called gauge symmetry, and it also guarantees that photons and gravitons are massless, which implies in turn that they travel at the same speed regardless of wavelength.
It's possible to have transverse sound waves: For instance, shearing waves in crystals are a form of sound. They typically travel at a different speed from longitudinal sound waves. No principle of gauge symmetry forbids longitudinal sound waves, and indeed they can be directly observed, along with their transverse cousins, in ordinary materials like metals. The gauge symmetries that forbid longitudinal light waves and longitudinal gravity waves are abstract, but a useful first cut at the idea is that there is extra information in electromagnetism and in gravity, kind of like an error-correcting code. A much more modest form of symmetry is enough to characterize the behavior of ordinary sound waves: It suffices to note that air (at macroscopic scales) is a uniform medium, so that nothing changes in a volume of air if we displace all of it by a constant distance.
In short, Maxwell's and Einstein's theories have a feeling of being overbuilt to guarantee a constant speed of propagation. And they cannot coexist peacefully as theories unless these speeds are identical. As we continue Einstein's hunt for a unified theory combining electromagnetism and gravity, this highly symmetrical, overbuilt quality is one of our biggest clues.
The transverse nature of gravitational waves is immediately relevant to the latest LIGO / Virgo detection. It is responsible for the existence of blind spots in each of the three detectors (LIGO Hanford, LIGO Livingston, and Virgo). It seems like blind spots would be bad, but they actually turned out to be pretty convenient: The signal at Virgo was relatively weak, indicating that the direction of the source was close to one of its blind spots. This helped localize the event, and localizing the event helped astronomers home in on it with telescopes. Gamma rays were just the first non-gravitational signal observed: the subsequent light-show from the death throes of the merging neutron stars promises to challenge and improve our understanding of the complex astrophysical processes involved. And the combination of gravitational and electromagnetic observations will surely be a driver of new discoveries in years and decades to come.
Curves are seductive. These smooth, organic lines and surfaces—like those of the human body—appeal to us in an instinctive, visceral way that straight lines or the perfect shapes of classical geometry never could. In this large-format book, lavishly illustrated in color throughout, Allan McRobie takes the reader on an alluring exploration of the beautiful curves that shape our world—from our bodies to Salvador Dalí's paintings and the space-time fabric of the universe itself. A unique introduction to the language of beautiful curves, this book may change the way you see the world.
Allan McRobie is a Reader in the Engineering Department at the University of Cambridge, where he teaches stability theory and structural engineering. He previously worked as an engineer in Australia, designing bridges and towers. The Little Book of Black Holes by Steven S. Gubser and Frans Pretorius takes readers deep into the mysterious heart of the subject, offering rare clarity of insight into the physics that makes black holes simple yet destructive manifestations of geometric destiny. Read on to learn a bit more about black holes and what inspired the authors to write this book.
Your book tells the story of black holes from a physics perspective. What are black holes, really? What's inside?
Black holes are regions of spacetime from which nothing can escape, not even light. In our book, we try to live up to our title by getting quickly to the heart of the subject, explaining in non-technical terms what black holes are and how we use Einstein's theory of relativity to understand them. What's inside black holes is a great mystery. Taken at face value, general relativity says spacetime inside a black hole collapses in on itself, so violently that singularities form. We need something more than Einstein's theory of relativity to understand what these singularities mean. Hawking showed that quantum effects cause black holes to radiate very faintly. That radiation is linked with quantum fluctuations inside the black hole. But it's a matter of ongoing debate whether these fluctuations are a key to resolving the puzzle of the singularity, or whether some more drastic theory is needed.
How sure are we that black holes exist?
A lot more certain than we were a few years ago. In September 2015, the LIGO experiment detected gravitational waves from the collision of two black holes, each one about thirty times the mass of the sun. Everything about that detection fit our expectations based on Einstein's theories, so it's hard to escape the conclusion that there really are black holes out there. In fact, before the LIGO detection we were already pretty sure that black holes exist. Matter swirling around gigantic black holes at the core of distant galaxies form the brightest objects in the Universe. They're called quasars, and the only reason they're dim in our sight is that they're so far away, literally across the Universe. Similar effects around smaller black holes generate X-rays that we can detect relatively nearby, mere thousands of light years away from us. And we have good evidence that there is a large black hole at the center of the Milky Way.
Can you talk a bit about the formation of black holes?
Black holes with mass comparable to the sun can form when big stars run out of fuel and collapse in on themselves. Ordinarily, gravity is the weakest force, but when too much matter comes together, no force conceivable can hold it up against the pull of gravity. In a sense, even spacetime collapses when a black hole forms, and the result is a black hole geometry: an endless inward cascade of nothing into nothing. All the pyrotechnics that we see in distant quasars and some nearby X-ray sources comes from matter rubbing against itself as it follows this inward cascade.
How have black holes become so interesting to non-specialists? How have they been glorified in popular culture?
There's so much poetry in black hole physics. Black hole horizons are where time stands still—literally! Black holes are the darkest things that exist in Nature, formed from the ultimate ashes of used-up stars. But they create brilliant light in the process of devouring yet more matter. The LIGO detection was based on a black hole collision that shook the Universe, with a peak power greater than all stars combined; yet we wouldn't even have noticed it here on earth without the most exquisitely sensitive detector of spacetime distortions ever built. Strangest of all, when stripped of surrounding matter, black holes are nothing but empty space. Their emptiness is actually what makes them easy to understand mathematically. Only deep inside the horizon does the emptiness end in a terrible, singular core (we think). Horrendous as this sounds, black holes could also be doorways into wormholes connecting distant parts of the Universe. But before packing our bags for a trip from Deep Space Nine to the Gamma Quadrant, we've got to read the fine print: as far as we know, it's impossible to make a traversable wormhole.
What inspired you to write this book? Was there a point in life where your interest in this topic was piqued?
We both feel extremely fortunate to have had great mentors, including Igor Klebanov, Curt Callan, Werner Israel, Matthew Choptuik, and Kip Thorne who gave us a lot of insight into black holes and general relativity. And we owe a big shout-out to our editor, Ingrid Gnerlich, who suggested that we write this book.
Steven S. Gubser is professor of physics at Princeton University and the author of The Little Book of String Theory. Frans Pretorius is professor of physics at Princeton.
This first-year, graduate-level text and reference book covers the fundamental concepts and twenty-first-century applications of six major areas of classical physics that every masters- or PhD-level physicist should be exposed to, but often isn't: statistical physics, optics (waves of all sorts), elastodynamics, fluid mechanics, plasma physics, and special and general relativity and cosmology. Growing out of a full-year course that the eminent researchers Kip S. Thorne, winner of the 2017 Nobel Prize in Physics, and Roger D. Blandford taught at Caltech for almost three decades, this book is designed to broaden the training of physicists. Its six main topical sections are also designed so they can be used in separate courses, and the book provides an invaluable reference for researchers.
This book emerged from a course you both began teaching nearly 4 decades ago. What drove you to create the course, and ultimately to write this book?
KST: We were unhappy with the narrowness of physics graduate education in the United States. We believed that every masters-level or PhD physicist should be familiar with the basic concepts of all the major branches of classical physics and should have some experience applying them to real world phenomena. But there was no obvious route to achieve this, so we created our course.
RDB: Of course we had much encouragement from colleagues who helped us teach it and students who gave us invaluable feedback on the content.
The title indicates that the book is a "modern" approach to classical physics (which emphasizes physical phenomena at macroscopic scales). What specifically is "modern" in your book's approach to this subject?
KST: Classical-physics ideas and tools are used extensively today in research areas as diverse as astrophysics, high-precision experimental physics, optical physics, biophysics, controlled fusion, aerodynamics, computer simulations, etc. Our book draws applications from all these modern topics and many more. Also, these modern applications have led to powerful new viewpoints on the fundamental concepts of classical physics, viewpoints that we elucidate—for example, quantum mechanical viewpoints and language for purely classical mode-mode coupling in nonlinear optics and in nonlinear plasma physics.
Why do you feel that it is so important for readers to become more familiar with classical physics, beyond what they may have been introduced to already?
KST: In their undergraduate and graduate level education, most physicists have been exposed to classical mechanics, electromagnetic theory, elementary thermodynamics, and little classical physics beyond this. But in their subsequent careers, most physicists discover that they need an understanding of other areas of classical physics (and this book is a vehicle for that).
In many cases they may not even be aware of their need. They encounter problems in their research or in R&D where powerful solutions could be imported from other areas of classical physics, if only they were aware of those other areas. An example from my career: in the 1970s, when trying to understand recoil of a binary star as it emits gravitational waves, I, like many relativity physicists before me, got terribly confused. Then my graduate student, Bill Burke—who was more broadly educated than I—said "we can resolve the confusion by adopting techniques that are used to analyze boundary layers in fluid flows around bodies with complicated shapes." Those techniques (matched asymptotic expansions), indeed, did the job, and through Bill, they were imported from fluid mechanics into relativity.
RDB: Yes. To give a second example, when I was thinking about ways to accelerate cosmic rays, I recalled graduate lectures on stellar dynamics and found just the tools I needed.
You also mention in the book that geometry is a deep theme and important connector of ideas. Could you explain your perspective, and how geometry is used thematically throughout the book?
KST: The essential point is that, although coordinates are a powerful, and sometimes essential, tool in many calculations, the fundamental laws of physics can be expressed without the aid of coordinates; and, indeed, their coordinate-free expressions are generally elegant and exceedingly powerful. By learning to think about the laws in coordinate-free (geometric) language, a physicist acquires great power. For example, when one searches for new physical laws, requiring that they be geometric (coordinate-free) constrains enormously the forms that they may take. And in many practical computations (for example, of the relativistic Doppler shift), a geometric route to the solution can be faster and much more insightful than one that uses coordinates. Our book is infused with this.
RDB: We are especially keen on presenting these fundamental laws in a manner which makes explicit the geometrically formulated conservation laws for mass, momentum, energy, etc. It turns out that this is often a good starting point when one wants to solve these equations numerically. But ultimately, a coordinate system must be introduced to execute the calculations and interpret the output.
One of the areas of application that you cover in the book is cosmology, an area of research that has undergone a revolution over the past few decades. What are some of the most transformative discoveries in the field's recent history? How does classical physics serve to underpin our modern understanding of how the universe formed and is evolving? What are some of the mysteries that continue to challenge scientists in the field of cosmology?
RDB: There have indeed been great strides in understanding the large scale structure and evolution of the universe, and there is good observational support for a comparatively simple description. Cosmologists have found that 26 percent of the energy density in the contemporary, smoothed-out universe is in the form of "dark matter," which only seems to interact through its gravity. Meanwhile, 69 percent is associated with a "cosmological constant," as first introduced by Einstein and which causes the universe to accelerate. The remaining five percent is the normal baryonic matter which we once thought accounted for essentially all of the universe. The actual structure that we observe appears to be derived from almost scale-free statistically simple, random fluctuations just as expected from an early time known as inflation. Fleshing out the details of this description is almost entirely an exercise in classical physics. Even if this description is validated by future observations, much remains to be understood, including the nature of dark matter and the cosmological constant, what fixes the normal matter density, and the great metaphysical question of what lies beyond the spacetime neighborhood that we can observe directly.
KST: Remarkably, in fleshing out the details in the last chapter of our book, we utilize classical-physics concepts and results from every one of the other chapters. ALL of classical physics feeds into cosmology!
The revolution in cosmology that you describe depends upon many very detailed observations using telescopes operating throughout the entire electromagnetic spectrum and beyond. How do you deal with this in the book?
RDB: We make no attempt to describe the rich observational and experimental evidence, referring the reader to many excellent texts on cosmology that describe these in detail. However, we do describe some of the principles that underlie the design and operation of the radio and optical telescopes that bring us cosmological data.
There is has also been a lot of excitement regarding the recent observation by LIGO of gravitational waves caused by merging black holes. How is this subject covered in the book, and how, briefly, are some of the concepts of classical physics elucidated in your description of this cutting-edge research area?
KST: LIGO's gravitational wave detectors rely on an amazingly wide range of classical physics concepts and tools, so time and again we draw on LIGO for illustrations. The theory of random processes, spectral densities, the fluctuation-dissipation theorem, the Fokker-Planck equation; shot noise, thermal noise, thermoelastic noise, optimal filters for extracting weak signals from noise; paraxial optics, Gaussian beams, the theory of coherence, squeezed light, interferometry, laser physics; the interaction of gravitational waves with light and with matter; the subtle issue of the conservation or non conservation of energy in general relativity—all these and more are illustrated by LIGO in our book.
What are some of the classical physics phenomena in every day life that you are surprised more people do not fully understand—whether they are lay people, students, or scientists?
KST: Does water going down a drain really have a strong preference for clockwise in the northern hemisphere and counterclockwise in the south? How strong? What happens as you cross the equator? How are ocean waves produced? Why do stars twinkle in the night sky, and why doesn't Jupiter twinkle? How does a hologram work? How much can solid objects be stretched before they break, and why are there such huge differences from one type of solid (for example thin wire) to another (a rubber band)?
RDB: I agree and have to add that I am regularly humbled by some every day phenomenon that I cannot explain or for which I have carried around for years a fallacious explanation. There is, rightly, a lot of focus right now on climate change, energy, hurricanes, earthquakes, and so on. We hear about them every day. We physicists need to shore up our understanding and do a better job of communicating this.
Do you believe that some of your intended readers might be surprised to discover the deep relevance of classical physics to certain subject areas?
KST: In subjects that physicists think of as purely quantum, classical ideas and classical computational techniques can often be powerful. Condensed matter physics is an excellent example—and accordingly, our book includes a huge number of condensed-matter topics. Examples are Bose-Einstein condensates, the van der Waals gas, and the Ising model for ferromagnetism.
RDB: Conversely, quantum mechanical techniques are often used to simplify purely classical problems, for example in optics.
Writing a book is always an intellectual journey. In the preparation of this tremendously wide-ranging book, what were some of the most interesting things you learned along the way?
KST: How very rich and fascinating is the world of classical physics—far more so than we thought in 1980 when we embarked on this venture. And then there are the new inventions, discoveries, and phenomena that did not exist in 1980 but were so important or mind-boggling that we could not resist including them in our book. For example, optical-frequency combs and the phase-locked lasers that underlie them, Bose-Einstein condensates, the collapse of the World Trade Center buildings on 9/11/01, the discovery of gravitational waves and the techniques that made it possible, laser fusion, and our view of the universe at large.
Kip S. Thorne is the Feynman Professor Emeritus of Theoretical Physics at Caltech. His books include Gravitation and Black Holes and Time Warps. Roger D. Blandford is the Luke Blossom Professor of Physics and the founding director of the Kavli Institute of Particle Astrophysics and Cosmology at Stanford University. Both are members of the National Academy of Sciences.
Fill in the blank: Some people speak English, some speak French, and some speak ____. I doubt you said "math." Yet, as I will argue, the thought should have crossed your mind. And moreover, the fact that mathematics being a language likely never has, speaks volumes about how we think of math, and why we should start thinking of it—and teaching it—as a language.
To make my point, consider the following fundamental characteristics shared by most languages:
A set of words or symbols (the language's vocabulary)
A set of rules for how to use these words or symbols (the language's rules of grammar)
A set of rules for combining these words or symbols to make statements (the language's syntax)
Now think back to the math classes you have taken. I bet you will soon remember each of these characteristics present throughout your courses. (For instance, when you learned that 𝑎2 means 𝑎 × 𝑎, you were learning how to combine some of the symbols used in mathematics to make a statement—that the square of a number is the number multiplied by itself.) Indeed, viewed this way, every mathematics lesson can be thought of as a language lesson: new vocabulary, rules of grammar, or syntax is introduced; everyone then practices the new content; and the cycle repeats. By extension, every mathematics course can be thought of as a language course.
Now that I have you thinking of mathematics as a language, let me point out the many benefits of this new viewpoint. For one, this viewpoint helps dispel many myths about the subject. For instance, travel to any country and you will find a diverse set of people speaking that country's language. Some are smarter than others; some are men and some women; perhaps some are Latino and some Asian. Group them as you wish, they will all share the capacity to speak the same language. The same is true of mathematics. It is not a subject accessible only to people of certain intelligence, sex, or races; we all have the capacity to speak mathematics. And once we start thinking of the subject as a language, we will recognize that learning mathematics is like learning any other language: all you need are good teachers, and lots of practice. And while mastering a language is often the endpoint of the learning process, mastering the language that is mathematics will yield much larger dividends, including the ability to express yourself precisely, and the capacity to understand the Universe. As Alfred Adler put it: "
Mathematics is pure language – the language of science. It is unique among languages in its ability to provide precise expression for every thought or concept that can be formulated in its terms." Galileo—widely regarded the father of modern science—once wrote that Nature is a great book "written in the language of mathematics" (The Assayer, 1623). Centuries later, Einstein, after having discovered the equation for gravity using mathematics, echoed Galileo's sentiment, writing: "pure mathematics is, in its way, the poetry of logical ideas" (Obituary for Emmy Noether, 1935). Most of us today wouldn't use words like "language" and "poetry" to describe mathematics. Yet, as I will argue, we should. And moreover, we should start thinking of—and teaching—math as a language.
The Global Math Project has a goal of sharing the joys of mathematics to 1 million students around the world from October 10th through the 17th. As we watch the ever-increasing number of lives that will share in math's wonders, let's talk about counting, which is fundamental to reaching this goal.
Let's count. Suppose we have five objects, like the plus signs below. We easily enough count five of them.
You could put them in a hat and mix them up.
If you take them out, they might be jumbled but you'd still have five.
Easy enough! Jumbling can induce subtle complexities, even to something as basic as counting.
Counting to 14 isn't much more complicated than counting to five. Be careful as it depends what you are counting and how you jumble things! Verify there are 14 of Empire State Buildings in the picture below.
If you cut out the image along the straight black lines, you will have three pieces to a puzzle. If you interchange the left and right pieces on the top row, then you get the configuration below. How many buildings do you count now? Look at the puzzle carefully and see if you can determine how your count changed.
Can you spot any changes in the buildings in the first versus the second pictures? How we pick up an additional image is more easily seen if we reorder the buildings. So, let's take the 14 buildings and reorder them as seen below.
Swapping the pieces on the top row of the original puzzle has the same effect as shifting the top piece in the picture above. Such a shift creates the picture below. Notice how we pick up that additional building. Further, each image loses 1/14th of its total height.
Let's look at the original puzzle before and after the swap.
This type of puzzle is called a Dissection Puzzle. Our eyes can play tricks on us. We know 14 doesn't equal 15 so something else must be happening when a puzzle indicates that 14 = 15. Mathematics allows us to push through assumptions that can lead to illogical conclusions. Math can also take something that seems quite magical and turn it into something very logical — even something as fundamental as counting to 14.
Want to look at counting through another mathematical lens? A main topic of the Global Math Project will be exploding dots. Use a search engine to find videos of James Tanton introducing exploding dots. James is a main force behind the Global Math Project and quite simply oozes joy of mathematics. You'll also find resources at the Global Math Project web page. Take the time to look through the Global Math Project resources and watch James explain exploding dots, as the topic can be suitable from elementary to high school levels. You'll enjoy your time with James. You can count on it!
What hope do we have of solving ciphers that go back decades, centuries, or even all the way back to the ancient world? Well, we have a lot more hope than we did in the days before the Internet. Today's mathematicians form a global community that poses a much greater threat to unsolved problems, of every imaginable sort, than they have every faced before.
In my Princeton University Press book, Unsolved! The History and Mystery of the World's Greatest Ciphers from Ancient Egypt to Online Secret Societies, I collected scores of the most intriguing unsolved ciphers. It's a big book, in proper proportion to its title, and I believe many of the ciphers in it will fall to the onslaught the book welcomes from the world's codebreakers, both professionals and amateurs. Why am I making this prediction with such confidence? Well, I gave a few lectures based on material from the book, while I was still writing it, and the results bode well for the ciphers falling.
Here's what happened.
Early in the writing process, I was invited to give a lecture on unsolved ciphers at the United States Naval Academy. I was surprised, when I got there, by the presence of a video camera. I was asked if I was okay with the lecture being filmed and placed on YouTube. I said yes, but inside I was cursing myself for not having gotten a much needed haircut before the talk. Oh well. Despite my rough appearance, the lecture went well.[1] I surveyed some of the unsolved ciphers that I was aware of at the time, including one that had been put forth by a German colleague and friend of mine, Klaus Schmeh. It was a double transposition cipher that he had created himself to show how difficult it is to solve such ciphers. He had placed it in a book he had written on unsolved ciphers, a book which is unfortunately only available in German.[2] But to make the cipher as accessible as possible, he assured everyone that that particular bit of writing was in English.
VESINTNVONMWSFEWNOEALWRNRNCFITEEICRHCODEEA
HEACAEOHMYTONTDFIFMDANGTDRVAONRRTORMTDHE
OUALTHNFHHWHLESLIIAOETOUTOSCDNRITYEELSOANGP
VSHLRMUGTNUITASETNENASNNANRTTRHGUODAAARAO
EGHEESAODWIDEHUNNTFMUSISCDLEDTRNARTMOOIREEY
EIMINFELORWETDANEUTHEEEENENTHEOOEAUEAEAHUHI
CNCGDTUROUTNAEYLOEINRDHEENMEIAHREEDOLNNIRAR
PNVEAHEOAATGEFITWMYSOTHTHAANIUPTADLRSRSDNOT
GEOSRLAAAURPEETARMFEHIREAQEEOILSEHERAHAOTNT
RDEDRSDOOEGAEFPUOBENADRNLEIAFRHSASHSNAMRLT
UNNTPHIOERNESRHAMHIGTAETOHSENGFTRUANIPARTAOR
SIHOOAEUTRMERETIDALSDIRUAIEFHRHADRESEDNDOION
ITDRSTIEIRHARARRSETOIHOKETHRSRUAODTSCTTAFSTHCA
HTSYAOLONDNDWORIWHLENTHHMHTLCVROSTXVDRESDR
Figure 1. Klaus Schmeh's double transposition cipher challenge.
When the YouTube video went online, it was seen by an Israeli computer scientist, George Lasry, who became obsessed with it. He was not employed at the time, so he was able to devote a massive amount of time to seeking the solution to this cipher. As is natural for George, he attacked it with computer programs of his own design. He eventually found himself doing almost nothing other than working on the cipher. His persistence paid off and he found himself reading the solution.
I ended up being among the very first to see George's solution, not because I'm the one who introduced him to the challenge via the YouTube video, but because I'm the editor-in-chief of the international journal (it's owned by the British company Taylor and Francis) Cryptologia. This journal covers everything having to do with codes and ciphers, from cutting edge cryptosystems and attacks on them, to history, pedagogy, and more. Most of the papers that appear in it are written by men and women who live somewhere other than America and it was to this journal that George submitted a paper describing how he obtained his solution to Klaus's challenge.
George's solution looked great to me, but I sent it to Klaus to review, just to be sure. As expected, he was impressed by the paper and I queued it up to see print. The solution generated some media attention for George, which led to him being noticed by people at Google (an American company, of course). They approached him and, after he cleared the interviewing hurdles, offered him a position, which he accepted. I was very happy that George found the solution, but of course that left me with one less unsolved cipher to write about in my forthcoming book. Not a problem. As it turns out there are far more intriguing unsolved ciphers than can be fit in a single volume. One less won't make any difference.
Later on, but still before the book saw print, I delivered a similar lecture at the Charlotte International Cryptologic Symposium held in Charlotte, North Carolina. This time, unlike at the Naval Academy, Klaus Schmeh was in the audience.
One of the ciphers that I shared was fairly new to me. I had not spoken about it publicly prior to this event. It appeared on a tombstone in Ohio and seemed to be a Masonic cipher. It didn't look to be sophisticated, but it was very short and shorter ciphers are harder to break. Brent Morris, a 33rd degree Mason with whom I had discussed it, thought that it might be a listing of initials of offices, such as PM, PHP, PIM (Past Master, Past High Priest, Past Illustrious Master), that the deceased had held. This cipher was new to Klaus and he made note of it and later blogged about it. Some of his followers collaborated in an attempt to solve it and succeeded. Because I hadn't even devoted a full page to this cipher in my book, I left it in as a challenge for readers, but also added a link to the solution for those who want to see the solution right away.
Figure 2. A once mysterious tombstone just south of Metamora, Ohio.
So, what was my role in all of this? Getting the ball rolling, that's all. The work was done by Germans and an Israeli, but America and England benefited as well, as Google gained yet another highly intelligent and creative employee and a British owned journal received another great paper.
I look forward to hearing from other people from around the globe, as they dive into the challenges I've brought forth. The puzzles of the past don't stand a chance against the globally networked geniuses of today |
MATHEMATICAL NUMBERS
Fundamental Numbers
p =
3.1415926535 8979323846 2643383279 5028841971 6939937510
e = 2.7182818284 59045
Prime Numbers
Prime numbers are those numbers that have no factors other than
one and themselves. These numbers now play a very important
part in cryptography, and also in SETI (the Search for Extra-Terrestrial
Intelligence).
Fibonacci was an Italian mathematician who lived from about 1175 to 1240 AD.
His book Liber Abaci (the Book of the Abacus) helped introduce Hindu-Arabic numerals (0,1,2,3,...) to the western world. In it he also explained the number
sequence that is now named after him. These numbers, particularly the first
seven, have an intimate relationship to the natural world. |
Pi Song
Some people say,
3's the magic number,
But that's not gonna' roll here anymore...
I'd say they're irrational,
If their thinking's mathematical,
They'll know the magic's with the constant,
3.14
(Did you say 3.14?)
That's what I said,
Pi's the coolest constant,
24-7 satisfaction,
22/7th as a fraction,
Say pi's the coolest constant,
A non-repeating decimal,
Pi's the magic irrational...
Archimedes tried to solve it,
A Greek dude, a real smart guy,
Searched for ratios in parts of circles,
He tried to pin down pi!
ENIAC, determined more decimals, (actually 2037 digits)
Now there's millions in the mix,
I guess computing's a little quicker than it was in, 1946…
Some people say,
3's the magic number,
But that's not gonna' roll here anymore…
I'd say they're irrational,
If their thinking's mathematical,
They'll know the magic's with the constant,
3.14
Circumference of all circles,
Like round wheels on a car,
To find how far those wheels have rolled,
Just use 2πR!
The ratio of circumference,
To diameter's straight-straight line,
Pi pinpoints that ratio,
3.14...it's pi time! |
Elwes
Dr Richard Elwes is a writer, teacher and researcher in Mathematics and a Visiting Fellow at the University of Leeds. He contributes to New Scientist and Plus Magazine and publishes research on model theory. Dr Elwes is a committed populariser of mathematics which he regularly promotes at public lectures and on radio. He is the author of Mathematics 1001 published by Quercus.
The Maths Handbook
Richard Elwes
Mathematics
Richard Elwes
Authors:
Richard Elwes
Mathematics in 100 Key Breakthroughs presents a series of essays explaining the fundamentals of the most exciting and important maths concepts you really need to know. Richard Elwes profiles the important, groundbreaking and front-of-mind discoveries that have had a profound influence on our way of life and understanding. From the origins of counting over 35,000 years ago, right up to breakthroughs such as Wiles' Proof of Fermat's Last Theorem and Cook & Wolfram's Rule 110, it tells a story of discovery, invention, gradual progress and inspired leaps of the imagination.
Maths in 100 Key Breakthroughs
Richard Elwes
Chaotic Fishponds and Mirror Universes
Richard Elwes
Authors:
Richard Elwes
What can we learn from fish in a pond? How do social networks connect the world? How can artificial intelligences learn? Why would life be different in a mirror universe? Mathematics is everywhere, whether we are aware of it or not. Exploring the subject through 35 of its often odd and unexpected applications, this book provides an insight into the 'hidden wiring' that governs our world. From the astonishing theorems that control computers to the formulae behind stocks and shares, and from the foundations of the internet to the maths behind medical imaging, Chaotic Fishponds and Mirror Universes explains how mathematics determines every aspect of our lives - right down to the foundations of our bodies. |
Abstract mathematical models can help businesses optimise their manufacturing equipment, new study suggests.
Screenshot of one of the mathematical models used to optimise industrial machinery. (Photo: Joakim Juhl, 2011)
The equipment that businesses use in their production isn't always optimally designed in terms of effect and profit.
The design of this machinery is often more or less based on random results from decades of development and adaption of old ideas.
Mathematical models, it now appears, can help solve this problem.
"My research illustrates how mathematical modelling can contribute to the improvement of exisiting machinery in industrial production. It's a question of approaching science in a new way," says engineer Joakim Juhl, who recently defended his PhD thesis at the University of Aalborg, Denmark.
Reduced energy consumption
The thesis, 'Models in Action – realising abstractions', describes how mathematics was used to optimise the 'intelligence' of the industrial machinery at the Danish company Daka, which produces bone meal for animal feed from waste products from pig production.
This optimisation resulted in a reduction in the company's energy consumption.
However, the thesis represents research that's a bit different from the usual procedure in which vast data is collected, eventually resulting in a more or less clear-cut conclusion.
A different approach
Instead of collecting data, Juhl carried out a case study in which a team of experts described their attempts at optimising Daka's machinery using mathematical modelling.
The team, consisting of mathematicians, physicists and engineers were first asked to identify the physical principles 'behind' the equipment so that they could simulate its functions on a computer.
They were then asked to develop concrete optimisation solutions and then implement them into the machinery.
From theory to reality
Scientists shouldn't always seek to develop general and universal theories, even though that's usually what's regarded the 'purest' science. If you want your research to have any use in society, it's important that you also base your research on real-life problems.
Joakim Juhl
The scientists used some basic laws of nature as the starting point, since industrial machinery relies just as much on e.g. gravity as everything else in the world.
One problem with these so-callled laws of nature, however, is that they only describe idealised phenomena. In other words, these laws only apply under specific circumstances, for instance in laboratory tests where all confounding factors can be strictly controlled. But with real-world production machinery, things are a bit more complicated.
Model can fine-tune machinery
To overcome this hurdle, the experts incorporated several different physical theories into their mathematical models.
This enabled them to get as close as theoretically possible to how the machinery works in practice.
One of the main challenges at Daka is that the composition and the amount of pig remains are constantly changing.
The team managed to develop new 'regulations' of the control signal in the dryer – i.e. new ways of responding to the raw material that's fed into it. This made it possible for the machine to 'intelligently' regulate how the amount of pig remains it receives any given day.
The more contrete, the more useful
"Scientists shouldn't always seek to develop general and universal theories, even though that's usually what's regarded the 'purest' science," says Juhl. "If you want your research to have any use in society, it's important that you also base your research on real-life problems." |
The Mysteries of Benford's Law
Benford's Law, in the most elementary form of understanding, states that the number "1" transpires as the leading digit 30% of the time compared to higher digits such as 9 which occurs 5% of the time. This occurs for all kinds of data sets ranging from electricity bills, street addresses, stock prices, to even physical and mathematical constants. Yes, that's right, the physical and mathematical constants of the universe follow this mysterious law.
This graph showcases the percentage to be expected based on the results of Benford's Law. The number 1 representing 30% frequency rate as the leading digit to the number 9 representing a mere 4.6%. Even more strange, the percentage decrease in order from 1 to 9.
The law has been used in court cases to detect fraud based on the 'plausible' assumption that people who make up numbers evenly distribute them. It has been used to detect election/voting frauds, fraudulent macroeconomic data, and even scientific fraud.
This graph showcases Benford's law in relation to physical constants. The results are absolutely remarkable to me. Under what logic does this occur? How does a seemingly random measurement such as physical constants end up following the rules of the Benford's law? Well, the explanation (albeit, the best possible) is that many data sets seem to follow logarithmic scaling, thus allowing for a distribution of leading numbers as found in the law. To understand how this works, the following image showcases how numbers are distributed in logarithmic scaling.
Probability Distribution
Clearly, there may lie some logic behind this seemingly random occurrence in life. |
Finding number PI - Research Paper Example
Extract of sample Finding number PI
In modern times, however, with the advent of computers, the emphasis has shifted to the speed at which the value of Pi can be determined together with increasing the number of decimal places. This paper traces the history of Pi and the efforts made by mathematicians and astronomers to get closer and closer to the "precise" value of π, and then discusses two methods for determining the value of Pi – one ancient method and one modern method.
The very first attempts to determine the value of π date back to around 2000 B.C., when the Babylonians and Egyptians approached the problem in their own ways. While the Babylonians obtained the value of 3+1/8, the Egyptians obtained the value as (4/3) ^4 for π. About the same time, Indians used the value of square root of 10 for Pi. All these values were based, essentially, on measurement of circumferences and diameters of circles of different sizes (Beckmann, 12-15 and 98-106). The first major step towards determining the value of Pi is attributed to the great Greek mathematician and physicist, Archimedes around 250 B.C. The ancient Greeks, with their penchant for precision, were interested in precise mathematical proportions in their architecture, music and other art forms, and hence were curious about better precision in determining the value of Pi. Thus Archimedes developed a method using inscribed and circumscribed polygons for calculating better and better approximations to the value of π and came to the conclusion:
Subsequently, around 150 A.D., the Egyptian mathematician Ptolemy (of Alexandria) gave the value of 377/120, and around 500 A.D., the Chinese Tsu-Ch'ung-Chi gave Pi the value of 355/113. Many others like Ptolemy and Tsu-Ch'ung-Chi continued to use Archimedes's method to calculate the vale of Pi to better approximations. Ludolph von Ceulen used this method with a 2^62-sided polygon to calculate Pi to 35 decimal
...Show more
Summary
Defined as the constant ratio between the circumference of a circle and its diameter, this number has drawn the attention of mathematicians of all periods…
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It is necessary to stress that Conan Doyle's stories are not real, although there are some characters taken from everyday life. For example, the character of Sherlock Holmes was created by the author not by chance. However, still Conan Doyle's Sherlock Holmes Detective Stories are analyzed by critics as fictional ones.
Over the years, it has been proven that this value is an infinite number, meaning that the number of decimal places can never be completely exhausted. Pi is a complex number that can never be completely explained whether mathematically or otherwise, since it has an infinite number of applications.
Find ratio: 299 / 334 = 0.895 = 89.5 % 2. Round up 89.5 % to the whole number. The answer is 90 % 3. The student will get A grade. Question: 2. Use your answer in part A1 to explain whether Student 1 will receive an A for the class if the teacher truncates the percentage to a whole number.
Psychological testing in the field of psychology refers to the use of certain samples of behavior so as to assess issues such as the emotional and cognitive functioning of an individual. A psychological test can be described as a standardized and objective measure of an individual or a group's behavioral or mental characteristics.
Number 3 is one place right after the decimal point, and it is called "tenths". Number 4 is two places right after the decimal point, and it is called "hundredths". Thus, place of 7 is "thousandths", and
there are a number of schools that have not been performing to their full potential and the introduction of the PI program has helped solved this particular problem. This program has still not been established as the solution to poor performing schools and research needs to be
Managers must consider that these projects affect the company in the long term. Four capital budgeting techniques are net present value (NPV), internal rate of return (IRR), payback period, and profitability index
The basis for major events in the story is when Nouf, a sixteen-year-old girl goes missing thereby triggering her family to organize a search. The family seeks for a lead from Nayir Sharqi, regarded as a desert guide, to help the search party. After 10 days of search, anonymous desert pilgrims discover her body just when Nayir was to give up. The test carried by coroner's office indicates that the cause of death was drowning.
c procedure, that involves summation and subtraction of numbers between 100 and 999 whose first and last digits have a difference of more than two, the result is always 1089. This has attracted a lot of curiosity in mathematics hence the wide use of the number in magical tricks |
Club Feature: Math Club
In 1974, Edinboro University held the first induction ceremony of Pi Mu Epsilon, the national mathematics society, and the Math Club was born.
The function of the Math Club is to promote mathematics to the Edinboro community.
"Mathematics is all around us and having a club where we can show the appreciation for it and how it's acknowledged in daily life is what we love to do," said club president Angela Toth.
There are currently 14 students in the club, but the club is actively recruiting new members.
The club participates in various math and computer science events such as Pi Day and Stem Day. At these events, high schools from around the area come and attend lectures and take part in activities related to math and computer science.
If you like math and would like to share that appreciation with other people who enjoy math, Math Club may be for you. Club members get the chance to do projects and research, as well as participate in campus events.
"We as Math Club members volunteer to help out with these events and even give talks to students about some research that we have done," Toth said.
"We also get the opportunity to go out to conferences each year to give talks about research that we have done related to math."
She continued, "Overall, this club has given me a reason to express myself and my appreciation for math to others. If I wasn't in this club, I wouldn't have been able to experience all that I have and I'm glad I got the opportunity to be a part of it."
The club will be hosting a pizza and movie night on Saturday, April 30. If you are interested in attending this event, please contact Angela Toth at at120731@ scots.edinboro.edu.
"We may not be one of the most popular clubs on campus, but we have fun and enjoy doing math along the way," Toth said.
There are $5 dues each semester, and the club meets every other Tuesday in Ross Hall 113 at 5 p.m |
7-8
Sunday, November 11, 2012
True Math
A few words about middle school math at Summers-Knoll. Our objective is to graduate students who are numerate. This is precisely analogous to literate. A literate person can read, but that isn't the whole picture. A truly literate person can use words precisely, practically, powerfully, and with an exciting sense of possibility. A truly numerate person can do all that with numbers. This is a critically, even criminally undervalued skill. It is why Tom Magliozzi of Car Talk fame--he of the PhD in from MIT--wrote, chagrined: The purpose of learning math . . . . . is only to prepare us for further math courses.
In 2011, journalist, author and social entrepreneur David Bornstein opined, in The New York Times:
Imagine if someone at a dinner party casually announced, "I'm illiterate." It would never happen, of course; the shame would be too great. But it's not unusual to hear a successful adult say, "I can't do math." That's because we think of math ability as something we're born with, as if there's a "math gene" that you either inherit or you don't.
It's hard to imagine what society might look like if we could undermine the math hierarchies that get solidified in grade school. These patterns tend to play out across society, often in negative ways. Wasn't it the whiz kids who invented financial derivatives and subprime mortgages? And how many adults got themselves into hot water with their mortgages because, at bottom, they didn't really understand the risks?
How will we produce thoughtful, numerate graduates at SK? There are three corners to this program: math in the world; math on the page; and project-based math. IN THE WORLD Math in the world asks real questions. Why was the 2012 Presidential election primarily fought in seven states? Why is zero a revolutionary concept? How much money should an adult save per paycheck? How does one build a safe playground structure? How much mulch is enough mulch underneath a set of monkey bars? Michael Paul Goldenberg has been posing such questions with students of all ages for three decades, and we are fortunate to have him driving these questions in the SK middle school. Answering them requires hard-core, gory, flat-out arithmetical axioms, theorems, operations, and principles. But answering them also requires the ability to identify which skills and operations are relevant. ON THE PAGE Those skills have historically been practiced at SK via Singapore Math, an innovative approach to basic skills that gained global credibility and popularity approximately a decade ago, after it prompted rapid advances in test results from that Asian city-state's elementary students. Singapore has been joined in recent years by the uber-viral website Khan Academy. Khan and Singapore are terrific practice, and we expect our students to spend fifteen minutes per day with their noses in the book or screen, because, in mathematics, as in second-language study, there is no substitute for exercise. Not for nothing do our students attend math on a regular basis, even in a project-based curriculum. Jason DePasquale is masterminding our students' work in these programs, and in a school the size of SK, we are able to individualize these work programs and goals to an unusual degree. PROJECT-BASED In keeping with the theme of Ancient Civilizations, students in my section of math have been divided into Mesopotamians and Egyptians. Each will build a pyramid in the style of its civilization. The math will emerge as necessary as each group decides what materials it will use, the dimensions of its building, the geometry required, the physics of architectural design. These projects should be on display at the performances of 'Gilgamesh' in mid-December. Other projects will evolve as our school service and theme-based work in Global Citizenship and Circle of Life develops over the course of the academic year. NEXT STEPS We have settled upon the work of an innovative scholar, teacher, mentor, and gamer, Henri Picciotto of Beirut and San Francisco, to help shape the direction of our seventh and eighth grade students. Henri is the author of a highly regarded algebra textbook--after whose publication, he founded a professional organization called Escape the Textbook. For more on his approach, and more of his plentiful materials, go to Henri is also a former colleague of mine from the Urban School of San Francisco. His vision of math education, in part: Mathematics education is not just about preparing students for "practical" matters and helping the economy. It is an important part of human culture of sense-making, and should be introduced as such to all students from a young age. In addition to being useful, math is fun and beautiful. We should not lose sight of this as we attempt to make the curriculum more relevant through greater reliance on applications. Finally, another word from David Bornstein. Even deeper, for
children, math looms large; there's something about doing well in math that
makes kids feel they are smart in everything. In that sense, math can be a
powerful tool to promote social justice. "When you have all the kids in a class
succeeding in a subject, you see that they're competing against the problem,
not one another," says [Canadian mathematician and teacher John] Mighton. "It's like they're climbing a mountain
together. You see a very healthy kind of competition. And it makes kids more
generous to one another. Math can save us. |
FedEx employee discovers largest known prime number
FedEx Employee Discovers Largest Prime Number Ever at 23 Million Digits Long
The discovery of a 1,000,000 decimal digit prime number netted the discoverer $50,000, and the 10,000,000 decimal earned $100,000.
However, the search requires complicated computer software and collaboration as the numbers get increasingly harder to find.
The newly found number has 23,249,425 digits, almost a million more than the previous record holder.
The world's largest prime number, with 23 million digits, has been discovered by a maths enthusiast from the United States as part of a collaborative prime-number hunting project.
Mersenne primes - named after the 17th century French monk Marin Mersenne who studied them - are calculated by multiplying together many twos and then subtracting one. Mersenne primes set number theorists' hearts aflutter because they can be used to generate "perfect numbers", those whose factors add up to their value.
Jonathan Pace, a 51-year-old electrical engineer from Germantown in the southern US state of Tennessee, uncovered the elusive number after running a special software for six full days. The GIMPS (Great Internet Mersenne Prime Search) worldwide research team then used four computers of other volunteers to confirm the resultThe primality proof for the new prime number took six days of non-stop computing. To prove there were no errors in the prime discovery process, the new prime was independently verified using four different programs on four different hardware configurations. One such volunteer, Jonathan Pace, discovered M77232917 on his machine the day after Christmas. Thousands, armed with "reasonably modern" computers and Great Internet Mersenne Prime Search (GIMPS) software, volunteer on the project, according to the GIMPS website. While there are applications for smaller prime numbers, Caldwell explained that the larger ones fill a different need.
The person who discovers such a number will be awarded a $150,000 by the Electronic Frontier Foundation for their efforts.
The numbers seem to occur sporadically, because researchers don't fully understand the pattern they follow, he said.
Mathematics professor Curtis Cooper of the University of Central Missouri in the USA, who had found the previous highest number in January 2016, said "a bit sad that lost the record so fast" but also "really happy about GIMPS and the particular researcher who found the new number after 14 years of hard work, as hard as I did".
He emphasised the pure excitement that searching for prime numbers brings, describing the latest discovery as "a museum piece as opposed to something that industry would use". |
History papers written
Thales used geometry to solve problems such as calculating the height of pyramids and the distance of ships from the shore. He is credited with the first use of deductive reasoning applied to geometry, by deriving four corollaries to Thales' Theorem . As a result, he has been hailed as the first true mathematician and the first known individual to whom a mathematical discovery has been attributed. [36] Pythagoras established the Pythagorean School , whose doctrine it was that mathematics ruled the universe and whose motto was "All is number". [37] It was the Pythagoreans who coined the term "mathematics", and with whom the study of mathematics for its own sake begins. The Pythagoreans are credited with the first proof of the Pythagorean theorem , [38] though the statement of the theorem has a long history, and with the proof of the existence of irrational numbers . [39] [40] Although he was preceded by the Babylonians and the Chinese , [41] the Neopythagorean mathematician Nicomachus (60–120 AD) provided one of the earliest Greco-Roman multiplication tables , whereas the oldest extant Greek multiplication table is found on a wax tablet dated to the 1st century AD (now found in the British Museum ). [42] The association of the Neopythagoreans with the Western invention of the multiplication table is evident in its later Medieval name: the mensa Pythagorica . [43]
*AP is a registered trademark of the College Board, Which was not involved in the production of, and does not endorse, this product.
We provide services for students around the world; that's why we work without a break to help you at any time, wherever you are located. Contact us for cheap writing assistance. |
JUST FOR FUN
Cartoon Corner
Studying?? ... (click picture to see)
A mathematician and ...
The following sketches show our dedication to abstract thinking
in the most unusual situations and strong belief in the universality of
mathematical methods. Mathematicians are always impatient and
intelligent.
A mathematician, a physicist, an engineer went to the races and
laid their money down. Commiserating in the bar after the race, the engineer
says, "I don't understand why I lost all my money. I measured all the horses
and calculated their strength and mechanical advantage and figured out how
fast they could run..."
The physicist interrupted him: "...but you didn't take individual
variations into account. I did a statistical analysis of their previous
performances and bet on the horses with the highest probability of
winning..."
"...so if you're so hot why are you broke?" asked the engineer. But
before the argument can grow, the mathematician takes out his pipe and they
get a glimpse of his well-fattened wallet. Obviously here was a man who
knows something about horses. They both demanded to know his secret.
"Well," he says, "first I assumed all the horses were identical and
spherical..."
An engineer, a physicist and a mathematician are staying in a hotel.
The engineer wakes up and smells smoke. He goes out into the hallway and
sees a fire, so he fills a trash can from his room with water and douses the
fire. He goes back to bed. Later, the physicist wakes up and smells
smoke. He opens his door and sees a fire in the hallway. He walks down the
hall to a fire hose and after calculating the flame velocity, distance,
water pressure, trajectory, etc. extinguishes the fire with the minimum
amount of water and energy needed. Later, the mathematician wakes up and
smells smoke. He goes to the hall, sees the fire and then the fire hose. He
thinks for a moment and then exclaims, "Ah, a solution exists!" and then
goes back to bed.
A physicist and a mathematician are sitting in a faculty lounge.
Suddenly, the coffee machine catches on fire. The physicist grabs a bucket
and leap towards the sink, filled the bucket with water and puts out the
fire. Second day, the same two sit in the same lounge. Again, the coffee
machine catches on fire. This time, the mathematician stands up, got a
bucket, hands the bucket to the physicist, thus reducing the problem to a
previously solved one.
Another version:
A mathematician
and an engineer are on desert island. They find two palm trees with one
coconut each. The engineer climbs up one tree, gets the coconut, eats. The
mathematician climbs up the other tree, gets the coconut, climbs the other
tree and puts it there. "Now we've reduced it to a problem we know how to
solve."
A biologist, a physicist and a mathematician were sitting in a street
cafe watching the crowd. Across the street they saw a man and a woman
entering a building. Ten minutes they reappeared together with a third
person. - They have multiplied, said the biologist. - Oh no, an
error in measurement, the physicist sighed. - If exactly one person
enters the building now, it will be empty again, the mathematician
concluded.
Top Ten Math Major Pick-Up Lines
10. You fascinate me more than the Fundamental Theorem of Calculus. 9.
Since distance equals velocity times time, let's let velocity or time approach
infinity, because I want to go all the way with you. 8. My love for you is
like a concave up function because it is always increasing. 7. Let's convert
our potential energy to kinetic energy. 6. Wanna come back to my room....and
see my 733mhz Pentium? 5. You and I would add up better than a Riemann sum.
4. Your body has the nicest arc length I've ever seen. 3. Why don't you
come up to my place to see my slide rule collection? 2. I hope you know set
theory because I want to intersect you and union you. 1. Would you like to
see my log?
A Mathematician named Klein ...
A mathematician named Klein
Thought the Möbius band was divine
Said he: If you glue
The edges of two
You'll get a weird bottle like mine.
A tragedy ... ... in mathematics is a beautiful conjecture ruined by an ugly
fact. |
Mathematics for the 21st century
Why are mathematics taught? From Aristotle, Plato, Al-Khawarizmi, and Al-Kindi, to John Allen Paulos (Temple U.), Paul Ernest, (U. of Exeter), and Eleanor Robson (U. of Oxford), maths thinkers have stated three types of reasons: emotional, cognitive and practical. Setting aside the emotional and cognitive reasons, let's discuss the implications of the practical reasons. Mathematical understanding is crucial for high performance in our personal, public, and work lives. At home, we may want to understand the results of a medical test, or rekindle our child's love of math. As citizens, we may want to judge the rise in carbon-dioxide levels in the air, or the proportion of tax dollars that should go to health, education, or war. At work, we may need to estimate the money, time, and employees for a large project. Finally, mathematics underlies our science, technology, and engineering. OECD countries spend $236 billion per year on mathematics education yet most countries report shortages in Science, Technology, Engineering and Maths (STEM) talent. How is the breadth of mathematical application reflected in PISA? There are four contexts assessed: personal (self, family and peer groups), societal (one's community), occupational (the general world of work) and scientific (application to science and related issues and topics). These contexts are outstanding choices. Furthermore, by weighting them equally, PISA ameliorates the misconception that mathematics is useful only in the scientific context. How do we make maths relevant for all occupations, and for new occupations? The synthesis of research by the OECD and the Royal Society highlights the need to rebalance traditional mathematics (geometry, algebra and calculus) with new branches (statistics and probabilities, applied maths and discrete maths) which are relevant for a wide swath of occupations. The OECD Global Science Forum Report on Mathematics in Industry describes the needs for different types of mathematics: statistics & probabilities; complex systems; computational maths. Additionally, the Royal Society's ACME 2011 "Mathematics in the workplace and higher education" highlights requirements such as: mathematical modelling (e.g. energy requirement of a water company; cost of sandwich); use of software and coping with problems (e.g. oil extraction; dispersion of sewage); costing (allocation; dispute management) (e.g. Contract cleaning of hospital; management of railway); performance and ratios (e.g. Insurance ratios; glycemic index); risk (e.g. clinical governance; insurance); and quality/SPC control (e.g. furniture; machine downtime; deviation of rails). How are maths used in personal and societal contexts? Again, personal and societal uses highlight the need to rebalance traditional mathematics (geometry, algebra and calculus) with new branches (statistics and probabilities, complex systems) and deepening the understanding of basic arithmetic (number sense and proportionality). John Allen Paulos, Mathematician at Temple University, and Author of "A Mathematician reads the newspaper" has stated: "Gullible citizens are a demagogue's dream… almost every political issue has a quantitative aspect". In PISA, Personal uses, mostly arithmetic and spatial, encompass: personal finance, proportional reasoning, understanding technical documents (plans, charts, etc.), mental maths (percentages, four operations, mental calculating including estimating, etc.), estimation (measures/references/distances such as navigation, etc.), basic geometry (billiards, parking, etc.), and spatial reasoning. Societal uses - related to data, logic, scale, chance, relationships – are defined in PISA as: structured logical arguments, understanding data (statistical), chance/risk/uncertainty (probabilities), visualization and presenting data, magnitude of numbers (budgets, taxes, etc.), rate of change (exponential, logarithmic, S-curve, etc.), understanding systems and scale (ecology, etc.) including identifying relations between objects. How can we achieve a more numerate society? Shockingly perhaps, none of this is particularly new! A 1982 US National Science Foundation report stated: "more emphasis on estimation, mental maths… "less emphasis on paper/pencil execution…" "content in… algebra, geometry, pre-calculus and trigonometry need to be… streamlined to make room for important new topics." "discrete mathematics, statistics/probabilities and computer science must be introduced". The Center for Curriculum Redesign's Stockholm Declaration has stated: "We call for a far deeper and reconceptualized understanding of mathematics by the entire population as a critical right, requiring:
a new vision of mathematics education that anticipates needs and reinforces the role of mathematics in society, economies, and individuals, and strengthens gender equity,
changes to existing Mathematics standards as presently conceived, through a significant rethinking of what branches, topics, concepts and subjects should be taught in Mathematics for human, economic, social and career development…" |
Complex Numbers
In mathematics, a complex number is an expression of the form a + bi,
where a and b are real numbers, and i represents the imaginary number defined as i2 = -1.
(In other words, i is the square root of -1.)
The real number a is called the real part of the complex number, and the real number b is the imaginary part.
For example, 3 + 2i is a complex number, with real part 3 and imaginary part 2.
Complex numbers are often used in applied mathematics, control theory, signal analysis, fluid dynamics and other fields.
Complex plane
A complex number can be viewed as a point or a position vector on a two-dimensional coordinate system called the complex plane.
If you project the complex number z = a+bi as a vector in the complex plane,
the angle of this vector is given by it's argument and the length is given by it's modulus (the absolute value).
By mirroring z in the real axis you will get the complex conjugate of z.
You can read more about the functions for complex values on the page Complex Functions. |
Pi Math Proof
Now that we have physically measuredthe true value of Pi — see the three Pi Videos in the Pi Measurementsection of this web site — it's time to learn about my new discovery regarding the "special squaring" of each side of Kepler's Golden Ratio Right Triangle to mathematically solve for the true value of Pi.
First, let's get rid of the erroneous notion that (1) Pi must be a transcendental number, and (2) one cannot square the circumference of a circle to the perimeter of a square. The error made by many mathematicians is that they first claim that Pi = 3.141592654… by the old Archimedes polygon to circle limit equations which are only approximations and the use of erroneous built-in trig functions and, depending on how they invoked their "proof," their erroneous Pi value = 3.141592654… is transcendental. That can be true, that their value of 3.141592654… is transcendental, BUT this value is not the true value of Pi. Then these same mathematicians make a quantum leap that, therefore, all proofs for the true value of Pi must be transcendental (even though theirs is wrong), and further, that nobody can use a straight edge and compass to square the circumference of a circle to the perimeter of a square. It is quite disingenuous to claim that Pi = 3.141592654… and that it is a transcendental number and, therefore, all proofs for Pi must be transcendental and not, for example, algebraic. This web site clearly shows that (1) by physical measurement, Pi cannot possibly be 3.14159… or even 3.141… and that it is at least 3.1446… , and (2) by algebra and geometric proofs, Pi does equal 3.144605511… (which I have carried out to 40 places), and, in addition, Pi is only one of many universal constants related to the Golden Ratio Phi phenomenon. All of the squared circle circumferences and their squares at this web site Measuring Pi Squaring Phi were drawn by using only straight edge and compass, including the squaring of a circle's area to the area of a rectangle comprised of 8 Kepler Triangles. So let's dump this notion that the proof of Pi must be "transcendental" (when we can show that it is algebraic) and that nobody can "square" the perimeter of a circle to a square.
So now, let's find out what mathematician Johannes Kepler did not discover about his Golden Ratio Right Triangle back in the 16th century, i.e. how the sides of his 1, sqrt Phi, and Phi right triangle hold the simple key to unraveling the true value of Pi: (Play the video) |
What are numbers?
Numbers are the central object of study in mathematics. Most humans, and certainly all humans in modern, industrial societies, are familiar with numbers and, indeed, use them regularly in their daily lives. Despite this, non-mathematicians will probably find it difficult to give a proper explanation of what numbers actually are. I don't mean that people actually misunderstand what numbers are—in fact most people do understand what they are—it's just that it's difficult to put this understanding into words. A lot of the concepts that are fundamental to our understanding of the world are like this—easy to understand implicitly, hard to explain explicitly. As an example, think about the concept of 'colour'. Five-year-old kids can understand this concept. But can you explain it in a way that a five-year-old kid could understand? I don't think I can.
One problem with explaining basic concepts like this is that explanations necessarily make use of concepts themselves. In order for an explanation of a concept to be a proper explanation it has to not make use of the concept itself (for example, it is not very helpful to explain that 2 is the second number since the explanation of the meaning of 'second' relies on knowledge of the meaning of 2). So it is impossible to properly explain a concept without relying on other concepts being understood. In this sense, it is actually impossible to explain a concept completely. This is why if you repeatedly ask someone 'Why?', they always end up not being able to give a satisfactory answer. It's not just that the person you're asking isn't smart enough: no matter how intelligent the being you ask is, they will be stumped after a given number of 'Why?'s.
The fact that nothing can be completely explained is not quite as disturbing as it may sound: really, it's just an indication that maybe we should define what it means for an explanation to be complete in a different way, or do away with the concept altogether. It is unhelpful to think of explanations as complete or not in an absolute sense, but an explanation can be complete given a set of concepts if it does not rely on any concepts other than those in the set. The concepts in the set can be referred to as the fundamental concepts used in the explanation. As long as the fundamental concepts are understood, the explanation is perfectly satisfactory.
Now we can state the reason why it's hard to explain basic concepts: for complex topics, you can usually find an explanation which relies fundamentally only on basic concepts, and these explanations seem more satisfying because you can assume the basic concepts are already understood. But if you're trying to explain a basic concept, there aren't many concepts which are even more basic; it's likely that some of the fundamental concepts in the explanation will not be much more basic than the one you're trying to explain. That makes it harder not to feel like those concepts need an explanation too.
The reason I say all this is to warn you that you might have this problem with the explanation of what numbers are given in this post. This explanation relies fundamentally on the meaning of the word set. So whether you'll be happy with it depends on whether you think your understanding of what a set is relies on your understanding of what a number is. If you think it does (and I don't think this is a ridiculous position to hold) then this explanation might seem pretty useless!
You probably do understand what a set is, even if you haven't come across the term in its mathematical sense. The meaning of the word 'set' in mathematics is basically the same as the normal English meaning of the noun 'set': a set is a group or collection of objects. The objects contained in a set are called its members. The important thing to realise about how mathematicians use the word 'set' is that mathematical sets do not have any additional structure to them beside their members. To see what I mean, consider a fairly concrete example of a set: the set of items in a shopping bag. If you wanted to describe the shopping bag, there is a lot more you can say about it than just saying what items it contains. For example, you can describe the nature of the bag itself, or you can describe the arrangement of the items within the bag. But these aspects of the bag are not aspects of the bag as a set. When we think about the bag as a set, all those details are abstracted away, and we are only concerned with which objects are in the set.
If you are familiar with formal logic, it might help if I state the following rule which gives a condition under which two rules are equal.
For every object and every set , is the statement that is a member of . Therefore, in words, this rule states that
For every pair of sets and , if it is the case that for every object , is a member of if and only if is a member of .
It might also be helpful to see that this can be expressed equivalently as
For every pair of distinct sets and , there is an object which is in one of and but not both.
This is just a more precise way of saying what I was talking about above, about how sets have nothing more to them than their members.
So, how do sets help with defining what numbers are? Well, first let's note that it is possible to speak of the number of members a set contains. This number is called the cardinality of the set. For example, there is a set with no members at all (which is called the empty set); that set has cardinality 0. And sets with just a single member have cardinality 1. Note that some sets have infinitely many members, so we cannot assign those sets a cardinality. If a set can be assigned a cardinality, its cardinality will always be a natural number (a non-negative integer such that 0, 1, 2 or 3).
Perhaps, then, we could define the natural numbers in terms of cardinality. We would have to find some way of distinguishing sets that have different cardinalities that does not rely on the notion of number already. This would allow us to classify the sets according to their cardinalities. Then each natural number can be defined as a symbol representing one of the classes. For example, 0 would represent the class of all empty sets, 1 would represent the class of all sets with a single member, and so on.
There is, in fact, quite a simple way of distinguishing sets with different cardinalities. But before I can articulate this I'm going to have to introduce a couple more concepts. The first concept is that of a correspondance from one set to another. A correspondance from a set to a set is simply a way of distinguishing certain pairs of objects and where is a member of and is a member of . The objects in the distinguished pairs are thought to correspond to each other. Correspondances between finite sets can easily be represented as diagrams: just write down the members of each set in two separate vertical columns and draw lines linking the distinguished pairs. Here are some examples of correspondances, represented by diagrams, between the sets and (i.e. the set of the first three positive integers and the set of the first three capital letters; the curly-bracket notation used here is a standard one for writing sets in terms of their members). I'll call these correspondances , and , going from left to right.
The last correspondance, , is the kind we're interested in. By , every member of corresponds to exactly one letter, and every member corresponds to exactly one number. In terms of the diagram, each object is linked to exactly one object in the other set by a line. Correspondances like are called bijections. and are not bijections, because 3 corresponds to no letter by and 1 corresponds to both A and B by .
It's no coincidence that and both have three members. In fact, there is a bijection between two sets if and only if they have the same number of members. This may be pretty clear to you from the definition of "bijection". If it isn't, consider this: when you count the members of a set like , you assign 1 to A, 2 to B and 3 to C, and so you are actually showing that the correspondance exists. Of course you could also count the other way and assign 3 to A, 2 to B and 1 to C; that would still be a perfectly valid way of counting, because the correspondance thus established is still a bijection. But if you were to "count" by establishing the correspondance , i.e. by assigning 1 to A, 1 to B and 2 to C, and then conclude that has only 2 members because you only got up to 2, this would obviously be wrong. The ways of counting which work are exactly those which establish a bijection between and .
This, then, is how we can distinguish sets with different cardinalities. So then we can define each natural number, as explained above, as a symbol representing the class of all sets with a given cardinality. Of course, to be totally sure that this definition works we have to prove that all the things we expect to be true about natural numbers are true when they are defined in this way. Now, in the 19th century, Giuseppe Peano gave an axiomatisation of the natural numbers: a set of self-evidently true statements (called axioms) which could be used to prove all true statements about the natural numbers1. These axioms are as follows.
For every natural number , is not 0.
For every pair of natural numbers and such that , .
For every set , if contains 0, and has the property that for every member of , is a member of , then for every natural number , contains .
In order to check that the axioms are satisfied we'll have to define 0 and 1 and explain how natural numbers can be added using this definition. Clearly, 0 should represent the class of all sets with no members and 1 should represent the class of all sets with exactly one member. Now, suppose and are natural numbers and let and be sets from the respective classes which they represent. and can be chosen so that they have no members in common, and since has members and has members, the union of these two sets, —the set which contains the members of both and —has members. So we define as the class of all sets equal in cardinality to .
Now you have the tools you need to prove that the axioms are satisfied. I'm going to end this post here, although perhaps I'll post the proofs some other time.
Footnotes
However, this is not actually true: there are true statements about the natural numbers which cannot be proven by Peano's axioms. One example is Goodstein's theorem. Goodstein's theorem is known to be true because it can be proven if you develop the theory of the ordinal numbers, a larger class of numbers which contains the natural numbers. There are, however, still true statements about the ordinal numbers which cannot be proven within that theory. In fact, any consistent theory stronger than that given by Peano's axioms is incomplete in this way—this is the famous incompleteness theorem of Kurt Gödel. |
Essay The Golden Ratio
995 Words4 Pages
The Golden Ratio
Certain pictures, objects, and animals appeal to the human mind more than others. Proportions and images of symmetry often contribute to our fascination with them. Often, when examined carefully, you may find a common "coincidence" between man made objects and those found naturally in nature. This fluke, however, may be used to ascertain various mathematical relationships between these objects. This paper will introduce the golden ratio and weigh its significance on math, art, and nature.
1.6180339887…. has been given many names varying from the "golden ratio" first coined by the Greeks, to the "golden rectangle" and "golden section", "phi" named after Phidias a renowned Greek sculptor, as well as the "divine…show more content…
He studied at various places including Milan and Florence and the Vatican. It is in these cities that he became famous. He masterfully uses the golden ratio in the Mona Lisa framing her head as well as the rest of her body parts in exact proportion to the golden rectangle. Furthermore, he goes on in such works as the Vitruvian Man and Virgin and Child with St. Anne to incorporate the golden rectangle into everything he possibly can. He was by no means enthralled in art.
Instead, his great passions were mathematics and the natural world, and he compiled volumes of detailed drawings and notes on anatomy, botany, geology, meteorology, architectural design, and mechanics. (Stokstad, 693)
Toward the end of his life math, particularly the golden ratio, began to dominate everything he created. Leonardo da Vinci died in 1519.
Another painter that used, primarily, golden rectangles was Piet Mondrian. In, Composition with Gray and Light Brown virtually every rectangle has the "pleasing" dimensions.
Still, more possibilities abound. It has been proven that famous composers such as Bach, Beethoven and Bartok have used the golden ratio between intervals of their masterpieces. (Fibonacci…
The golden ratio can be seen in anywhere. It is also captured in many books and articles. It is also aesthetically shown in literature, art and ancient architectural buildings. The golden ratio can be seen in the way trees grow, in the ratios of both human and animal bodies. The ratio is approximately 1.618. The golden ratio is only can be seen by the one of the most complex organ in our body and it can be directly seen. The golden ratio has been discovered and used since ancient times. Our eye analyzes…
Throughout history the length to width ratio for rectangles was one to 1.61803 39887 49894 84820. This ratio has always been considered most pleasing to the eye. This ratio was named the golden ratio by the Greeks. In the world of mathematics, the numeric value is called "phi", named for the Greek sculptor Phidias. The space between the columns form golden rectangles. There are golden rectangles throughout this structure which is found in Athens, Greece.
He sculpted many things including…
Financial statement and ratio analysis is also used to drill down within the larger financial performance of the company as a whole to evaluate various divisions and product or service lines. These analyses are critically important as they are often used to enhance the firm's credibility in the larger marketplace; assist in determining its own creditworthiness; and comparing its performance to that of potential competitors.
In examining company reports the focus is primarily on revenues (gross…
RATIO ANALYSIS AS A TOOL FOR DETERMINING CORPORATE PERFORMANCE ( A STUDY OF SELLECTED BANKS IN NIGERIA)
RATIOS ANALYSIS AS A TOOLS FOR DETERMINING CORPORATE PERFORMANCE :( A STUDY OF SELECTED BANKS IN NIGERIA)
BEING A RESEARCH PROJECT SUBMITTED TO THE POSTGRADUATE SCHOOL IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE AWARD OF THE DEGREE OF MASTER OF BUSINESS ADMINISTRATION (MBA) OF AHMADU BELLO UNIVERSITY,ZARIA NIGERIA
DEPARTMENT OF BUSINESS ADMINISTRATION,…
Golden Gate Creamery Inc.
I. INTRODUCTION
Golden Gate Creamery Inc. started as a distributor of food products that eventually went into manufacturing and marketing of ice cream. They introduced two ice cream brands in the Filipino market namely, The American Dream and Pistahan. The third brand is up for launch and is still in planning process.
The author of this case study aims to help the company to come up with a good plan and proposal for the said product with the gathered information from…
Competitors (Economic factor)
In the Zurich region, there were 17 new hotels and two extensive enhancements were planned, currently under construction or already finished. They would be the new and potential competitors that Golden Arch was facing. As a four star hotel, Golden Arch has to consider the price of rooms as well as comparing to others.
Higher expectation on four-star hotel (social-culture Factor)
The Concept and brand of the McDonald's which is cheap fast-food shop and it does…
Golden Rule Of Interpretation-Comparision Between English Law And Indian Law
INTRODUCTION
The golden rule is that the words of a statute must prima facie be given their ordinary meaning. It is yet another rule of construction that when the words of the statute are clear, plain and unambiguous, then the courts are bound to give effect to that meaning, irrespective of the consequences. It is said that the words themselves best declare the intention of the law-giver. In law, the…
and the golden door was Ellis Island. Daniels's work is particularly striking when detailing the story of Asian immigration to the United States. For example, in the 19th century Chinese immigrants were nearly entirely male; it was not until after World War II that large numbers of Chinese women were admitted, many as war brides. The book is a survey written in two parts providing a chronological account of immigration policy, law, and politics in the American Century. Part 1, "The Golden Door Opens…
1). Liquid ratios include current ratios, quick ratios, debt to equity ratios, financial leverage, and key ratios, (Morning Star, 2013, p. 1). Ford has been maintaining a current ratio of 2.11 from 2012 to the latest quarter. This ratio increased during the period of 2004, however, decreased in 2008. Ford reported a decrease in their current ratio from 2010 to 2011, and again in 2012 (Morning Star, 2013, p. 1). The same can be said about quick ratios; the ratio stabilized at 2 for two years. The…
The Golden Gate Bridge is "considered to be one of the best and most beautiful examples of bridge design" (Poel and Royakkers 110). Unfortunately, this bridge is also "the US's most popular place to commit suicide" (110). Due to this fact, bridge designers decided that they needed to consider the option of installing some sort of suicide prevention system. Before any decision was made, the ramifications of both implementing a system and not implementing one had to be considered. Deciding whether… |
The Unfinished Game: Pascal, Fermat, and the Seventeenth-Century Letter that Made the World Modern
Book format:An electronic version of a printed book that can be read on a computer or handheld device designed specifically for this purpose.
Publisher: Basic Books (23 Mar. 2010)
By:Keith Devlin (Author)
In the early seventeenth century, the outcome of something as simple as a dice roll was consigned to the realm of unknowable chance. Mathematicians largely agreed that it was impossible to predict the probability of an occurrence. Then, in 1654, Blaise Pascal wrote to Pierre de Fermat explaining that he had discovered how to calculate risk. The two collaborated to develop what is now known as probability theory&:#151:a concept that allows us to think rationally about decisions and events.
In The Unfinished Game, Keith Devlin masterfully chronicles Pascal and Fermat&:#8217:s mathematical breakthrough, connecting a centuries-old discovery with its remarkable impact on the modern world.
Read online or download a free book: The Unfinished Game: Pascal, Fermat, and the Seventeenth-Century Letter that Made the World Modern.pdf |
The shapes of everyday things, like a tangle of string or a coffee mug, don't seem to require sophisticated math to understand. But there's an entire field of study, called topology, that examines how different shapes are related. Amazingly, some of this same math applies to quantum behavior that emerges near absolute zero. And this year's physics prize goes to three researchers that identified this relationship.
The basic concepts of topology are deceptively simple. Let's say you have a tangle of string. If you find the two ends and pull to remove any slack, how many knots will end up in the string? And how many different configurations of tangles will produce the same number of knots? Answering those questions mathematically is where topology comes in.
Similar math can be applied to three-dimensional items. For example, a bowl shape can be transformed into a variety of other different shapes, but not a coffee mug, since the latter has a single hole in it. Neither of those can be transformed into a traditional pretzel, which has three. Again, topology can help identify equivalent shapes and the means of transforming one into another. |
GMAT Question of the Day
October 17A)B)
Contrary to common belief, Euclid's Elements is more a compilation than a composition, repeating work by scholars such as Eudoxus, Pythagorus, and Theaetetus, who clarified earlier proofs and corrected weak ones.
(C)
Contrary to common belief, Euclid's Elements is more compilation than composition that repeats the work of Eudoxus, Pythagorus, and Theaetetus, accomplished by clarifying and correcting earlier and weak proofs.
(D)
Euclid's Elements, contrary to common belief, is more a compilation than a composition; repeating the work of scholars such as Eudoxus, Pythagorus, and Theaetetus; it often clarifies earlier proofs and corrects weak ones.
(E)
Euclid's Elements is more a compilation than a composition; contrary to common belief, the work of scholars such as Eudoxus, Pythagorus, and Theaetetus are repeated by it, which clarifies earlier proofs and corrects weak ones |
Friday, April 11, 2014
The three most basic shapes -- squares, triangles and circles -- are all around us, from the natural world to the one we've engineered. Full of fascinating facts about these shapes and their 3D counterparts, Shapes in Math, Science and Nature introduces young readers to the basics of geometry and reveals its applications at home, school and everywhere in between. Puzzles and activities add to the fun factor.
MY TAKE:
I'm usually not interested in learning about shapes, but I figured that this would be at least a little bit interesting since it is a children's trivia book.
In Shapes in Math, Science and Nature Squares Triangles and Circles, readers get to learn all sorts of trivia about squares, triangles and circles. There are also activities, puzzles and other fun stuff for kids to learn and try.
This book was chockful of interesting stories, a number of which were new to me. For example, I had no idea that the cylindrical structures at the corners of some castle walls were there because cannonballs tended to roll off them instead of destroying them. There were also some legends and stories about things like famous landmarks.
What makes the book worth buying, I think, are the number of fun puzzles, brainteasers and activities for kids to try, like creating a tetrahedral gift box. There were also tips for things like doubling a square, and there were a list of formulas for quick reference.
However, there were moments when I skipped around and lost interest. I think it's because the book is rather text-heavy and because I'm not too keen on the illustration style.
Thanks to NetGalley and Kids Can Press for the e-ARC.
THE GOOD:
There are lots of activities and puzzles to keep kids busy.
There are a lot of interesting trivia.
Kids can learn a lot of things they can use in school.
THE BAD:
Some kids may get bored easily because of the amount of text.
FAVORITE QUOTE/S:
The best things about cones is you can put jamocha grape ice cream in them |
You must remember this - Mnemonics can take maddening memory matters out of maths
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The other day I re-introduced the idea of co-ordinates to my advanced skills group. How to remember that (2, 3) means two across, three up, rather than the other way around? I reached into my own mathematical childhood and recalled the phrase which echoes across the years still - "You go IN the house, then UP the stairs." Daniel immediately stuck his hand up with a glint in his eye - "What if you are a burglar?"
He made me reflect that one's person's aide-memoire could be another's mental stumbling block. But are there not times when we need a crutch for useful facts? As educators we prefer memory based on understanding, but if that's tough to achieve, might a piece of mental jogging be allowed to come to our rescue?
It is Mnemosyne, the Greek goddess of memory, who gives us the word "mnemonic", and I remember one of my first. Aged 11, I recalled that "lt;" stood for "less than" by telling myself that "lt;" looked quite like a squashed "L".
Years later I twigged that "lt;" had a small left hand end, and a big right hand end, so the notion the symbol was trying to convey was embedded in the mark itself. It was its own mnemonic.
I asked: in statistics, what does "negative skew" mean? It means the hump of the distribution is to the right, and the tail is to the left. Positive skew means the hump is to the left and the tail is to the right. Come up with a mnemonic for this, please.
Claire suggested "positive" and "port" (= left) shared the same two starting letters. Angus offered that the second letter of "plus" is "l" for "left". But then, if you remember these incorrectly as "a positively skewed distribution has a TAIL to the left", you are in trouble. James offered this improvement - imagine a p on its side, bubble up, and a g on its side, bubble up. Yes!
Jonny Griffiths teaches at Paston College in Norfolk
What else?
Resources
Try some of Jonny Griffith's own RISPS resources, which are a big hit with maths teachers on TES and highly recommended by advanced skills teacher Craig Barton
In the forums
Teachers discuss the use of the grid method for long multiplication and share innovative ways to digitise maths solutions using tablets and |
Thursday, May 19, 2016
PARCC Practice Test Question 22 (Day 165)
Chapter 22 of Morris Kline's Mathematics and the Physical World is "The Differential Calculus." We apparently can't avoid Calculus forever!
"It's enough if you understand the Propositions with some of the Demonstrations which are easier than the rest." -- Newton's advice to prospective readers of his Principia Mathematica
Kline begins,
"Through the study of the motion of projectiles, planets, pendulums, sound, and light the scientific world of the seventeenth century became conscious of the pervasiveness of change. It had also become aware of the usefulness of a function to represent relationships between variables and to deduce new scientific laws."
Kline's opening quote is from one of the co-inventors of Calculus, Isaac Newton. As we can see so far, Calculus deals with functions and change. Kline writes:
"Next to the creation of Euclidean geometry the calculus has proved to be the most original and the most fruitful concept in all of mathematics."
I said that Newton was one of the co-inventors of Calculus. The other was Gottfried Wilhelm Leibniz, who created the subject independently from Newton. Here's a little of what Kline says about Leibniz:
"This man of universal gifts and interests, son and grandson of professors, was born in Leipzig [Germany] in 1646. At the age of fifteen he entered the University of Leipzig with the announced intention of studying law and the unannounced intention of studying everything. His brilliance so excited the jealousy and the envy of his teachers that they never granted him his doctor's degree. During the years in which he acquired his legal training Leibniz was also busily studying mathematics and physics."
Kline's first example of Calculus is to find the instantaneous velocity of a ball three seconds after it has been dropped. He writes the following calculation, starting with the formula Galileo had found for the distance traveled by an object in free fall:
d = 16t^2 d_3 = 144 d_3 + k = 16(3 + h)^2 (That is, in an additional h seconds the ball travels an extra k feet.) d_3 + k = 16(9 + 6h + h^2) d_3 + k = 144 + 96h + 16h^2 k = 96h + 16h^2 (Here Kline substitutes in the value of d_3 found earlier and subtracts.) k/h = (96h + 16h^2)/h k/h = 96 + 16h
Kline writes that to find the instantaneous velocity, we should plug in h = 0 into the penultimate equation in this list -- but this gives us 0/0. So we plug it in to the last equation instead, and we end up with k/h (distance divided by time, or velocity) equals 96 feet per second. According to Kline, the great insight by Newton and Leibniz is that this is justified using limits -- as h approaches 0, the value of 96 + 16h approaches 96, and so this is taken to be the instantaneous velocity.
The author goes on to demonstrate that at an arbitrary time t = x, we perform the same calculation to y = 16x^2 to obtain 32x as the instantaneous velocity. This is what we now know as the derivative of the expression 16x^2. Kline writes that Leibniz used the notation dy/dx to denote the derivative of y with respect to x, but he prefers Newton's notation, y-dot (that is, a dot above y). This is difficult for me to show in ASCII, so I'll use another commonly used symbol for derivative -- the apostrophe, often pronounced "prime" (just as we do with transformation images in Geometry). So we have:
y = 16x^2 y' = 32x
Kline also works out the velocity of a bob on a spring, which follows the formula y = sin x, at the initial time t = 0. He shows that this equals the limit of the expression (sin h)/h as h -> 0. He gives the following argument -- when h is a small central angle of a circle, we can let r be the radius of the circle, and then we draw a right triangle with hypotenuse r and opposite leg a (since sine equals opposite over hypotenuse, or a/r). Then s is the arc subtended by the central angle h, and so the measure of h in radians is the arclength s divided by the radius r, or s/r. Then (sin h)/h works out to be a/s, as the r's cancel. But as h approaches 0, a approaches the arclength s (which is why a regular polygon with many sides approaches a circle), so a/s is approaching 1. Nowadays we would write:
y = sin x y' = cos x y'_0 = cos 0 y'_0 = 1
but we must actually prove that the derivative of sine is cosine.
Here are a few more amusing examples. Kline begins with the area of a circle and differentiate it with respect to the radius:
A = pi r^2 A' = 2pi r
That's funny -- the derivative of the area is the circumference! Kline explains why:
"This result is intuitively clear, for as the radius increases, one might say that 'successive' circumferences are added to the area."
And of course, we do the same thing with the volume of a sphere:
V = (4/3) pi r^3 V' = 4pi r^2
which is the surface area of a sphere.
Actually, the volume of a sphere is a great place to segue into PARCC. Question 22 of the PARCC Practice Exam is on the volume of cylinders and -- you guessed it! -- spheres.
22. Hank is putting jelly candies into two containers. One container is a cylindrical jar with a height of 33.3 centimeters and a diameter of 8 centimeters. The other container is spherical. Hank determines that the candies are cylindrical in shape and that each candy has a height of 2 centimeters and a diameter of 1.5 centimeters. He also determines that air will take up 20% of the volume of the containers. The rest of the space will be taken up by the candies.
Part A
After Hank fills the cylindrical jar with candies, what will be the volume, in cubic centimeters, of the air in the cylindrical jar? Round your answer to the nearest cubic centimeter.
Part B
What is the maximum number of candies that will fit in the cylindrical jar?
Part C
The spherical container can hold a maximum of 260 candies. Approximate the length of the radius, in centimeters, of the spherical container. Round your answer to the nearest tenth.
Part D
Hank is filling the cylindrical candy container using bags of candy that have a volume of 150 cubic centimeters. Air takes up 10% of the volume of each bag, and the rest of the volume is taken up by candy. How many bags of candy are needed to fill the cylindrical container with 260 candies?
To answer this question, we obviously have to calculate volume. For Part A, we calculate the volume of the cylinder as pi(4)^2 (33.3) = 1673.8 cm^3. Only 20% of this is air, so this gives us an air volume of 334.8 cm^3, which rounds off to 335 cm^3.
For Part B, we see that 1339 cm^3 is left for the candy. Now the volume of a single candy is pi(0.75)^2 (2) = 3.53 cm^3, and so we divide the volumes to give a value of 378.88 candies. (This last value is exact -- the factors of pi cancel out to leave a rational number.) Notice that there isn't quite enough room for 379 candies -- so the answer is 378 candies.
For Part C, we begin by multiplying 3.53 cm^3 for each candy by 260 to obtain 918.92 cm^3. But notice that this doesn't include the air. The candy only takes us 80% of the space, so we must divide this volume by 0.8 to obtain the total volume of the sphere as 1148.64 cm^3. This is the volume of a sphere and we want to know the radius, so we use the sphere volume formula that we found in Kline:
For Part D, first of all, I'm wondering, why does the question have Hank put 260 candies in the cylinder when we already know that exactly 260 candies fit in the sphere? Wouldn't it have been more logical to ask, "How many bags of candy are needed to fill the spherical container?" (Then there wouldn't have been any need to say "260 candies" again -- "fill the sphere" would've been sufficient.)
At this point, it's probably easier to notice that since air takes up only 10% of the 150 cm^3-bag, the remaining 135 cm^3 is candy. We already calculated the volume of 260 candies earlier (when trying to find the radius of the sphere) as 918.92 cm^3, so we only need to divide this by 135 to obtain an answer of 6.8 bags. Six bags aren't enough -- we need 7 bags.
This question has several places where student error can creep in. This includes how to handle the air factor in both cases. Students must know the formulas for both the cylinder and the sphere without mixing the two up, as well as plug in the radius (not the diameter) correctly and solve for it properly (for example, don't take the square root instead of the cube root of r^3).
As for rounding error, according to the PARCC answer key, both 334 and 335 are acceptable for Part A, and both 6.4 and 6.5 are acceptable for Part C. However, both Part B and Part D have only one acceptable answer, as there isn't enough room for the 379th candy, and six candy bags aren't enough to fill the sphere.
One problem with questions that include many values that need to be rounded off is that rounding error may accumulate throughout the problem. (I assume this is the reason that Parts A and C have two acceptable answers.) But notice that since both the cylinder and sphere have pi in their formulas, the irrational value of pi cancelled out. This means that for Parts A, B, and C, we could have simply ignored pi altogether (or used a crude value like 3.14) and obtained the correct answer. Only Part D requires a value for pi as it doesn't cancel out (since the volume of the bag is 150 cm^3, not anything in terms of pi). But even a crude value like 3.14 still produces seven bags as the correct answer. (In fact, the approximation pi = 3 gives an answer of exactly six and a half bags.)
With so many possibilities for error, I expect most students to get at least one part, if not several parts, of this question wrong.
PARCC Practice EOY Question 22 U of Chicago Correspondence: Lessons 10-5 and 10-8, Volumes of Cylinders and Spheres Key Theorem: Volume Formulas The volume V of any prism or cylinder is the product of its height h and the area B of its base. V = Bh The volume V of any sphere is (4/3)pi times the cube of its radius r. V = (4/3)pi r^3 Common Core Standard: CCSS.MATH.CONTENT.HSG.GMD.A.3 Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Commentary: There are no problems just like this one in the U of Chicago text. Instead, I combine parts of two questions -- one from the SPUR section of Chapter 10 about the volume of a cylindrical cup, the other from Lesson 10-8 on the volume of a spherical tank -- and ask about how many cupfuls of water fit in the tank. |
Posts tagged with 'japanese abacus'
October last year, I have introduced how to represent numbers using the Japanese abacus. In this post, I am going to teach you how to add and subtract using it. If you are not familiar on how to represent numbers in abacus, please read the first part of this series by clicking the link above. […] Continue reading…
Before computers, there were abaci (plural of abacus). They were used as a calculation tool for hundreds of years. They were popular to the Greeks, Romans, Chinese, Japanese and Koreans. However, history tells us that there were way earlier versions of the modern abaci we know. According to Wikipedia, they existed in Sumeria as early […] Continue reading… |
Category: Helpers
If someone had come up to my 11-year-old self and told me "You should really study your math because one day, when you grow up and are able to grow something that resembles a mustache, you are going to have… |
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Fun
A surprisingly large number of academic studies—as in, more than one—have applied mathematical modeling to the concept of human-vampire co-existence. Using the depiction of bloodsuckers in various forms of media, from Bram Stoker's Dracula to True Blood, these papers look at whether Earth's vampire population would inevitably annihilate humanity, and, if so, how long it would take.
This article of Atlas Obscura presents an interesting history of mathematics dealing with the transylvanian problem of renewable resources. Great lecture!
In regard of Star Wars there are quite some things to say, always. For the time being, I want to keep it short with those two interesting facts. They concern the in-time chronological first three movies:
As the end of the year is coming closer I wanted to share my new favorite palindrome with you. As you may know, a palindrome is a word or a sentence (or "string of characters ") which reads the same backward or forward. Unfortunatley, it is in German. But if you are not able to understand it, at least be impressed by this very special palindrome. I provide a translation afterwards so you might see, that it is not totally foolish text. And yes, my favorite English palindrome stays "A man, a plan, a canal - Panama"...
Some additional remarks:
1) As a German I seem to be obliged to mention that "Reliefpfeiler" is a) one of the longest German one word-palindromes and was b) "invented" by Goethe (although Wikipedia states I) it was Schopenhauer and II) that this is not the truth). I am not sure if this is true or interesting but several teachers in my life seem to care about this.
3) There are also Palindrome novelles. According to Wikipedia, there is e.g. the novel "Dr Awkward & Olson in Oslo" by Lawrence Levine from 1986 containing 31,954 bidirectional words, take a look at it here at DigitalCommons.
4) Regarding palindromic dates, according to Gnudung the next one we will encounter is 21.12.2112 at 21.12.
Natural languaes (and some planned languages as well) bring forth strange flowers from time to time. For example, in many languages there exist sentences that are built of the same word or syllable all over. Let's call it a "repetion play" and take a closer look:
Chinese
The following is a Chinese poem that tells the story of a poet who is craving for lion flesh while living in a cavern. This is an incredible example of those repetition plays and only possible due to the Chinese distinguishment of word by tone pitch. The following table shows the poem in Traditional Chinese, in Pinyin transliteration and as a translation, on the Wikipedia page you can also hear a native speaker reading it out.
《施氏食獅史》
石室詩士施氏,嗜獅,誓食十獅。
氏時時適市視獅。
十時,適十獅適市。
是時,適施氏適市。
氏視是十獅,恃矢勢,使是十獅逝世。
氏拾是十獅屍,適石室。
石室濕,氏使侍拭石室。
石室拭,氏始試食是十獅。
食時,始識是十獅屍,實十石獅屍。
試釋是事。
« Shī Shì shí shī shǐ »
Shíshì shīshì Shī Shì, shì shī, shì shí shí shī.
Shì shíshí shì shì shì shī.
Shí shí, shì shí shī shì shì.
Shì shí, shì Shī Shì shì shì.
Shì shì shì shí shī, shì shǐ shì, shǐ shì shí shī shìshì.
Shì shí shì shí shī shī, shì shíshì.
Shíshì shī, Shì shǐ shì shì shíshì.
Shíshì shì, Shì shǐ shì shí shì shí shī.
Shí shí, shǐ shí shì shí shī shī, shí shí shí shī shī.
Shì shì shì shì.
« Lion-Eating Poet in the Stone Den »
In a stone den was a poet called Shi Shi, who was a lion addict, and had resolved to eat ten lions.
He often went to the market to look for lions.
At ten o'clock, ten lions had just arrived at the market.
At that time, Shi had just arrived at the market.
He saw those ten lions, and using his trusty arrows, caused the ten lions to die.
He brought the corpses of the ten lions to the stone den.
The stone den was damp. He asked his servants to wipe it.
After the stone den was wiped, he tried to eat those ten lions.
When he ate, he realized that these ten lions were in fact ten stone lion corpses.
Try to explain this matter.
Japanese
In contrast, the Japanese example works not due to same syllables with different pitch but with different ways to read the same Kanji 子. There is a story around this sentence and the scholar Ono no Takamura meeting the emperor Saga Tennō. Here you can see the sentence as a seemingly meaningless repetition of the Kanji, the way to pronounce it correctly next to the way to write it normally as well as the translation.
English
My favorite blog on nerdy things io9 came up with this some days ago with the english-centric title The most confusing sentence in the world uses just one word. But I have to admit: It is really confusing. Here, neither graphemes nor sounds are the source of confusion, but classical homonymy, i.e. the same word bears several meanings. This special sentence has its own website hosted by its inventor, linguist William J. Rapaport from the State University of New York at Buffalo with a complete history, many examples and discussions. Here you see the sentence, a (shortened) parse tree visualization of its parts of speech and a "translation" to understand the somewhat constructed meaning.
Buffalo who live in Buffalo, and who are buffaloed (in a way unique to Buffalo) by other buffalo from Buffalo, themselves buffalo (in the way unique to Buffalo) still other buffalo from Buffalo.
German
In most cases, German needs a small introduction in order to get a repetition play working, as in "Wenn Fliegen hinter Fliegen fliegen fliegen Fliegen Fliegen nach." which means thas flies flying behind other flies are flying behind other flies. But I have also found an example that comes without other words and makes also use of the homonymy. The content, however, is even weirder than in the English example...
The man, who leads the construction of the bridge that is going over the river that conducts cold water, has an exciting life.
Ook!
Ook! is a so called esoteric programming language and is a derivate of another one called (rightly) brainfuck. As programming languages can be understand as planned languages and as Ook! was designed in order to be understood at least by orang-utans I think it is only fair to consider it here. I present an example code to write the famous "Hello World" program next to the basic programming cocepts and the omitted output:
Ook. Ook!
Read a character from STDIN and put its ASCII value into the cell pointed at by the Memory Pointer.
Ook! Ook.
Print the character with ASCII value equal to the value in the cell pointed at by the Memory Pointer.
Ook! Ook?
Move to the command following the matching Ook? Ook! if the value in the cell pointed at by the Memory Pointer is zero. Note that Ook! Ook? and Ook? Ook! commands nest like pairs of parentheses, and matching pairs are defined in the same way as for parentheses.
Ook? Ook!
Move to the command following the matching Ook! Ook? if the value in the cell pointed at by the Memory Pointer is non-zero.
Via Twitter I stumbled upon the Reverse OCR bot yesterday. The bot itself states its mission like this:
I am a bot that grabs a random word and draws semi-random lines until the OCRad.js library recognizes it as the word.
And indeed, it is pretty interesting to follow those inexplicable lines which the OCRad algorithm identifies as words. You get a glance on why it is so difficult to write a good OCR system although most of us can parse text without any afford.
Isaac Asimov's First Law of Robotics states that "a robot may not injure a human being or, through inaction, allow a human being to come to harm." That sounds simple enough — but a recent experiment shows how hard it's going to be to get machines to do the right thing.
In shaping his vision of Middle-earth, Tolkien sometimes ran into "difficulties" of a scientific nature. One he frankly and openly admitted was in the area of biology: "Elves and Men are evidently in biological terms one race, or they could not breed and produce fertile offspring – even as a rare event: there are two cases only in my legends of such unions, and they are merged in the descendents of Eärendil." Given his mindfulness of natural laws, one may ask did Tolkien incorporate astronomical lore and fact into his universe? The answer is, in far more ways than can possibly be explored in a talk of this length. Astronomy helped, and haunted, Tolkien as he set out to develop his universe – or Eä, as the Elves would say.
Regardless, judging from the relatively low number of shops available (and considering that some of them even had their entrances hidden from first sight), it's assumed that the heavy presence of monsters and wild creatures has severely affected the different businesses. And even after Link manages to defeat Ganon, Hyrule still has a long way to go before restoring itself (as revealed by the instruction manual of The Adventure of Link. In this second game, Rupees have disappeared completely; no shops are seen in any of the current eight towns of the land, not even money-based trades are made, and therefore Link doesn't collect the legendary and long-lasting, yet ultimately deceased currency. |
An Invitation to Mathematics
Description: Mankind may be in a better position to deal with the baffling problems which confront it in the modern world if an understanding of mathematics were the rule rather than the exception. The author's aim is to give a reader who has but little knowledge of the technique of mathematics, an insight into the character of at least some of the important questions with which mathematics is concerned, to acquaint him with some of its methods, to lead him to recognize its intimate relation to human experience and to bring him to an appreciation of its unique beauty.
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Suanpan – 算盘
The extent of which one influenced the other remains unclear: if the trade between China and the Roman Empire motivated the exchange of ideas, or if their parallelisms are mere coincidences from counting with five fingers.
Its structure and rod composition depends on what kind of maths it will be used for: it can do from basics to square and cube root operations… all at high speed!
Early electronic calculators could handle only 8 to 10 digits, but suanpans can be built to almost limitless precision: in fact, they were used in the calculations for the development of the first Chinese atomic bomb. In the early days of handheld calculators, news of suanpan beating electronic calculators in arithmetic competitions in both speed and accuracy often appeared in the media.
Suanpan arithmetics were being taught in schools in Hong Kong, Taiwan and China until the late 90s, and accountants and other financial staff had to pass a test in bead arithmetic before they were qualified for their jobs until late 2004.
Nowadays, some parents still hire private tutors to teach their kids the bead arithmetic as a learning aid and a stepping stone to faster and more accurate mental arithmetic, as a matter of cultural preservation, as well as face through suanpan speed competitions. |
Easily Count to 899 On Your Fingers
By Matthew Canning, Become Better at Everything Founder
Tools you'll need: Ten fingers and working wrists.
A speedy counting method known as Chisenbop was developed in Korea centuries ago. It provides an easy method for counting to ninety-nine on your fingers. We're first going to learn basic Chisenbop and then extend it using some simple customizations I developed.
I've seen other methods that allow you to count beyond ninety-nine, but they involve learning binary or hexadecimal notation. They are difficult to learn and aren't worth all the work involved. Chisenbop relies only on knowledge of our current (decimal) counting system, and my extension just takes advantage of the fact that our hands can point in different directions, allowing us to represent different increments of one hundred.
Step 2: 10 through 99
Take your left hand, and hold it with your palm facing down. This hand represents the tens place. Ball your fist so that no fingers are extended. This represents zero, so if your left hand is in a ball, anything you do with your right hand will simply represent zero through nine. Now, follow the same system you used for your right hand, but know that each digit represents ten. An index finger represents ten, index and middle represent twenty, etc. Your thumb represents fifty.
Example: A five (just the thumb) on your left hand and a seven (thumb plus two fingers) on your right hand would together represent the number fifty-seven.
Everything you learned so far is standard Chisenbop. Spend a few minutes getting familiar with this.
Extension
Let's grow beyond basic Chisenbop.
Step 3: 100+
From your perspective, both hands palms-down means the number you're representing with your fingers is between zero and ninety-nine. This means, for example, that your left and right index fingers extended with your palms facing down equals 11. Basic Chisenbop.
However, with your left palm down but your right hand turned ninety degrees clockwise (so that your right thumb faces the sky), you represent a number between 100 and 199. In such a case, both index fingers extended would represent 111.
See where I'm going with this? Let's continue:
With your left palm down and your right hand turned 180 degrees clockwise (so that your right palm faces up): 200 – 299.
With your left hand turned 90 degrees counterclockwise (so that your left thumb faces the sky) and your right hand palm down: 300-399.
With both hands turned 90 degrees, so that both thumbs face the sky: 400-499.
Spend a few minutes going through this quickly so that you get used to the pattern. With both fists closed, quickly represent 0, 100, 200, 300, 400, 500, 600, 700, and 800. Once you feel comfortable with the hand placements, practice by representing the following numbers as quickly as possible: 12, 88, 127, 321, 193, 685, 339. Once you represent a number, freeze and check your answer against the instructions above.
You can also randomly extend fingers and choose a hand placement, and then attempt to "read" the number you've represented as quickly as possible. Check yourself, and, if you made an error, spend a moment trying to figure out where things went wrong. Most of the time, there will be a pattern to your mistakes, and by addressing them, you minimize future mistakes of the same type.
A quick way to further reinforce this is by practicing with real-world examples. If you see a three-digit number in your daily life (the time on a digital clock, for example), attempt to represent it using your fingers as quickly as possible. While I understand if you'd like to do this discreetly when in public, it's important to actually make the physical motion, as simply picturing your hands in your mind doesn't help build muscle memory.
That's it. Nice and simple. Happy counting |
Maths of spirituality
The rules of addtion applicable in maths very much hold good in spirituality too-
1. Two positives - resultant is addition - increased positivity
2. Two negatives - resultant is addtion - increased negativity
3. One postive and one negative - resultant is subtraction - who so ever is strong or bigger remains as charge
This shows its important to understand that never loose hope on goodness or positivity. Make it bigger and stronger then the negative so that the result is always positive. In todays world, it has become a challenge to remain positivite and believe in goodness because everywhere around, we may see a lot of examples where negative people are flourishing and a simple man who is livinglife simply, is struggling for basic needs of life and starts giving up and becomes weak.
However it is imperative now, that the goodness wins and positivity prevails. |
Pages
Monday, January 11, 2016
The Joy of Sectors: Getting our Galileo on...
"For the eye is always in search of beauty, and if we do not gratify its desire for pleasure by a proportionate enlargement in these measures, and thus make compensation for ocular deception, a clumsy and awkward appearance will be presented to the beholder."
Of course, a major part of the "rebirth" heralded by the renaissance was a revival of the mathematics and geometries of the Arabs and the ancients. By harkening back to the glories of their Hellenic ideal with their domes and pillars, the Renaissance brought with it a new and almost slavish devotion to finding the sacred in geometry and symmetry. Not just buildings, but furniture and textiles began to push painted, woven, and carved decorations to ostentatious heights.
I'm not particularly well known for being good at math and certainly didn't receive high enough marks in school to give one the feeling I would go on to write fluently about engineering and architecture. Thankfully, our typical renaissance artisan wasn't particularly well known as a mathematician either.
Please note that here I am drawing a line between the theory and the application of maths. Although the loftier theories may have passed him by, the practical maths of proportion and symmetry were alive and well in 16th century workshops. The average Elizabethan joiner may or may not have known who Euclid or Pythagoras was, but he could apply their theories well enough to please the eye and the customer.
We've discussed some basics of dividers before, when we were coopering. Add a sector and by their powers combined, you can accomplish an amazing number of tasks with very little actual number-crunching.
I first learned the magic of the sector in the same math class where I learned about the Fibonacci and the various permutations of the Golden Mean. Then I didn't think about it much for several decades.
Like most woodworkers, I've always kept a set of dividers. Dividers are handy for drawing circles and arcs for those fantastically symmetrical carvings I mentioned, also transferring dimensions from a ruler or a drawing to the wood. I've used them for laying out dovetails and for finding center and a host of other simple tricks.
But when they're accompanied by a sector, they can do much, much more.
My geometry teacher knew that the wickedly-sharp compasses we were equipped with as part of our standard kit were capable of more than stabbing us through our canvas bookbags. When paired with a sector, they could be used to accomplish great feats of proportion and scale
And she had no less a personage than Galileo Galilei backing her up on that.
I didn't care, I was nine; I wanted to draw circles and stab ants with the damn thing. Education is wasted on the young. Sometimes, I think adults should be required to repeat primary school periodically to pick up all the sharing and math and social studies that we missed, never mind the history. We seem so determined to keep repeating our history anyway, it might as well be in a classroom.
"I'm sorry, boss, I can't come in today, I have geometry class and then detention because I said I was thinking about voting for Donald Trump..."
Anyway... flash forward to a 2011 issue of Popular Woodworking magazine I picked up at the newsstand because of a cool cover article about Thomas Jefferson's stacking bookcases. Inside was an article by Jim Tolpin on the use of the dividers combined with a sector (see the video below) to derive a host of useful proportions and measurements for cabinetry design.
Like my teacher before him, Jim attributed the invention of the the sector to Galileo. I'm a big Galileo fan, going way back, and ere the end of things, we might even get into some of his experiments with optics because I enjoy that sort of thing.
They were both likely wrong about the inventor. The basic principles were first proposed by Euclid and put to various uses since. It seems more likely that he was the Bill Gates or Steve Jobs of the late Renaissance. He was a technological entrepreneur who envisioned new and popular uses by combining existing technologies and concepts in unique ways. That said, who initially turned a compass into a more complex instrument matters little, because ere the end of the 16th century, the concept broke out in a Big Way in the manner that technological leaps always seem to.
The sector as Galileo created it is partly well known because of who he was, and partly because it was enormously successful as a commercial product. The sales of the instruments made his fortune long before he started tweaking the beards of the Inquisition with his planetary models.
Galileo primarily sold his sector as a military tool, an instrument which in addition to its more basic Euclidean functions carried additional scales useful for the gunner in the trenches.
I have no use at the moment for determining powder loads and trajectories. There just aren't that many armies out there right now that need that sort of thing done the old fashioned way. I will be making a simpler, significantly less schmancy, workingman's sector along the same lines as Jim Tolpin's.
If nothing else, I have a lot of period carving and surface decoration on my project list, so we can look forward to seeing great granddad's dividers and sectors come out for that.
And for now -- since sectors weren't all that widely used until the 17th century anyway -- that will be the soft limits for our use for the things. I'll make a couple in different sizes and we shall see what use can be made of them without gunpowder getting involved.
That said, the Honorable Artillery Companywas knocking about, but they weren't really what you'd call a trade guild. Nevertheless, I picked up a copy of Galileo's instruction book that was sold alongside his sector because you never know when you might need to hit something a long way away with a ball of something fired out of a tube full of grey powder |
Jane Street Estimathon: KAIST
"What's an Estimathon" you ask?! It's a team contest where the goal is to create confidence intervals to difficult math and science questions. e.g., what's the volume of the earth's oceans (in cubic km); or, how many prime number contain strictly increasing digits. It's a very interactive game, and focuses on some ideas that are central to what we do at Jane Street: thinking about hard problems, assessing confidence levels, trying to strike a balance between quick-and-rough estimates versus more refined solutions. The game is pretty mathematically driven, but we welcome anyone interested in giving it a shot! |
Mathematics The Arabs developed the concept of irrational numbers, made algebra an exact science, founded analytical geometry, plane and spherical trigonometry, and incorporated into mathematics the...
Established in 1998, Access California Services (AccessCal) is a culturally and linguistically sensitive health and human services organization. We provide economic and social resources to local Arab- and Muslim-Americans, refugees and immigrants, yet we are non-sectarian, serving families and individuals of any faith or ethnicity. |
A lot of people are familiar with British mathematician John Horton Conway's "Game of Life" – an algorithm to simulate cellular growth and decay, first published in Scientific American in 1970.
The concept is simple. Imagine an infinite grid of empty squares. Each of these squares can either be empty or occupied by a "cell". The are only two rules. One rule for empty squares (the birth rule) and one rule for squares occupied by a cell (the survival rule).
Birth: If an empty square has exactly three neighboring cells a new cell is born in that square.
Survival: An existing cell will only survive if it has either two or three neighbors.
These two rules implies a time axis along which the grid will project forward in discrete phases.
But why do we have these specific rules? Why is exactly three neighbors required to give birth to a new cell?
Conway's classical rules are sometimes abbreviated to the code B3/S23.
The reason for these particular rules to have become so popular isn't that strange. Anyone who has experimented with a simulation engine using these rules knows that they often create very complex and interesting situations.
But this being a fact doesn't exclude the possibility of other rules also having the potential of creating complex and interesting scenarios.
Conway himself and others have explored a lot of other possible rules for Birth and Survival.
I played around two days ago with my own javascript-powered game of life engine where I can chose any set of rules for birth and survival. I wanted to explore an idea I had to use the game of life cellular automaton concept to create a cryptographic hashing algorithm. So, by a mere coincidence, I tried the B345/S45 rules and was fascinated by the way that an initial small colony of cells transformed during hundreds of phases only to later die out. This was almost always the fate that awaited small colonies occupying an initial eight times eight square. I tried out new random shaped colonies, one after another and very seldom a shape would pop up that wouldn't die out but instead grow and grow and never stop growing. How come, I wondered?
I dealt with this question systematically and found out that there are only 36 symmetrical shapes out of a total of 37888 that will grow indefinitely. All the other symmetrical 8×8-shapes will eventually be either totally annihilated or get stuck in small for ever repeating loops of shapes (so called pulsars).
Here is the total set of miracle seeds – the smallest possible symmetrical shapes that I found that will grow forever. All the tens of thousands of others are destined to perish or stagnate.
I have shown that there exist no smaller symmetrical shapes, for example 7×7-shapes or 6×6-shapes, that will grow indefinitely.
Add or remove a single pixel anywhere and they will die – that's how sensitive they are (try it out for yourself).
So these are the symmetrical ones. What about the asymmetrical ones?
I don't know, but I guess that you can't find an asymmetrical shape smaller that 8×8 that will grow indefinitely.
It could be tested systematically but the number of all shapes that occupy a maximum space of 8 times 8 squares are much larger than just the symmetrical ones.
Click on this link or on the picture above to go to my B345/S45 simulation engine. |
Zero
This article/section deals with mathematical concepts appropriate for notedscholar.
Zero is an incredibly useful mathematical concept — particularly for place-value numbering systems — obtained by the West from the heathens of the East in the closing stages of the 1st millennium CE, which raises the question of why a benevolent god would leave his followers in the dark on such an important matter for so long.
Contents
If you are having trouble locating zero, try this. Pick a number, any number. No, don't tell me what it is! Now, take your number, and subtract it from itself. This is your new number. No, I said, don't tell me what it is! Let me allow the aether to flow between our minds… your new number is zero!
Alternatively zero is the count of objects in the empty set (∅).[1] Too dry and mathematical for you? Zero is the number of chocolates in an empty box. (Sadly.)
""Imagine that you have zero cookies and you split them evenly among zero friends. How many cookies does each person get? See? It doesn't make sense. And Cookie Monster is sad that there are no cookies, and you are sad that you have no friends. |
Zero is such an incredibly abstract invention of the human mind that neither the Babylonions nor
the Greeks or Romans had any idea of it. (The Mayas and Chinese had some notion of zero
without really any rules of how to apply it properly). The first man to establish strict rules for the use of zero was
Brahmagupta (A.D. 598 - 660).
To understand the difficulty of zero consider how to define it: Zero cannot be defined as 'nothing' since adding 'nothing' to 'one' would not change the value from one to ten.
An even stranger thing happens if we calculate with zero.
Consider first 2x3=6. We can turn that around and divide 6:3=2.
Now let's use zero:
3x0=0. If we turn that around it reads 0:0=3. But if we multiply 5x0=0 then if we turn that
around it reads 0:0=5 ! Which means that zero divided by zero equals any number from zero to infinity - not
very useful in a science that demands an exact result ! It required a supreme intellect to establish rules in order to handle calculations with Zero. |
A recently discovered set of original Nikola Tesla drawings reveal a map to multiplication that contains all numbers in a simple to use system. The drawings were discovered at an antique shop in central Phoenix Arizona by local artist, Abe Zucca.
Trees in Celtic Mythology: Trees were hugely significant to the ancient Celts. They believed different kinds of trees served different mystical purposes that helped them through their lives. Possible wood hardness chart? |
Can you solve history's most mind-boggling puzzles?
Ever since the Sphinx asked his legendary riddle of Oedipus, paradoxes, conundrums, and puzzles of all kinds have kept humankind perplexed and amused. Why is this so? What do puzzles reveal about the human mind? Do they have implications for the study of mathematics?
The Liar Paradox and the Towers of Hanoi answers these questions, taking you on an interactive tour of the world's most enduringly intriguing brain twisters-ingenious puzzles that have played a pivotal role in shaping mathematical history. Marcel Danesi introduces you to ten masterpieces, explaining the math behind them and including exercises and answers-as well as the chance to try your hand at similar puzzles. As you navigate the maze of labyrinths, bridges, maps, and baffling problems, you'll see how certain ideas in mathematics originated in the form of puzzles, from optical illusions to sequences to impossibility theory. From die-hard puzzle mavens to math aficionados, this kaleidoscope of conundrums is sure to enlighten, entertain, and impress.
Review
""With the proliferation of puzzle books, one looks for something different, and here it is! ... This treatment will arouse interest, ally suspicion and banish fear."" (""Mathematical Association of America Online"")
""Delightful."" (""Mathematics Teacher"")
Synopsis
A walk through history's most mind-boggling puzzles
Ever since the Sphinx asked his legendary riddle of Oedipus, riddles, conundrums, and puzzles of all sizes have kept humankind perplexed and amused. The Liar Paradox and the Towers of Hanoi takes die-hard puzzle mavens on a tour of the world's most enduringly intriguing braintwisters, from K?nigsberg's Bridges and the Hanoi Towers to Fibonacci's Rabbits, the Four Color Problem, and the Magic Square. Each chapter introduces the basic puzzle, discusses the mathematics behind it, and includes exercises and answers plus additional puzzles similar to the one under discussion. Here is a veritable kaleidoscope of puzzling labyrinths, maps, bridges, and optical illusions that will keep aficionados entertained for hours.
About the Author
MARCEL DANESI is the author of Increase Your Puzzle IQ (Wiley) and The Puzzle Instinct, among many other books on semiotic topics. He is Professor of Semiotics and Linguistics at the University of Toronto and Director of the Program in Semiotics and Communication Theory. Danesi has also been cross-appointed as a professor of education, having established a continuing studies mathematics program for students with difficulties in this subject. |
Tuesday, February 15, 2011
It always intrigued me how one could find the square-root of a number. In school I was taught a digit-by-digit method to calculate square-root. I did implement it using BASIC in Std IX, but the solution did not seem elegant enough. By Std XI I had actually figured out the Bisection method by myself and used it to find any root of a number. This again was implemented in C++ (... and I still have the code !).
Then in my B.Sc. I was formally introduced to the Newton-Raphson method and I have been using that ever since. At about this time only a classmate told me of the logarithm based method, which (if Wikipedia is to be believed) is the method of choice for most implementation of sqrt().
Recently I have been going through the MIT open course, Structure and Interpretation of Computer Programs. While watching the original 1981 videos, I came across this brilliant algorithm which was supposedly invented by the Babylonians around 2000 years back. It is also often attributed to Heron of Alexandria, who was also a great inventor (the link to his biography in Wikipedia has a section on his inventions ... looks impressive to me).
Heron's Algorithm can be derived from the Newton-Raphson method and can be considered to be a special case of that. The brilliance however lies in the fact that it precedes Newton-Raphson by 1600+ years. The algorithm is very simple:
Guess a solution for sqrt(x), say g
Check whether the current value of guess is "good enough" (square roots can be irrational, so exact value is often not achievable)
If the guess is not "good enough", compute the new guess by updating g=(g+x/g)/2
Go back to step 2
You know from my last post that I have recently been learning Python. So here is a python implementation of the above algorithm.
Once again, please let me know if you find any bugs in the implementation. I am a Python newbie and your suggestions can help me improve. |
Zero
Some say it comes from the gambling expression 'love or money' – you can play a game for money (stakes) or love (nothing).
Others claim it's because in French 'l'oeuf' means 'the egg' and in 2-dimensions an egg looks like a zero. Ridiculous? Maybe, but in the sport of cricket a batsman who scores zero runs is said to have scored 'a duck' – which is meant to be short for 'a duck's egg' – the shape of which looks like a zero!
One
1 is significant in fraud detection. Benford's law shows us that in real life situations, 1 appears as the first digit in numbers more often than 2, which appears more often than 3 etc.
For example, 145, 1189 and 1590 will appear more often than 245, 2189 and 2590, which will appear more often than 345, 3189 and 3590, etc.
About a third of numbers in many real life situations – including scientific data and financial accounts – should begin with 1. Otherwise fraudulent manipulation may be suspected.
Two
Only one prime number is even, and no doubt you've guessed by now that it's 2.
Three
Take any number and multiply it by 3. Now add up the digits of the new number. Whatever number you begin with, the result will always be divisible by 3. For example, take the number 1587:
1587 × 3 = 4761
4 + 7 + 6 + 1 = 18
And 18 can be divided by 3 to leave a result with no remainder.
Four
Four colors are sufficient to color any map. This conjecture by Francis Guthrie in 1853 was the first major mathematical theorem to be proved using a computer. The honors went to programmers Kenneth Appel and Wolfgang Haken in 1976.
4 color map of contiguous USA
Five
There are only five platonic solids: tetrahedron (4 faces); cube (6 faces); octahedron (8 faces); dodecahedron (12 faces); icosohedron (20 faces). The platonic solids are completely regular, and so can be used as fair dice.
The Platonic Solids
Six
6 is the smallest perfect number, meaning it can be made by summing its divisors
Eight
Nine
For 76 years our solar system was said to have nine planets. Pluto became the ninth planet following its discovery by Clyde Tombaugh on February 18, 1930. It lost its status on August 24, 2006 when the International Astronomical Union formally defined the word planet in a way that excluded Pluto, now defined as a dwarf planet.
Hubble Telescope Image of Pluto and its Satellites
Ten
Pythagoras and his followers believed 10 was a divine number. Their holy symbol the tetractys or decad consisted of 10 points; the number symbolized the harmony of the cosmos, a greater unity than 1. |
maths is more than simply sums
The picture above, by artist Ricardo Solís, gives an idea for how a tropical fish came to have its distinctive patterns. The real-life process is just as exciting and mysterious! Can we use mathematical models to understand how these patterns are formed? Can the biological processes that we know and trust be combined to produce the stripes on a fish?
We journey back to the year 1952 and to the University of Manchester, where a famous mathematician was also interested in these questions. Alan Turing is better known for breaking the Enigma codes, especially with the recent movie "The Imitation Game". But in 1952 he published an important piece of work into the formation of patterns: "The Chemical Basis of Morphogenesis". In this, he describes a mathematical model for how patterns can arise.
First, what exactly do we mean by a "pattern"? There are some images that come to mind when we think of what a pattern is, but how can we make this mathematically precise? It's important to think about why and how patterns form, and to predict their structure.
Pattern design from White Stuff clothing, see below for info on the maths equations.
We define a pattern to be a temporally stable but spatially heterogeneous mixture of substances. Let's pick apart what this means in two stages:
Spatially heterogeneous means "different colours in different places" – it's not much of a pattern if it's the same colour everywhere
Temporally stable means it stays that way – we wouldn't call something a pattern if it started out patterned but the colours rapidly mixed into one
To make this definition mathematically precise we have to quantify these factors. For example, we might say that the colours have to be distinguishable by the average human eye and the pattern has to have no visible changes over the course of one day.
Turing showed that patterns can emerge from an initially homogeneous mix of substances by a reaction-diffusion system with one activator and one inhibitor. Equations for this process were given in the later paper from 1972 "A Theory of Biological Pattern Formation" by Gierer and Meinhardt. The actual equations are in the picture above.
The equations are pretty simple. We have a stable system, and then two diffusion terms and added on, and somehow we create patterns. Turing's insight is summarized in this article by Philip Maini:
One can take a system which has stabilizing reaction kinetics, add to it diffusion (which we also think of as stabilizing) and the resultant system is unstable!
The biological relevance of the model today may be disputed. But, from a mathematical perspective, it is fascinating to see how such a simple system of two equations can produce some of the patterns we observe in nature: |
Anonymous, that reminds me of some anecdote by Feynman where he has complex mathematical ideas described to him by young students. He wouldn't fully understand them, but he would imagine a shape, and for each new concept he'd add an extra bit, like a squiggly tail or other appendage. When something didn't fit in right, it would be instantly obvious to him, even if he couldn't explain exactly why. |
Kaso - Abstract Graffiti
Tuesday, April 10, 2007
Fibonacci Graffiti - The Project
Fibonacci Graffiti is an experimental project that brings mathematics schemes in the world of Graffiti Art. It has the intent to create new visual pattern and harmony in the urban space. Fibonacci Graffiti is also a tribute to Leonardo Pisano.
Fibonacci Graffiti - Information
Who is Fibonacci?
Leonardo di Pisa aka Fibonacci (Pisa 1170 to 1250) was an italian mathematician whom studied the "Nine Indian Figures" and their arithmetic as used in various countries around the Mediterranean. Fibonacci sequences appear in biological settings, such as branching in trees, the curve of waves, the fruitlets of a pineapple.
In Fibonacci numbers after two starting values, each number is the sum of the two preceding numbers. The first Fibonacci numbers (sequence A000045 in OEIS), also denoted as Fn, for n = 0, 1, … , are: |
For a while my Gtalk status was "pi^2=~10". Which is true, pi times itself is roughly ten. But of course I was probably hoping someone would ask what that was all about, and JZ obliged. This was my response.
it comes from a physics class
walking us through a problem, the teacher rhetorically asked permission to replace pi^2 with 10
we were aghast, because in most of the problems, you keep the pi in there and it cancels out later
and also it seemed like a crude approximation
but he showed us that since we use "10" for the force of gravity instead of the more typically correct 9.8, just because it makes the math easier, that pi^2 was even closer to ten than that
Mr. Reno was one of those "been around the block" science teachers and I admired how obviously (in restrospect) he was totally prepared for our objection, even though he played it innocent.
Similarly Mr. Von Banken, our chemistry teacher, would develop a reputation for entertaining demos in class, often explosive (but fun with liquid nitrogen, from shattering things to sending a little puddle of flaming liquid natural gas across the highway floor was memorable too) and had his class management well in hand as well: |
The aim of this book is to explain, carefully but not technically, the differences between advanced, research-level mathematics, and the sort of mathematics we learn at school. The most fundamental differences are philosophical, and readers of this book will emerge with a clearer understanding of paradoxical-sounding concepts such as infinity, curved space, and imaginary numbers. The first few chapters are about general aspects of mathematical thought. These are followed by discussions of more specific topics, and the book closes with a chapter answering common sociological questions about the mathematical community (such as "Is it true that mathematicians burn out at the age of 25?") It is the ideal introduction for anyone who wishes to deepen their understanding of mathematics.
Product Details
Table of Contents
About the Author
Timothy Gowers is Rouse Ball Professor of Mathematics at Cambridge University and was a recipient of a Fields Medal for Mathematics in 1998, awarded for 'the most daring, profound and stimulating research done by young mathematicians'.
Reviews
a marvellously lucid guide to the beauty and mystery of numbers * Gilbert Adair * |
Social networks, algorithms and life
Bouba/Kiki effect and the number 4
I happened to see an article on Bouba/Kiki effect. Basically researchers show people two geometric shapes, and want them to guess which shape is Bouba and which shape is Kiki. You can see two exemplary shapes on the right. Btw, which one do you think is Bouba?
Majority of humans choose the shape with many corners (jagged, the left one) to be Kiki, and the other to be Bouba, suggesting that the human brain is somehow able to extract abstract properties from the shapes and sounds[read the article].
This reminded me my ill founded hypothesis about the number 4. My native language is Turkish, and I know English. In the past I have tried to learn Arabic, Dutch, German, Russian and French. Right now I am living in Italy and learning Italian.
As you somehow always learn the numbers in a foreign language first, I noticed that in all languages the number 4 had an 'r' letter in it; dört, çar, četyre(четыре) , four, vier, vier, quatre, quattro, arba(أربعة) and so on. My hypothesis was that all languages had a letter 'r' in number 4.
Bouba/Kiki effect can be seen here. Four, I think, reminds a shape with sharp four edges, and it is fairly reasonable to say that except three and four, we cannot really associate shapes with numbers. How does a figure with 6 edges look like? Or seven?
After I started talking about this, a friend told me about an egzotic language which did not have 'r' in number 4.
Interestingly, for the Bouba/Kiki effect, majority does not mean all of the humans. Around 3% of them still chose Bouba for the Jagged shape. So I guess some exceptional languages may have no 'r's in number 4. It would be interesting to learn if they have some common properties. By the way, that friend was from Mahabharat of several hundred languages.
I used Google translate to see which languages do not have 'r'. Here are some; Finnish, Basque, Chinese. I am sure the list is long. |
In the memory of Maryam Mirzakhani, I have created maths4maryams.org not as a memorial site, but as a social platform for connecting mathematically like-minded people. Here is what Timothy Gowers says about […]
This post has nothing to do with Michael Artin Algebra! Artin of this post is my son, and this is the story of him learning algebra. He is 11 years old now, and I guess one of few students on the planet that still […]
"Read Euler, read Euler, he is the master of us all" written by Robin Wilson or "Euler: the master of us all" written by William Dunham are to show us how great and multifaceted Euler was as a mathematician. […]
Look at this equality: ( (a + b) + c = a + (b + c) ) , or this one: ( a . (b + c) = a.b + a.c ) . They are true structurally. In principle, You can just replace one side of the equality with the other side […]
"Plan for the mess" is one of my favourite teaching ideas. The idea is to bring students to a point where after a heavy messy work (most of the time calculations and symbol pushing) they say "why I didn't see that […] |
Wednesday, November 4, 2009
People appear symmetrical, but even the most perfect human face shows irregularities if we compare the left side with the right. Perhaps this is why the absolute, rigid symmetry of crystals seems beautiful yet alien to us. Unlike DNA's soft spiral, a crystal's molecular bonds align themselves to form regular three-dimensional structures.,which the Greeks considered math- ematically pure. The most fundamental of these shapes are known as the five Platonic solids.
If you assemble equal-sided triangles -- all the same size, with the same angles to each other -- you can create three possible solids: a tetrahedron (with 4 faces), an octahedron (8 faces), and an icosahedron (20 faces). If you use squares instead of triangles, you can create only a hexahedron, commonly known as a cube. Pentagons create a dodecahedron (12 faces), and that's as far as we can go. No other solid objects can be built with all- identical. equal-sided, equal-angled polygons.
The Platonic solids have always fascinated me. My favorite is the dodecahedron. which is why I used it in this project as the basis for a table lamp. By extending its edges to form points, we make something that looks not only mathematically perfect, but perhaps a little magical. |
Learn from a vibrant community of students and enthusiasts,
including olympiad champions, researchers, and professionals.
Crunchy Coconuts
A positive number \(n\) is called a coconut if the logarithm of \(n\) to the base 10 is in the interval \([2,3) \).
Moreover, a natural number \(n\) is called crunchy if it suffices the following condition
\[\displaystyle \text{SOD}(3+n)=\dfrac{\text{SOD}(n)}{3}\]
,where \(\displaystyle \text{SOD}(n)\) is an operator which tells the sum of digits of the number \(\displaystyle n\). |
Matrices for the stupid
A good friend of mine has been trying to explain Matrices to me as a mathematical tool. His description of the function and rules was wonderful but I couldn't get my head around the applications in the real world. I'm a bit of a dunce in this department. So I went hunting and found the most basic example I could to explain the use of Matrices to the layman. – Deskarati
Let's say we want to find the final grades for 3 girls, and we know what their averages are for tests, projects, homework, and quizzes. We also know that tests are 40% of the grade, projects 15%, homework 25%, and quizzes 20%.
Here's the data we have:
Let's organize the following data into two matrices, and perform matrix multiplication to find the final grades for Alexandra, Megan, and Brittney. To do this, you have to multiply in the following way:
Just remember when you put matrices together with matrix multiplication, the columns (what you see across) on the first matrix have to correspond to the rows down on the second matrix. You should end up with entries that correspond with the entries of each row in the first matrix.
For example, with the problem above, the columns of the first matrix each had something to do with Tests, Projects, Homework, and Quizzes (grades). The row down on the second matrix each had something to do with the same four items (weights of grades). But then we ended up with information on the three girls (rows down on the first matrix).
So Alexandra has a 90, Megan has a 77, and Brittney has an 87. See how cool this is? Matrices are really useful for a lot of applications in "real life"! Via shelovesmath |
For all - beginning VTF about algorithms with the Universe and finishing nanotechnology of
In further VTF - the great theorem of Fermat.
In tm. iatp. net are considered algorithms:
- creations of the ideal Universe and Earth - paradise;
- clonings of eternal and ideal intelligence;
- creations of uniform science, education, a calendar and language.
Ideal intelligence: spiritual, corporal and intellectual developments are identical. Among Mankind there are no orphans, disabled people, deaf-mutes, cripples, giants, Liliputians, blind people and others. Each intelligence it is not dependent on a monad - there are winners of a beauty contest. Reaching, the 25th summer age does not grow old, does not know an illness, has the only couple.
Modern life of Mankind is inconceivable without language, natural numbers and counts - reached everyone the hour of triumph. The advantage them was found when the communication channels containing forms of placement, storage, processing, receptions, and transfers, real-life information in any language by means of figures and words - counts are open. If any problem is reduced to columns, then it is reduced to problems of integer linear programming of the size n x n.
These properties of counts and natural numbers show that they are engaged:
- as unite (develop) and then as also two family of a body are multiplied one;
- calculation and expressions of the forms created and the created uniform bodies;
- to measure and show everything that gives in to measurement and a look - columns;
- quantities and sizes which increase or decrease;
- dismantling (representations of a cube - a rubik in the form of cubes of the identical size and color, and a sphere - a ball in the form of a piece) and assembly (representations of cubes of the identical size and color in the form of a cube - a rubik, and a piece in the form of a ball - a sphere);
- decisions is reduced to system of 2 linear equations with 2 variables - addition and multiplication of two natural numbers - one of 6 sides of a cube - a rubik of an order of n = 2:
Each natural numbers and the created bodies have a name, monads (the yin and Yan), images - counts (Counts and contess) and colors - manifestations of their general harmony as:
- in a combination of figures 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 looks through a code - number to each body;
- in a combination of letters of alphabets and figures looks through coordinates and time - passports;
- in a combination of counts, alphabets and figures looks through images - appearance;
- in a combination of flowers looks through perfection of biological forms.
Theoretical fundamentals of cloning of eternal and ideal intelligence, with assignment to everyone 12 - a digit code - number proceeds from the great theorem of Fermat. It turns out that it is not up to the end formulated. The proof of the theorem shows that we do not know property of natural numbers.
Natural numbers in a case of infinity loses force the situation which is from time immemorial approved by mathematics and all life experience of mankind that " part not equally whole ". Really, everyone positive, is also negatively multiple-valued an integer - the single copy. Each created uniform ideal body consists of definitely quantity of cages, and they can be counted by means of integers. We know that natural numbers through each 3 categories pass into other numeral systems. Then the account begins from 3n zero to 3n the nine.
using properties of counts, grammar and natural numbers Now And also allows to synthesize effective algorithms and on this basis to make a complex of programs.] |
Tuesday, May 16, 2017
Indian logic attracted the attention of many Western scholars, and had an influence on pioneering 19th-century logicians such as Charles Babbage (1791-1871), Augustus De Morgan, and particularly George Boole, as confirmed by Boole's wife Mary Everest Boole in an "open letter to Dr Bose" titled "Indian Thought and Western Science in the Nineteenth Century" written in 1901:[10][11] De Morgan himself wrote in 1860 of the significance of Indian logic: "The two races which have founded the mathematics, those of the Sanscrit and Greek languages, have been the two which have independently formed systems of logic."[12] Mathematicians became aware of the influence of Indian mathematics on the European. For example, Hermann Weyl wrote: "Occidental mathematics has in past centuries broken away from the Greek view and followed a course which seems to have originated in India and which has been transmitted, with additions, to us by the Arabs; in it the concept of number appears as logically prior to the concepts of geometry. [...] But the present trend in mathematics is clearly in the direction of a return to the Greek standpoint; we now look upon each branch of mathematics as determining its own characteristic domain of quantities |
"Pythagoras of Samos (570–495 BC) was an Ionian Greek philosopher, mathematician, and putative founder of the Pythagoreanism movement. He is often revered as a great mathematician and scientist and is best known for the Pythagorean theorem which bears his name.
It was said that he was the first man to call himself a philosopher, or lover of wisdom, and Pythagorean ideas exercised a marked influence on Plato, and through him, all of Western philosophy."
"Philolaus, one of "the three most prominent figures in the Pythagorean tradition", was the precursor of Copernicus in "moving the earth from the center of the cosmos and making it a planet".
Aristotle records:
It remains to speak of the earth, of its position, of the question whether it is at rest or in motion, and of its shape. As to its position there is some difference of opinion. Most people–all, in fact, who regard the whole heaven as finite–say it lies at the centre. But the Italian philosophers known as Pythagoreans take the contrary view. At the centre, they say, is fire, and the earth is one of the stars, creating night and day by its circular motion about the centre. "
"The circled dot was used by the Pythagoreans and later Greeks to represent the first metaphysical being, the Monad or The Absolute.
For the Pythagoreans, the generation of number series was related to objects of geometry as well as cosmogony.."
"Gottfried Wilhelm (von) Leibniz (1646-1716) was a German polymath and philosopher who occupies a prominent place in the history of mathematics and the history of philosophy, having developed differential and integral calculus independently of Isaac Newton. Leibniz's notation has been widely used ever since it was published.
Leibniz's best known contribution to metaphysics is his theory of monads, as exposited in Monadologie. According to Leibniz, monads are elementary particles with blurred perceptions of one another. Monads can also be compared to the corpuscles of the Mechanical Philosophy of René Descartes and others. Monads are the ultimate elements of the universe By virtue of these intrinsic instructions, each monad is like a little mirror of the universe."
Fifty years later, in 1766, hydrogen was discovered.
"Hydrogen is a chemical element with. Non-remnant stars are mainly composed of hydrogen in the plasma state. The most common isotope of hydrogen, termed protium (name rarely used, symbol 1H), has one proton and no neutrons.
The universal emergence of atomic hydrogen first occurred during the recombination epoch.... Hydrogen gas was first artificially produced in the early 16th century by the reaction of acids on metals. In 1766–81, Henry Cavendish was the first to recognize that hydrogen gas was a discrete substance, and that it produces water when burned, the property for which it was later named: in Greek, hydrogen means "water-former"."
In three dimensions the electron orbitals of hydrogen are described by toroidal geometry:
Although modern science has moved beyond ancient philosophy, stripping away erroneous mysticism, it is easily seen that the preoccupation of the Pythagoreans with mathematical relationships had a predictive power, leading thousands of years later to inevitable discoveries about the universe we live in.
Today, one might hypothesize math as an extant abstract, or "lateral", state of the universe, not always directly observable, but manifesting observable effects in all things.
"Commentary from Sir William Smith, Dictionary of Greek and Roman Biography and Mythology (1870, p. 620).
'Pythagoras resembled greatly the philosophers of what is termed the Ionic school, who undertook to solve by means of a single primordial principle the vague problem of the origin and constitution of the universe as a whole. But, like Anaximander, he abandoned the physical hypotheses of Thales and Anaximenes, and passed from the province of physics to that of metaphysics, and his predilection for mathematical studies led him to trace the origin of all things to number, this theory being suggested, or at all events confirmed, by the observation of various numerical relations, or analogies to them, in the phenomena of the universe. "Since of all things numbers are by nature the first, in numbers they (the Pythagoreans) thought they perceived many analogies to things that exist and are produced, more than in fire, and earth, and Avater; as that a certain affection of numbers was justice; a certain other affection, soul and intellect; another, opportunity; and of the rest, so to say, each in like manner; and moreover, seeing the affections and ratios of what pertains to harmony to consist in numbers, since other things seemed in their entire nature to be formed in the likeness of numbers, and in all nature numbers are the first, they supposed the elements of numbers to be the elements of all things". Brandis, who traces in the notices that remain more than one system, developed by different Pythagoreans, according as they recognised in numbers the inherent basis of things, or only the patterns of them, considers that all started from the common conviction that it was in numbers and their relations that they were to find the absolutely certain principles of knowledge, and of the objects of it, and accordingly regarded the principles of numbers as the absolute principles of things; keeping true to the common maxim of the ancient philosophy, that like takes cognisance of like. Aristotle states the fundamental maxim of the Pythagoreans in various forms.'"
Consider the mathematical nature of nature... Could the universe exhibit characteristics that can be described mathematically because math itself is an extant state of the universe, but one which is... more
basis of reality.... - gravity waves which suggest that space and time are real and act as a fluid medium of some sort - cosmic microwave background assumed to be emitted during recombination which... more
So I fall back my preferred model that the entire universe is a black hole: ... that the event horizon of a black hole with a mass equal to all the matter in the visible universe... is much larger... more
Complex Spacetime "In mathematics and mathematical physics, complex spacetime extends the traditional notion of spacetime described by real-valued space and time coordinates to complex-valued space... more
"Although Greek mathematician and engineer Heron of Alexandria is noted as the first to have conceived these numbers, Rafael Bombelli first set down the rules for multiplication of complex numbers in ... more
What if space-time is in polar coordinates instead: - angle a - angle b - radius R (a single spatial distance between any two particles) plus time t (a scalar time dimension that gives us relative... more
If there is only one spatial dimension, how can it lead to the curved space of general relativity? Well, if the other dimensions are angular (circular), then instead of being linear, maybe all the... more
So maybe mass determines the tightness of the spiral dimensions, which in turn determines whether the spiral manifests as an angular dimension (or imaginary, or lateral dimension) or whether the... more
More thoughts on how the universe is easier to understand if it is represented using polar coordinates, where there is only one dimension of space, and therefore space can only be curved if it is a... more
A string that vibrates is acting like a spring. Springs store energy. Hooke's Law: E = k . x^2 ... So when a particle travels through space it is transferring energy in & out of the spring, to & from ... more |
A Geometry of Early Islam – The Dynamic Circle Method arrangement (8th in the sequence) that was used to create the window design of the 1356CE Madrasa of Amir Salf al-din Sargatmish in Old Cairo, Egypt. Basically the method requires algorithmic steps where circle sizes and positions are changed in a step-by-step fashion. Once one understands the dynamics of the geometry then looking at the Madrasa window in old Cairo becomes a dynamic experience that couples with the numerology of the window itself, for example, 5, 6 and 7 (سنع; نزخ; نوذ) – a repository, a place of wealth. Connecting circle contact and center points with rosettes and straight lines will create surface designs or lattices of which some will have been used in the past. |
Former Kakamega Senator Boni Khalwale on Tuesday excited Kenyans online with his arithmetic skills after solving a class six-level trigonometric question.
Through his social media, the vocal National... |
4 counting mechanism Every counting wheel represents a digit. By rotating in positive direction it is able to add, by rotating in negative direction it is able to subtract.If the capacity of a digit is exceeded, a carry occurs.The carry has to be handed over the next digit.
6 Chapter 2: calculating machines bevore and after Leibniz 1623Wilhelm Schickard developes a calculating machine for all the four basicarithmetic operations. It helped Johann Kepler to calculate planet's orbits.1641Blaise Pascal developes an adding- and subtracting machine to maintainhis father, who worked as a taxman.Leibniz is working on his calculator. 1774Philipp Matthäus Hahn ( ) contructed the first solid machine.
7 Leibniz' calculating machine. Leibniz began in the 1670 to deal with the topic.He intended to construct a machine which could perform the four basic arithmetic operations automatically.There where four machines at all. One (the last one) is preserved.
8 stepped drumA configuration of staggered teeth. The toothed wheel can be turned 0 to 9 teeth, depending of the position of this wheel.
15 Multiplication (excampel) was possible by interated additions32.448*75Input of in the adjusting mechanism.Input of 5 in the rotation counter.Rotating the crank H once. The counting mechanism showsRotating the crank K. The adjusting mechanism is shifted one digit left.Input of 7 in the rotation counter.Rotating the crank H once. The counting mechanism shows |
Tuesday, October 11, 2005
What is fascinating about number theory is that it uses very deep methods to attack problems that are in some sense very "natural" and also simple to formulate. A schoolchild can understand Fermat's last theorem, but it took extremely deep methods to prove it. A schoolchild can understand what a prime number is, but understanding the distribution of prime numbers requires the theory of functions of a complex variable; it is closely related to the Riemann hypothesis, whose very formulation requires at least two or three years of university mathematics, and which remains unproved to this day. Another interesting aspect of number theory was that it was absolutely useless--pure mathematics at its purest.
In graduate school, I heard George Whitehead's excellent lectures on algebraic topology. Whitehead did not talk much about knows, but I had heard about them, and they fascinated me. Knots are like number theory: the problems are very simple to formulate, a schoolchild can understand them; and they are very natural, they have a simplicity and immediacy that is even greater than that of prime numbers or Fermat's last theorem. But it is very difficult to prove anything at all about them; it requires really deep methods of algebraic topology. And, like number theory, knot theory was totally, totally useless.
[...]
[F]ifty years later, almost to the day. It's 10 p.m., and the phone rings in my home. My grandson Yakov Rosen is on the line. Yakov is in his second year of medical school. "Grandpa," he says, "can I pick your brain? We are studying knots. I don't understand the material, and think that our lecturer doesn't understand it either. For example, could you explain to me what, exactly, are 'linking numbers'?" "Why are you studying knots?" I ask; "what do knots have to do with medicine?" "Well," says Yakov, "sometimes the DNA in a cell gets knotted up. Depending on the characteristics of the knot, this may lead to cancer. So, we have to understand knots."
I was completely bowled over. Fifty years later, the "absolutely useless"--the "purest of the pure"-- is taught in the second year of medical school, and my grandson is studying it.
Another quote:
You know, sometimes people make disparaging remarks about [game theorist Oskar] Morgenstern, in particular about his contributions to game theory. One of these disparaging jokes is that Morgenstern's greatest contribution to game theory is von Neumann. So let me say, maybe that's true--but that is a tremendous contribution.
And again:
In short, I have serious doubts about behavioral economics as it is practices. Now, true behavioral economics does in fact exist; it is called empirical economics. This really is behavioral economics. In empirical economics, you go and see how people behave in real life, in situations to which they are used. Things they do every day. |
Idea OO
by zenquaker
If you're new to the blog, or haven't been paying attention, I have been using roman numerals for the titles of all my blog posts: Waste I, Waste II, Waste III, Waste IV. It's just a style I carried over from an earlier blog. It has had its interesting side effects, to be sure. I got more hits than I expected off my 30th movie review. But it occurred to me today that roman numerals are incredibly outdated. Not even Wikipedia knows how old they are. Clearly a hip techophiliac like myself should have something more up to date. So I decided to create Ichabod Numerals.
Ichabod numerals work on the same principles as roman numerals, just more intuitive and mathematically improved. In roman numerals, you have an I for a one, which gets repeated for 1-4. Ichabod numerals do the same thing, but with an O for one. I mean, "one" starts with an O, right? It just makes more sense.
In roman numerals, you eventually stop repeating the I's, and use a V for five. Ichabod numerals similarly use an S. Yes, I know "five" does not start with an S. But "six" does. Why six and not five? Because while roman numerals are decimal (base ten), new and improved Ichabod numerals are dozenal (base 12, also called duodecimal).
One of the things that has always driven me nuts with metric proponents is that they talk about how it's better because it's a decimal system. How is 10 better than 12? Quite frankly it's not. For example what's a third of a meter? Not 33 centimeters. That's close, but it's off by at least 3 millimeters. Even if you add the 3 millimeters, you're still off by 333 micrometers, and 333 nano meters, and so on. A third of a meter does not exist. What's a third of a foot? Four inches. Exactly. 12 can be evenly divided by 2, 3, 4, and 6. 10 can only be evenly divided by 2 and 5, and when was the last time you needed a fifth of something besides whiskey? The problem with the Imperial system of measurements is not they're base 12, it's that they're inconsistent. [But that's another post] Remember that I'm not just a hip technophiliac, I'm a hip contrarian technophiliac.
Where were we? Right, base 12. Since Ichabod numerals are based on a dozenal system we use a D for dozen instead of a X for ten, again making more sense than roman numerals. To recap before moving on to higher numbers, the first dozen numbers in the Ichabod numeral system are:
O
OO
OOO
OOOO
OS
S
SO
SOO
SOOO
SOOOO
OD
D
One little quirk of the system is that thirteen (DO, which I guess should be called elezen in the dozenal system) is often abbreviated as B, for a baker's dozen. This is not to be confused with a banker's dozen, which is equal to eleven. Now, obviously a dozen dozen is a gross, which the Ichabod numeral system represents with a G. Of course, between X and C in the roman numeral system we have L for fifty. So we need a letter for six dozen in the Ichabod system. An L is sort of an upside down F (for fifty), so we'll go with a Z for six dozen.
But what's a dozen gross? The Dozenal Society of America uselessly calls this a great gross, which just doesn't work in my system. However, one of there articles mentions an egg delivery company that had a dozen gross to a crate, so we'll call a dozen gross a crate, and give it a C in the system. For six gross all the letters similar to a C are taken except for Q. A dozen crates would seem to be a truck load (T), and half a truck is a van (V). To recap, here is a table of the letters in the Ichabod system, with their equivalent in standard dozenal and decimal notation.
Numeral
Decimal
Dozenal
O
1
1
S
6
6
D
12
10
Z
72
60
G
144
100
Q
864
600
C
1,728
1,000
V
10,368
6,000
T
20,736
10,000
That gives us more symbols than are commonly used for roman numerals, and far more than I ever need to count my blog posts. Oooh, chance for estimation! I've written 168 blog posts in about five months (note that there are twelve months in a year), and my largest roman numeral is XXXII. We would therefore expect a truck load of movie reviews to occur around my 108,864th blog post. At the current rate (depending on whether you use the five month average or the past month's average) that will take between 77,001 and 98,496 days, which means it would occur sometime between December 28th, 2222 and November 3rd 2281.
I'll end this post with some describable numbers in the Ichabod numeral system: |
In which our heroine, having defeated the three-headed demon of supercomputing, optimization, and ill-posed problems in her quest for the elusive Ph.D., embarks upon her new career.
Friday, March 03, 2006
Adventures in Math
I know some folks come here just for the math, so here's another little piece of math to keep you coming back.
The other day I got an interesting e-mail from my fearless sister-in-law Ginger. (Ginger is a closet mathematician, but she just doesn't know it yet. I got her addicted to Sudoku, and now she's sending me math e-mails. It's only a matter of time before she starts doing calculus for fun.) Anyhow, here's what the message says:
1. GRAB A CALCULATOR. (YOU WON'T BE ABLE TO DO THIS ONE IN YOUR HEAD) 2. KEY IN THE FIRST THREE DIGITS OF YOUR PHONE NUMBER (NOT THE AREA CODE) 3. MULTIPLY BY 80 4. ADD 1 5. MULTIPLY BY 250 6. ADD THE LAST 4 DIGITS OF YOUR PHONE NUMBER 7. ADD THE LAST 4 DIGITS OF YOUR PHONE NUMBER AGAIN. 8. SUBTRACT 250 9. DIVIDE NUMBER BY 2
DO YOU RECOGNIZE THE ANSWER?
I tried it on my old phone number in Illinois, 367-5384. (As an aside, that was the coolest phone number ever. It spelled DORKETH. One can be so fortunate only once in life, I suppose.)
That is indeed a familiar number! It's the original phone number I started out with.
This trick will work with any phone number. I know you're all asking why. At least, the folks who are in this for the math are, and they're the ones who count (ha ha! a little math humor there!) so I'll now explain how this trick works.
First of all, remember from elementary school how starting from the decimal point and going to the leftt, we have the ones place, then the tens place, then the hundreds, etc.? In each of those slots, we can put any integer from 0 to 9, and this represents a number. The number is just the sum of each of those integers times the proper power of ten; for example, 436 = 6 x 1 + 3 x 10 + 4 x 100 = 6 + 30 + 400 = 436.
I thought this was a cute trick. I wrote all the steps out there, but you can see, if you gather like terms, that after step 5, we have 2 x a,bc0,000 + 250. (The 250 is just there to throw you off the track.) After step 7, we have 2 x a,bcd,efg + 250, and after we take away the extraneous 250, we obtain 2 x a,bcd,efg. Finally, in the last step, we divide by two and get back the original number.
I have to admit that I did follow step one and pull out the old calculator when I first tried this trick. Although perhaps it would have been more fun to do it with my trusty slide rule or my new abacus that I bought in San Francisco. Until next time, math fans!
2 comments:
without even thinking, I used my danish phone number... we have 8 digits, which means it didn't quite work the way it was supposed to. It was close tho, my number is 26484700, and it came out with 2644700 |
What would you like to search for?
Ancient Arrows
Whether in pre-historic society or while communicating with alien civilisations, is there any better symbol to depict position and direction? Hold's creative director Steve Hyland thinks not.
Words by Steve Hyland
When, in the early Seventies, the Pioneer 10 and 11 probes were launched with brilliantly conceived etched plaques depicting messages from mankind, one criticism of them was the use of an arrow to indicate trajectory. It was argued the origin of the arrow being rooted within a hunter-gatherer pre-history meant this symbol would be meaningless to a being with a different cultural heritage.
For me the criticism slightly missed the point. If an alien culture stumbles upon Pioneer, it's just as likely to be akin to a cuttlefish as to a human, thus deeming the entire exercise futile. As Wittgenstein sagely wrote, "If a lion could speak, we would not be able to understand him."
However, it did get me thinking. If not an arrow, then what? Shot from pre-history and now ubiquitous in modern cultures, there aren't many symbols as mutable in form yet still comprehensible. Without a Babel fish handy, how, for instance, can you communicate direction universally without it being lost in translation? What, if anything, could supersede the humble arrow?
The prevailing meaning of the arrow as a symbol is to indicate either a direction or a position. In the digital realm it is used to point and to navigate. It can also be found in maths notation, it bestows the concept of time with a direction and in some cultures it is even a sign of danger. It's been called a circumflex-hatted 'T'. It can appear in a myriad of forms, yet generally be understood if it follows the basic rules. A line with a triangular end.
What I find fascinating about the arrow is its versatility. It can be decorative or minimalist, dressed up or down, it just needs context to express its intent. Which brings us back to the Pioneer message. The plaques employed an ingenious schematic of the most abundant element in the universe, hydrogen, to represent a basic unit of measurement. This gave a baseline to understanding the message's other measurements. Is there a universal element analogous to a shooting arrow that contains the dual properties of direction and position, negating the need for a cultural context? For me the humble arrow will do. It reassures me that, as languages are forgotten, some things persist and will perhaps yet be understood by our distant cousins, hopefully without the need to insert a fish into their ears.
…is creative director of Hold, a Brighton-based studio which he founded with Stuart Langridge five years ago. Their clients include the Design Museum, the PRSF and Fabrica Gallery in Brighton. Interested in design for the cultural sector, Hyland previously worked for Sony Music Entertainment and HarperCollins and runs his own record label Concrete Plastic.
Pioneer 10
Launched in March 1972, Pioneer 10 was the first spacecraft to fly through the asteroid belt, first to fly close to Jupiter and first to cross Neptune's orbit. Until its distance from Earth was exceeded on February 17, 1998 by Voyager 1, Pioneer 10 was the most distant human-made object. It will take more than 2 million years for Pioneer 10 to pass Aldebaran, the nearest star on its trajectory |
Strength in Numbers
Discovering the Joy and Power of
Mathematics in Everyday Life
SHERMAN K. STEIN
What is the spell of cool numbers? Was the golden ratio used to build the Great Pyramid of Khufu? What do two goats and a car have to do with making good decisions? In Strength in
Numbers, awardwinning teacher and author
Sherman K. Stein offers an entertaining
exploration of the surprising ways in which the
language of mathematics can enhance our
understanding of the world around us.
"After finishing this book, you should have a clearer idea
of the importance of mathematics in the 'real' world and
the ability to read the language of mathematics. I hope, in
addition, you will have gained an appreciation of the
beauty of mathematics and the elegance of its reasoning." |
Monday, March 14, 2016
Pi Pie
Look at this delicious pie Mommy made today! It is Pi Day, 3/14. If you round up pi to 3.1416, then that matches the date of the day (3/14/16). But who really rounds up pi anymore? It's either 3.14 or some really long extended number that I can never seem to remember past 3.14159265 or something like that.
Daddy was getting the car taken care of this morning in Roswell, and thought to go to a Publix in town there, where it was more likely to find a gluten-free pie crust. Sure enough, there it was - and thus the whole family was able to enjoy this amazing pi pie to celebrate the day. This thing is so incredibly rich!
After ballet, we got home and were able to enjoy this and dinner as well, also tackling the usual gambit of homework and piano that sat before us. We didn't have much time after that, as ballet is from 4:00 - 5:15 each day, and by the time you get home, there's homework, eating and just enough time to squeeze in a short round of video games or something. We did do a bit of Disney Infinity 3.0, and after that there was bedtime, with reading and prayers, and dreaming of π. Or pie.
What is it about pi that gets it a holiday each year? There's nothing terribly deep here, but pi is that constant in all circles. It's a hidden number always there, always inside no matter the size of that circle. You could have a circle with the dimensions of the universe, or a microcosmic circle invisible to the naked eye, and yet both will have pi within. It's a constant, a dependable sort of thing that fascinates us. What is it about this magical number? If there is a circle, it must have within it 3.14. A circle without pi is quite simply not a circle.
Furthermore, pi is infinite. Certain people with much more time than I do have memorized as many as 70,000 digits of pi (again, Daddy gets to around 3.14159265 and that's it). But it goes beyond that, of course. I think currently, computers have figured out pi up to around 13.3 trillion digits. But it always goes beyond that, because the number is infinite, with the wise always searching for more, and there always being more to it.
These sorts of characteristics - infinite and always there, always constant, as well as an essential component - these are the things that fascinate us about pi, and at least in my case, remind me of our Creator and his perfect design for this universe. It's like a little Easter egg God placed in every circle that has ever existed |
The Glorious Golden Ratio (Alfred S. Posamentier) at Booksamillion.com. What exactly is the Golden Ratio? How was it discovered? Where is it found? These questions .The Glorious Golden Ratio by Dr.
Alfred S Posamentier starting at $4.42.Scribd is the world's largest social reading and publishing site.Designed for high-school students and teachers with an interest in mathematical
problem-solving, this stimulating collection includes more than 300 problems that are . |
The 2nd edition of Robert Lang's Origami Design Secrets is here! I have been a long time fan of his and his work is vastly different from mine, but despite the differences, I am in complete agreement with the questions and thoughts expressed in his book. Below are few favorites quotes from the book I found helpful when pursuing the world of origami design.
Origami as an art form: "Origami is, first and foremost, an art form, an expression of creativity, and it is the nature of creativity that it cannot be taught directly. It can, however, be developed through example and practice."
Origami can be approached in a very analytical and logic way: "Origami design can indeed be pursued in a systematic fashion."
Questions to ask when you are designing origami: "Does it flow naturally? Is the revelation of the finished form predictable or surprising? Does the use of folded edges contribute or detract from the appearance?"
Subsequent chapters teach the reader basic folds as well as more advanced folds that can transform a simple sheet of paper into a lizard, frog, pegasus, and a few furry eight legged insects. This book is perfect for math geeks who aim to apply their math in a very creative way. |
The date for the dumbasses who are visually impaired to the point that they can't see it is 3-14-2010, and it is PI day!
For the donut-eating elementary schoolers who don't know what PI is, it is a mathematical constant equaling approximately 3.14, or 22/7. You'll probably study it in your geometry, trig, and calculus classes. (and by probably I mean you will fools) Math people have been trying for centuries to find the exact value of PI, but it's quite the challenge!
3.14 ..... 3-14..... See the cheese?
PI is used in pretty much every field.. science, math, business, engineering, idk much about computer science but it's probably used in the hardware aspect of that too. Respect for the pie guys! |
The Math Clock
Dinner will be at 3×2 o'clock. Bed time is at 20÷2 o'clock because we have to get up at 10-3 o'clock. Now doesn't that make telling time more fun?
Here are some math clocks for you to consider. Whether you want math clocks with formulas or symbols or math sayings, you'll find them all here. These are the best deals on math clocks that you're going to find and there are tons to choose from.
If you like the math clock to the left, you can get it here from Zazzle: Formula Math Clock
Pop Quiz Math Clock
I know, no one told you there was going to be a test. It's ok, most of the questions aren't too tough.
These clocks give you a math problem at each hourly position. For example, 4X=8. Now solve for X and you find out it's 2 o'clock. See now, that wasn't so hard.
These clocks give you a little something to do in your spare time (no pun intended) and can make telling time a bit more fun and interesting.
Math Clock For Those Who LOVE Math
Are you a real math nerd? Oops, I mean math lover? Then these clocks are for you. Show everyone who looks at the clock in your home or office how much you love those symbols and digits. You can get one that simply declares that math rules or you can get one that bribes people to come over to your point of view by promising pi.
Math Clock For Those Who HATE Math
Do you hate math? Do you really hate math? These are some great clocks for displaying your math hating nature. Go ahead and announce your phobia of math and all math related things or call it what it really is: abuse.
The Just-Can't-Handle-Any-More-Math Clock
If you hate math so much that you're ready to scream and pull you hair out, then you need this clock. The numbers are just falling off of it.
If all the numbers just drop away, does that mean there is no more math? Maybe it'll just cease to exist? We can only hope.
"The Pis Have It" Math Clock
If only it were pie instead of pi. We'd all be a lot happier about it. But I guess we can pretend.
Go ahead and have a slice of pi. Take whichever slice you want. You can have the one between 12 o'clock and 1 o'clock. Or if you're really hungry, take the one between 3 o'clock and 6 o'clock.
Math Clock To The Infinite Power
Infinity is a mighty long time. Do you hate math infinitely? Are you willing to make that kind of commitment?
These clocks display the infinity symbol in various ways and colors and they can be customized. You can change the colors or add your name. You might say the possibilities for these clocks are infinite |
Euclid: Euclides of Megara, mathematician, 5th century BC. He was younger than Plato but older than Aristotle. His main works are the elements that develop his axioms. (See Der kleine Pauly, Lexikon der Antike, Munich 1979)._____________Annotation: The above characterizations of concepts are neither definitions nor exhausting presentations of problems related to them. Instead, they are intended to give a short introduction to the contributions below. – Lexicon of Arguments.
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Euclid/Kant/Russell: Kant rightly remarked that the Euclidean theorems cannot be deduced from the Euclidean axioms without the aid of numerals. Kant's doctrine of the a priori intuitions, by which he explained the possibility of pure mathematics, is completely useless for mathematics |
So then in my senior year… I was… I was majoring in physics mostly and with fairly poor grades because I was doing so many other things… and I thought I would make up for this by doing a good thesis, but it turned out at Harvard, physics students can't… they don't have an undergraduate thesis, but they do in math. So, I asked Andy Gleason why can't I just switch to math and do a thesis and he said: 'No problem,' signed something and I decided to do a thesis on fixed point theorems which was a beautiful little fragment of mathematics that… that I was very interested in. And there was a theorem that… Kakutani was a professor at Yale and here's an example of a fixed point theorem: Of… consider that at each point in the world on… on the earth, there's a temperature and there's a humidity, and these vary continuously over… over continuous functions. So there's two continuous functions on this sphere. Kakutani… somebody else had proved that… I forget who… Whitney, Professor Whitney, was a friend of mine at Harvard actually, had proved that you can find two points on the earth which are exactly opposite points which have the same temperature and the same humidity. This is rather surprising.
So that… that was a famous fixed point theorem. Or you could say there's, if you consider mountains… There's two points which have the same altitude and the same temperature, any two functions. It's very strange, why should that be? Anyway, Kakutani had proved that if you take a… you can find a triangle, an equilateral triangle, that is… 60º angles, the same thing is true, for an equilateral triangle you can find three points which will have the same temperature and pressure… or temperature and humidity. So that was Kakutani's fixed point theorem. He was a professor at Yale – I'd never met him actually – so that's very interesting because it shows that somehow if you have an extra point you can… you can do a little more because now they're equal at three points. Isn't that funny?
And so then I said: 'Well, what about other... why does it have to be an equilateral triangle?' And so I worked on that a lot and I managed to prove that if it were a right triangle, it was also true and also if it were a triangle which were three points of a regular pentagon, that would be… the same thing would be true. And that's as far as I got, I couldn't prove it for any old triangle. But, anyway, my proof of the thing for the pentagon impressed Gleason and that was probably why he told Princeton they had to admit me as a graduate student. That's the next step for a mathematician.
So it was a great adventure and… but it always haunted me that I couldn't show this thing for any old triangle.
Marvin Minsky (1927-2016) was one of the pioneers of the field of Artificial Intelligence, founding the MIT AI lab in 1970. He also made many contributions to the fields of mathematics, cognitive psychology, robotics, optics and computational linguistics. Since the 1950s, he had been attempting to define and explain human cognition, the ideas of which can be found in his two books, The Emotion Machine and The Society of Mind. His many inventions include the first confocal scanning microscope, the first neural network simulator (SNARC) and the first LOGO 'turtle'. |
Thinking in Numbers
On Life, Love, Meaning, and Math
In Inspired by the complexity of snowflakes, Anne Boleyn's eleven fingers, or his many siblings, Tammet explores questions such as why time seems to speed up as we age, whether there is such a thing as an average person, and how we can make sense of those we love. |
computer science and mathematics for the masses
Holiday Maths Puzzle #1 |
A view of the world from my own unique perspective
Let's begin with a definition. A palindrome is a word or a series of words that is spelled the same way backwards and forwards. For example: radar, level, "Madam, in Eden, I'm Adam", "Never odd or even". A palindrome can also be a sequence of numbers.
I've always thought that palindromes were pretty neat, but I hadn't given them much serious thought until I read a fascinating article about them by Richard Lederer, in the July 2011 issue of the Mensa Bulletin. In this article, Lederer included a quote by Alistair Reid, who stated "The dream which occupies the tortuous mind of every palindromist is that somewhere within the confines of the language lurks the Great Palindrome, the nutshell which not only fulfills the intricate demands of the art, flowing sweetly in both directions, but which also contains the Final Truth of Things".
I wish that this blog post could serve as a Palindromist's Guide to the Galaxy, and uncover the Final Truth of Things, but unfortunately I'm not nearly as talented as Douglas Adams. Instead, I will attempt to answer the question: why are palindromes so beguiling?
There is a link between palindromes and a quest to uncover a long-hidden cryptic message, in Jules Verne's novel Journey To The Center Of The Earth. The novel's protagonist, Professor Lindenrock, purchased a runic manuscript, and then spent days trying to decipher it. When the message was finally translated it instructed him to travel to the centre of the Earth, which is a palindromic journey. Starting at the Earth's surface, one would progress through the crust, the upper mantle, the lower mantle, the outer core and the inner core before finally arriving at the centre. At this point, being in the centre of a sphere, you will encounter everything in the reverse order during the return journey, no matter which direction you travel. After completing his palindromic journey, Lindenrock returned to Hamburg, and was hailed as one of the greatest scientists in history.
Humans are hard-wired to look for patterns in things. Babies can recognize faces better than adults, and although that ability diminishes with age, it is still very prevalent in adulthood. During the past few years, there have been numerous stories in the news about people claiming to see the face of Jesus in a piece of French toast, a chocolate bar, a fragment of their breakfast cereal or in a stain on a wall. We are hard-wired to recognize faces, so that's typically what we see when gazing at some nebulous design. Not only do we see patterns, but we feel that these patterns must have some inherent meaning – enough to convince us to change our inward-looking ways, or at least to sell the item on eBay for a tidy profit. Similarly, a pattern contained in a series of words makes us infer some deep significance beyond their semantic meaning.
I'm going to use a bit of interpretative license and assume that "The Final Truth Of Things" in Alistair Reid's quote, can also be construed as the secret of the universe or even the meaning of life itself. The ultimate palindrome may in fact reveal the secret of the universe – but only if the universe is bilaterally symmetrical. I don't imagine that our universe is bilaterally symmetrical, but we humans are, which is one reason why I believe that we find palindromes so intriguing. The bilateral symmetry of palindromes is found not only within our bodies – it permeates our society, as the following examples will show.
According to social anthropologist Desmond Morris, there is a strong correlation between beauty and symmetry. The more symmetrical one's face is, the more beautiful one is perceived to be. Therefore, the perfect bilateral symmetry in a palindrome implies an inherent beauty in the word or phrase.
Encountering a palindrome is much like gazing into a mirror and seeing the reversed image of ourselves. Historically, a mirror displays not only our physical appearance, but also our soul. Therefore, one way to gaze into our own soul and see our essence is to look in the mirror. While our bodies are bilaterally symmetrical, our entire being – body and soul, on either side of the mirror – is also bilaterally symmetrical. According to legend, vampires, who have no souls, are not visible in mirrors.
In the 1960s, a popular psychological test was the Rorschach ink blot test. Ink would be spread onto a sheet of paper, which was then folded in half, creating a bilaterally symmetrical design. The subjects were then asked to interpret the design. Based on their answers, the psychologist would be able to assess their personality.
If you write a palindrome on a piece of paper, and then bisect it with a mirror, you will see in the mirror, the entire word or phrase (although with the letters reversed in the second half). In 1967, this mirror technique was used to uncover a "secret message" in the cover of the Beatles' Sgt. Pepper album. At the time, some fans were convinced that Paul McCartney had died and was replaced by someone called Billy Shears, who sang on the album. As proof, they invited fans to place a mirror across the drum on the album's cover, so that it bisects the word "HEARTS" horizontally. A secret message is revealed – "HE DIE" – with a crude diamond shape (which was interpreted as an arrow) directly underneath Paul's image. An early example of steganography, perhaps…
Bilateral symmetry is also used in society to illustrate a sense of fairness, as shown in this diagram of the Scales Of Justice. Fairness and equality are achieved only when the scale is perfectly balanced and the diagram becomes symmetrical. This sense of balance is also present in Newton's Third Law of Motion: for every action, there is an equal and opposite reaction.
On a larger scale, a kind of bilateral symmetry extends to the entire universe. The writers of prime-time network television programs have created a parallel universe that is the opposite of ours. One example is a Seinfeld episode broadcast in 1996, called The Bizzaro Jerry. Another example is the Star Trek (Original Series) episode Mirror, Mirror, in which Kirk, Uhura, Scotty and McCoy are transported to an alternate universe, where their counterparts are barbaric instead of civilized. For non-Trekkies, this is the episode in which Spock has a goatee.
If there is a Final Truth of Things contained in the proverbial Great Palindrome, then I propose that there will someday exist, The Ultimate Computer Program, which when run, will output the secret of the universe. Naturally, its source code, when displayed as a series of binary numbers, will be a palindrome. I hate to disappoint Douglas Adams fans, but I already know that the program's output will not be 42. That's because 42 expressed in binary is: 101010.
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In the spirit of uncovering the Final Truth of Things, I'd like to share with you something that I consider an undiscovered gem – a song called Bob, written by (all of people), Weird Al Yankovic. In this song, he imitates the incomprehensible singing style of Bob Dylan. The video itself is a parody of Dylan's Subterranean Homesick Blues video, complete with two characters in the background who actually resemble Alan Ginsberg and Bob Neuwirth (who appeared in the background of Dylan's video). Bob is certainly an inspired title, but that's not why I like it. The most intriguing part of the song is its lyrics; every line is a palindrome, and every second line rhymes. However, no mention of this was made in the CD liner notes – it's as if the palindromes were a lyrical Easter Egg. Weird Al may act silly most of the time, but every now and then he does something that really impresses me!
And finally, a bit of trivia. According to the Urban Dictionary, the unofficial term for the irrational fear of palindromes is: aibohphobia. |
MathSierpinski's Triangle is an example of a self-repeating shape known as a fractal. Students will learn to create their own as well as extend this idea into other shapes, leading to interesting math-based artIn this video, students investigate a strange image that asks which has more sugar: a donut or a health drink? What about a salad? Using math and language arts skills, they'll determine if this image shows a complete picture or is misleading.
Twin Primes are prime numbers that have a difference of two. Mathematicians think there are an infinite number, but aren't sure yet. Have your students look for patterns as they dig into the Twin Prime Conjecture. |
God- The Great Geometer
Extracts from this document...
Introduction
God- The Great Geometer Since the dawn of mathematics, humans have tried to use it's methods to answer this question: What are we, and everything around us, made of? The ancients believed that the world was made up of four basic "elements": earth, water, air, and fire- "for the Creator compounded the world out of all the fire and all the water and all the air and all the earth, leaving no part of any of them nor any power of them outside"1. Around 350 BC, the ancient Greek philosopher Plato, in his book Timaeus, theorized that these four elements were all aggregates of tiny solids (in modern parlance, atoms). He went on to argue that, as the basic building blocks of all matter, these four elements must have perfect geometric form, namely the shapes of the five "regular solids" that so enamoured the Greek mathematicians -- the perfectly symmetrical tetrahedron, cube, octahedron, icosahedron, and dodecahedron. As the lightest and sharpest of the elements, said Plato, fire must be a tetrahedron. Being the most stable of the elements, earth must consist of cubes. Water, because it is the most mobile and fluid, has to be an icosahedron, the standard solid that rolls most easily. ...read more.
Middle
(Figure 1) A circle contained within that square has an area of half the original circle, so the light within it and the darkness around it are equal. Next, he separated the dry land from the ocean. To imitate that, you divide the inner circle into six parts by use of the compass. Inside it you can then draw a rhombus, a diamond shape made up of two equilateral triangles. The diagram now contains an inner circle and two concentric rings. Their respective areas, beginning at the centre, are 1, 2 and 3. This diagram has numbers and measures attached to it. The first, outer circle represents the whole universe and the 12 gods or astrological dominants that rule it. Its area is therefore 'factorial 12' - meaning the numbers from 1 to 12 multiplied together. From that you can calculate that the radius of the inner circle is 5,040, or factorial 7. This inner circle with radius 5,040 (or 1 x 2 x 3 x 4 x 5 x 6 x 7) represents the 'sublunary' world, or world beneath the moon, in which we live. It is called the cosmological circle or Holy City diagram. Even as recently as 1956, Paul Dirac, one of the founders of quantum mechanics, wrote "a physical law must possess mathematical beauty"5 When it ...read more.
Conclusion
We like easy and straightforward patterns. Humans attempt to comprehend the intricate geometry in nature, and each time a new type is discovered, we attempt to handle it, and point out imperfections- but god accepts them without doubt. So, we can see that god is a geometer, although he/she is much better at it that we are, or ever can be. As Kepler and the Greeks saw god as 'Geometer', and Newton saw God as 'Watchmaker', we now see god as a 'Computational Process'- the idea steadily gets more complex as the years go on and we realise the complexity of the universe. Figure 1 1 Plato, Timaeus. (2000). Hackett Publishing Co. (translated by Donald J. Zeyl ) pg15 2 Plato, Timaeus. (2000). Hackett Publishing Co. (translated by Donald J. Zeyl ) pg 25 3 4 Stewart, I. & Golubitsky, M. Fearful Symmetry. Is God a Geometer? (1992) Blackwell Publishers. Oxford. Pg 1 5 Stewart, I. & Golubitsky, M. Fearful Symmetry. Is God a Geometer? (1992) Blackwell Publishers. Oxford. Pg 1 6 Stewart, I. & Golubitsky, M. Fearful Symmetry. Is God a Geometer? (1992) Blackwell Publishers. Oxford. Pg 243 7 Stewart, I. & Golubitsky, M. Fearful Symmetry. Is God a Geometer? (1992) Blackwell Publishers. Oxford. Pg 244 8 So, what exactly is it that sets science, and religion at the opposite ends of the pole? Where religion accepts blindly that everything is created by God, and that is just how it is, science questions, what, when where, why, and is only then satisfied.
Religious believers would argue that God created humans with free will and all the suffering we get is deserved because humanity is in the debt of God. However non-believers would argue that the fact that babies suffer who have done nothing wrong is not fair and God cannot be all
Aristotle believed that to everything that exists there must be a cause, and if we cannot know that cause it must not exist. He does not intend to explain what he cannot know. Color, sound, tastes, touch, are all senses, a person is one who perceives existence so it must
Descartes even suggests that to doubt this would be to doubt he has hands or that the body that is there is his, for this would be thinking like a mad person, "The fact that I am here, sitting by the fire, wearing my dressing gown, holding this page in my hand and other things like that.
However, in order to discover truth we must be totally neutral. We cannot allow our senses to deceive us in the pursuit of truth. Descartes has a clear distinct picture of God, which he cannot, and will not doubt. He believes all other truths can be doubted, but not God.
the verses of the Qur'an is that the sky is made up of seven layers. "It is He Who created everything on the earth for you and then directed His attention up to heaven and arranged it into seven regular heavens. He has knowledge of all things." (The Qur'an, 2:29)
The further we trace the universe into the past, the faster we find its rate of expansion. As the rate of expansion increases, the curvature of the universe and the density of matter increase and the radius of the universe decreases, until a time is reached when the curvature of
Certain images of the uncreated cosmos often recur in different traditions: it may be represented as a void, as a state of chaos or of unformed elements, as a primeval sea, or a "cosmic egg" containing all things in embryonic form. |
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