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Sunday, June 27, 2004
Categories of knowing
Unknown unknowns: a five year old child
does not know that he does not know calculus. For him or her,
calculus is an `unknown unknown'.
Known unknowns: an older child know he that
he does not know calculus. For him or her, calculus is a
`known unknown'.
Unknown knowns: a high school student starts
to study calculus. He or she does not yet understand
calculus. For him or her, it is an `unknown known'. (You
will note that the focus of the first `unknown' shifts here
from lack of knowledge of the general to lack of knowledge of
the specific.)
Known knowns: after learning calculus, it becomes
a `known known': it is both known to exist and understood.
A part of life: after learning calculus and
using every day, a person may forget how much he or she knows. |
23 Hilbert's Problems
At the second International Congress of Mathematicians which was held in Paris in 1900, Hilbert posed 23 questions to the world mathematicians to solve in the next century. Some of them were general, such as the axiomatisation of physics (see on Euclid's elements and find out what axioms, and axiomatisation mean), but some were specific and had been solved reasonably quickly.
Hilbert's problems were designed to serve as examples for the kinds of problems which lead to mathematical researches that would advance mathematical knowledge and disciplines. Often such work would lead a researcher into something that was not obvious from the start.
Quite a few books have been written on these problems, and those mathematicians who managed to solve any of them are considered to be leading mathematicians of their day.
Just to get you a taste of the problems, here are the (very) simplified first ten of the total of twenty-three. You can also have a look at the original description of problems which are given in the translation of the original paper delivered by Hilbert in 1900.
Is there a number, which is larger than any finite number, between that of a countable set of numbers and the numbers of the continuum? [To think of a continuum, think of a number line - and ALL the numbers on it - without any gaps.] - This problem has been answered by Gödel.
Can it be proven that the axioms of logic are consistent? Gödel also answered this problem with his incompletness theorem which states that all consistent axiomatic formulations include some undecidable propositions. See the short history of Euclidean and non-Eucliden geometry to find more.
Give two tetrahedra that cannont be decomposed into congruent tetrahedra directly or by adjoining congruent tetrahedra. Dehn showed this could be done, but had to invent his own invariants (something that does not change under a set of transformations).
Find other geometries (with specific directions) which are close to Euclidean geometry. This has been solved by G. Hamel.
Generalisation of the Cauchy functional equation - was solved by John von Neumann.
If a is not 1 or 0, and b is irrational number, is a to the power of b transcendental number? [Transcendental number is a number that is not the root of any integer polynomial, meaning that it is not an algebraic number of any degree. Algebraic number is a number which is a root of a polynomial equation with rational coefficients.]
Prove or disprove Riemann hypothesis. If you solve this, you get $1 000 000. There are some other 'Millennium Problems' which bear a nice price, both in terms of an award and academic prestige. See the site of Clay Mathematics Insitute to learn more.
Construct generalisations of the reciprocity theorem of number theory. This hasn't been solved in general, although some cases have been solved by Gauss, Einstein and Kummer.
Does there exist a universal algorithm for solving Diophantine equations? - solved by Matiyasevic and Davis. |
There is a very good reason why e i and pi are related in such a way. It lets me know why it is convoluted. It is my decision that the form of matrix math must be ABCx=d instead of Ax=b and this is to remove the ambiguity of e and i and pi. I have heard it said that i or j as the imaginaries should not be used as it hides some deeper meaning or relationship and that is absolutely true.
In the nature of mathematics itself is the model of the universe and it serves for tribal societies and petty bureaucracy to mark their territory in two dimensions. The universe in 3D is more complex than that and it is quite obvious that though the math can be made to model it is not properly defined to give convenient accurate results to predict process.
Geometry started out as the Gaia-metry which is the measure of the Earth. The Earth is a spherical 2 dimensional surface and as such has a topology that is more complex than plane geometry.
The symbol and use of i arises due to the fact that certain aspects of relationships are not real and thus the imaginary is an appropriate name for the state. The interaction of systems with themselves indirectly is always a confusing process. Some things do interact with themselves to produce useful product. The formalization of mathematics without the present understanding of the universe has led to a very bad situation. It is much like the language we use to communicate. A kind of continuously more complex case statement of products. This is effective when you wish only to communicate a few things and serves well in nature as it can be kill, breed, eat, and it doesn't take much to solve that set of compound relationships in limited dimensional space.
I could just extend the quantum and relativistic equations as they exist with all their formalized flowery hieroglyphics, but it seems a bit of a joke on myself to take that which is easy to me and purposely make it difficult. So I will break with tradition and as a result I will speak a new language that represents the relationships that I know and can be proved to exist.
It is not possible to represent the relationship of gravity and charge and the forces of the nucleus with the system that exists. It simply serves to confuse the relationships of measure and product in a higher dimensional context. Everybody knows it is wrong, but nobody poses a useful solution , that I have seen. I am tired of looking, it just needs to be forked and rewritten to be more current and reflect the state of technology.
I started out to write a program and I still have that, but it is true that a project must be stripped apart and put back together ( again ) when it is obvious that foundational principles will not allow extension.
It seems that Linux and its foundation is a system of integrated extension and this is what makes it so usable. In the case of gimp and blender and other Linux programs, the fact that it has attachment to allow extension in the form of Python, shell script, perl, C, C++, or scheme is absolutely necessary for it to be able to adapt to new technology and understanding.
It doesn't matter to me if I am shunned and isolated by science because I do something that I feel is correct. This is what freedom is all about, the ability to change anything if that is what you want. So it will either work or it won't and I won't really know until I try.
So I know why eiπ, and for me it is y do the i π when it makes me lie in alge<bra|ket>. |
Math for Mystics: From the Fibonacci Sequence to Luna's Labyrinth to Golden Section & Other Secrets (New)
Description
A large portion of mathematics history comes to us directly from early astrologers who needed to be able to describe and record what they saw in the night sky. Notably, most mathematics history books refer to them as astronomers but in those days astronomers and astrologers were one and the same. Everyone needed maths: whether you were the king's court astrologer or a farmer marking the best time for planting, timekeeping and numbers really mattered. Mistake a numerical pattern of petals and you could poison yourself. Lose the rhythm of a sacred dance or the metre of a ritually told story and the intricately woven threads that hold life together were spoiled. Ignore the celestial clock of equinoxes and solstices, and you'd risk being caught short of food for the winter.
Math for Mystics unveils and demystifies some of the many concepts our distant ancestors knew and used, based on long generations of observation and record-keeping, sky-watching, folk wisdom and ever-more-complex calculations. Shesso' s friendly tone, delightful maths lore, and clear information makes the maths go down easy in the marvelous book which begins with the simplest lunary and planetary mathematics, and then tackles the most enigmatic of numerical esoterica such as magical squares, Golden Sections, Luna's Labyrinth, and Benjamin Franklin's favourite way to pass the time, the Durer Square |
Posts
I'm not a big fan of these various math tricks (such as how to multiply any two numbers between 11 and 19 or how to square some special numbers ending in certain digits) unless they increase students' understanding of the number system or math principles. Often they're just another thing to memorize that fades away quickly... at least with me.
But some people like them, and that's fine with me.
One particular kind of math "trick" is where you are told to think of a number, then do all sorts of manipulations with it, and then in the end... they know your number! How is that?
Here's one for you... it's neat, easy, amusing, and explains in the end how it works.
I just finished reading the third volume of Calculus Without Tears... worksheets designed to teach calculus concepts before high school, for those interested.
So I've gotten a refresher on calculus... which I've always loved, and so that's why in today's post I hope to explain to you ONE little point from calculus - a fascinating and important one - that is.... THE EXPONENTIAL FUNCTION and THE NUMBER E.
First you need to know a calculus concept, and something about e: The steeper the graph of a function, the faster it's growing (and vice versa) We can measure the steepness of a function at any point by drawing a tangent, and checking the slope of the tangent. e is a VERY special irrational number with the approximate value 2.718281828459...
Simple enough, huh? Okay, so here we go:
There is a function.... whose growth rate at any point is always its value at that point!
The function ex
The 'steepness' of the tangent shows us the growth rate of the function. Tangent dr…
I seem to get so involved in writing about math topics that I forget that I'm supposed to also write about my website updates. So here goes, some recent updates and additions:
1) I've reorganized the links and stuff on the reviews page and on the math curriculum reviews page, for easier navigating. (Not that my site couldn't use more improvement in that navigation... hopefully some day.)
2) Added a page for Thinkwell CDs. If you've used those, please leave a review.
My child is dealing identifying Triangles as SAS,SSS,AAS,ASA and HL Theroms. She is also having trouble with the Flow proofs and Column proofs on explaing why 2 triangles are congruent.
All of the theorems about proving that triangles (or other shapes) are congruent can be "translated" into a drawing problem:
If I have my 'secret' triangle and I give you THESE pieces of information, can you reproduce my triangle? Can you do that every time, no matter what my triangle?
We can describe a triangle using 6 pieces of information: the legths of the three sides, and the measures of the three angles. But you don't need all of those to be able to draw my secret triangle.
Can you draw a copy of my triangle if I tell you that....
my triangle has a side 5 cm long, another side 6 cm long, and the angle between those sides is 29 degrees (I've given you S - A - S)? my triangle has a 30 degree angle, a 60 degree angle, and a 90 degree angle (I've given you A - A - A)?
Summer is approaching, and you all my readers probably have all kinds of different plans. If you're a homeschooler, maybe you continue homeschool, maybe not. Some people take a vacation, some stay at home.
Anyway, however it might fit into your plans, I'd like to recommend a nice (British) math website Count On (Counton.org).
In my mind, Count On website is almost like a mathematical vacation, full of enjoyment with mathematical themes. Even the logo looks like the site is supposed to make you feel happy.
I especially like their "mathzines", or math related online magazines. There are two different ones: Meenie Minus for smaller kids, and Kaleidoscope for middle school age. The magazines have engaging, short games and activities,…
I hope you get the idea; in elementary books you often see these kind of missing addend problems with a little box:
15 + = 30 or 234 + = 700 or + 1,923 = 5,000.
Well, I decided for the x over the box! I don't think solving missing addend problems with subtraction is too difficult to learn on fourth grade; after all, students have been working with addition and subtraction connection from 1st grade on, right?
In my own old schoolbooks I actually see x from 3rd grade on.
I made several problems with charts such as
Write a missing addend sentence and a subtraction sentence to solve it
I want to blog again on Julie's great Living Math website. You can really get help there if you're one of those who don't care for math, or even hate math, but want to get rid of that feeling.
She has suggestions on how to start teaching "living math" - teaching math in a way that makes connections to real life, takes away the dryness of it, takes away the 'kill' from "drill and kill" (note that drill in itself doesn't have to be always bad - drill is a tool amongst many), etc.
On this page, Julie has book suggestions to many different situations... Consider finding one or a few (from library or bookstore). Reading a math book that's not a school book can do SO much good! For your kids too!
As you probably know, I wish everyone would get to know some of the interesting, fascinating, fun, curious aspects of mathematics. Or, get to know a bit of math history. You don't get those in school books. Or, finally learn why things work.
I felt very inspired by the story I posted last time, about the 12 kids who wanted to have a class to study arithmetic, and then finished 6 years worth of school math in 20 weeks - meaning they had 20 contact hours with the teacher, and who knows how many hours spent on homework.
They studied the four basic operations, fractions, decimals, percent, and square root. I figure they didn't go into algebra - just arithmetic.
Like it said in the article, the material itself is not incredibly difficult, once your mind has developed to handle these concepts.
The story shows how much motivation (and subsequent hard work) can do. So how could we increase our students' motivation to study math?
I feel it is important to PREVENT the student's feeling of, "I hate math" or "I don't like math" that the traditional math instruction seems to produce. Little kids usually like learning about different things. Somehow we must keep that enthusiasm going strong.
Just found this... decided that it's good reading for all. When you have lots of motivation, like these kids did, you can learn and will learn - math or anything else. It may not be easy at all times, but the motivation will make you persistent so you will eventually learn.
QUOTE "Because everyone knows," he answered, "that the subject matter itself isn't that hard. What's hard, virtually impossible, is beating it into the heads of youngsters who hate every step. The only way we have a ghost of a chance is to hammer away at the stuff bit by bit every day for years. Even then it does not work. Most of the sixth graders are mathematical illiterates. Give me a kid who wants to learn the stuff -- well, twenty hours or so makes sense."
I wanted to briefly touch some more on this topic of "grind" or daily grind that learning math might sometimes become that I mentioned briefly last time.
I didn't mean to imply that learning to add, subtract, multiply, and divide whole numbers, decimals, fractions, percents, numbers with exponents, and integers has to be such a boring task. You can avoid that type of feeling.
One way: Make connections between the concepts. Don't make math appear as fairly separate compartments of "fractions" and "decimals" and "percents" and "geometry".
I sometimes wonder how students feel when every year (on 4th, 5th, 6th, 7th, and 8th grade) they have a chapter on fractions, a chapter on decimals, a chapter on whole numbers, a chapter on geometry... Almost the same thing over and over.
We need to make sure they can see how these things connect.
An example: A jogging track is 3.5 km long. If you jog 2/5 of it, how far was that? How many percent of t… |
Two middle school students in Zhengzhou, central China's Henan Province managed to transform dull functions into beautiful figures, depicting hearts, apples and even a little girl with pigtails, according to Zhengzhou Evening Post.
Their math teacher, Yan Lina, was surprised when two of her students created a heart-shaped figure using functions. Yan wrote about the accomplishment on her blog, remarking that math looks so beautiful that way. Yan's blog post sparked a discussion on the best way to teach math. The method of combining mathematical, rational thinking with art and creativity may worth a try, said Yan |
The study of fractals has recently become a hot topic. Fractals can
model real world phenomena such as coastlines, trees, and crystals. In
this discussion, we will focus on some of the most basic ideas of two
dimensional fractals.
Affine Transformations
So far our transformations have all been linear transformations. We now
define a special transformation that is not a linear transformation.
You may be asking what affine transformations have to do with fractals.
We will consider successions of several affine transformation and see that the
total image under these produce fractal images.
Consider the line segment of length 1. Take out the middle third and
replace it with a square (without bottom) of side length 1/3. Then replace
the middle third of each horizontal segment with a square (without bottom of
side length 1/9). We can continue this process indefinitely to get a
fractal.
The big challenge is to figure out the transformations that create this
fractal. We notice that once a vertical line segment is created, it
remains in the picture. Hence we need to determine how the horizontal
segments are transformed. We see that each horizontal segment is converted
into five separate segments. We will need five different affine
transformations for each. Consider the first set (from the blue graph to
the green graph. We will need to first contract the segment by a factor of
three. The matrix that does this is
For
the horizontal line segments we translate by
b1 =
(0,0) b2=
(1/3, 1/3) b3 = (2/3, 0)
For
the vertical lines we need to rotate the segment by an angle of p/2.
The matrix that does this is
then
the vertical line segments are translated by the vectors
c1 = (1/3, 0) c2= (2/3, 0)
We
can summarize this by saying that there are five affiine transformations given
by
T1(x) = Ax T2(x)
= Ax + b1T3(x)
= Ax + b3
T4(x)
= BAx + c1T5(x)
= BAx + c2
For the next stage, we again can find the transformations needed.
Notice that each of the transformations will be compositions of a
translation followed by multiplication by A or AB
and a final translation. |
Learn from a vibrant community of students and enthusiasts,
including olympiad champions, researchers, and professionals.
Fallacies in mathematics
Mathematicians can be wrong at times and and sometimes this errors lead to interesting results.And sometimes is some intentional effort to find foolproof solutions and really some outstanding proofs that really amaze people by unbelievable conclusions and try to show that mathematics is pointless(no,,actually not coz every fallacy is based upon some error which only a good mathematican can notice).This is what is meant by fallacies
So under this discussion u can share some of fallacies u know or mayb u hav derived urself........
Some of the fallacies i know of till date are:-
The fallacy of the isosceles triangle.
The fallacy of the right angle.
The fallacy that 4 = o.
The fallacy that +1 = -1.
The fallacy that all lengths are equal.
The fallacy that the sum of the Squares on two sides of a triangle is never less than the square on the third.
7.The fallacy that every 2 consecutive numbers are equal
8.The fallacy of the radius of a circle is indeterminate
The fallacy that the four points of intersection of two conics are collinear.
and many more....list is endless...in every branch of mathematics
really learning some mathematical fallacies is fun....u can share some |
Georg Cantor A Genius Out of Time Term Paper
Excerpt from Term Paper :
Georg Cantor: A Genius Out of Time
If you open a textbook, in high school or college, in the first chapter you will be introduced to set theory and the theories of finite numbers, infinite numbers, and irrational numbers. The development of many theories of math took years upon years and the input of many mathematicians, as in the example of non-Euclidean geometry. This was the case with most math theories, however set theory was primarily the result of the work of one man, Georg Cantor. In his time, these hypotheses were considered greatly controversial by other mathematicians. However, now they are an integral part of the study of mathematics. Georg Cantor received more criticism than complement in his time and it eventually led him to mental illness. However, one must remember that many other things, once thought to be controversial are now considered to be fact. Georg Cantor should be considered one of the pioneers of modern mathematical theory.
Mathematics can be considered a language in its own right. It is the language that we use to describe our world. Math tells us vital information such as how big, how fast, and describes the relationship between two things. A set is a group of things that can be treated as a single unit. There are two ways to describe a set. The first method is to simply list the elements of a set. The second way is to describe the members of a set and define what characteristics determine which elements will be included or not be included in a particular set.
Family and Early Life
Georg Ferdinand Ludwig Phillip Cantor was born on March 3, 1845 in St. Petersburg, Russia. The family lived in Russia for eleven years until his father's failing health forced the family to move to the milder climate of Frankfort, Germany in 1856. It was here that Cantor would spend the rest of his life. Georg was the eldest of the three children. His father was a wealthy merchant, Georg Walsematr Cantor and his mother was a famous artist, Maria Boehm. The other children had exceptional artistic talents like their mother. Georg's brother, Constantine, was an army officer and also a fine pianist. His sister, Sophie Nobiling, was an accomplished designer. However, Georg excelled in Math (Johnson, 1997). Georg Cantor came from a family with a wealth of talent in math, physics, and philosophy. His brothers and sisters also displayed talent in math.
Cantor had a strict religious upbringing, and he carried a strong religious sense all through his life. His father was Jewish, but later converted to Protestantism around the time of Georg's birth. His mother was a devout Catholic. This difference of religious opinions did not sway Georg's own beliefs and he became a knowledgeable theologian as well as mathematician (Johnson, 1997). It was no doubt that this diverse religious background made him the type to question his surroundings and stand by his ideas, even when everyone else said he was wrong.
Education
Georg attended several private schools in Frankfurt, and in 1859, entered the distinguished Grossherzoglich Hessiche Provinzialrealschule in Darmstadt. He left this institution in 1860 with high recommendations in mathematics. His father discouraged the study of math due to the fact that he wished him to become an engineer, a job that paid considerably more than mathematics. He originally attended Grossherzogliche Hoehere Gewerbeschule (Grand-Ducal Higher Polytechnic, later changed to Technische Hochschule) at Darmstadt following his father's wishes and studying Engineering. Later, when Georg convinced his father that his heart was truly in math, his father relented and he began the study of Mathematics in 1862 (Johnson, 1997).
Cantor began his higher studies in Zurich, the fall of 1862. He left in Spring of 1963 due to the death of his father. In Fall of 1863, he entered the University of Berlin to study mathematics, physics, and philosophy. The University of Berlin was home to three famous mathematicians Ernst Eduard Kummer, Karl W.T. Weierstrass and Leopold Kronecker. These three men made the University of Berlin one of the top schools for the study of mathematics in the entire world. The student population was small and therefore the students were in close contact with these three great minds. Cantor was heavily influenced by the works of Weierstrass. Kronecker was also a great influence, but would later become one of his greatest critics. It was customary in Germany, at the time, to study at another University for a period time. He studied at the Cantor attended the University of Gottingen during the summer term of 1866 (Johnson, 1997). Cantor received the degree of doctor on December 14, 1867. His dissertation was based on a study of the Disquisitiones Arithmeticae of Carl Friedrich Gauss, another contemporary mathematician of his time, and on the number theory of Adrien-Marie Legendre (Johnson, 1997).
Cantor's thesis centered around one of the ideas that Gauss had left aside concerning the solutions in integers x, y and z of the indeterminate equation ax2 + by2 + cz2 = 0, where a, b and c are any given integers. The full title of the thesis was "De aequationibus secundi gradus indeterminatis" ("On indeterminate equations of the second degree") and as was customary, dedicated to his guardians, Eduard Flersheim and Bernhard Horkheimer. As was also the custom for Doctoral candidates, Cantor also defended three theses against opposing doctors. All fo which were translated from the original Latin, the theses were "In arithmetic merely arithmetic methods far surpass analytic methods," "Since it is disputed, the question of the absoluteness of space and time is more important than its solution" and "In mathematics the art of proposing a question must be held of higher value than solving it." (Johnson, 1997). Cantor's early works were considered to be excellent among his peers, but no on ever suspected the genius that would emerge in his later writings.
In Spring of 1869, Cantor began his career as a Privatdozent at the University of Halle on the basis of his paper "De transformatione formarum ternariarum quadraticarum" ("On the transformation of ternary quadratic forms") (Johnson,1997). Cantor specialized in Number theory. Cantor's became Extraordinarius at Halle in 1872 and Ordinarius in 1879. He was released from his official duties in 1905 and resigned his post altogether in 1913.
In 1874,Cantor published his first paper on the Theory of Sets. In that same year, he married Vally Guttman in the summer. They had two sons and four daughters, none of whom, it might be noted were gifted in mathematics. One of Cantor's daughters, Frau Gertrud Vahlen, was an important source of information for the biography of Cantor by A.A. Fraenkel, which was written in the latter part of his life. When Cantor received his professorship on the Theory of Sets in 1879 (Breen, 2000), not everyone agreed with and readily accepted Cantor's ideas. His lectures on the theory were not well attended. They criticized his set theory and believed that only numbers were integers and that negatives, fractions, and imaginary numbers did not belong in the field of mathematics proper, but rather as a type of metaphysics. Cantor's career at Halle would not be considered to be a success by many standards. He produced few researchers and doctorate candidates, unlike those who had influenced him. Cantor stood by his theories and now is considered to be one of the greatest mathematicians in history (Breen, 2000).
In 1874, Cantor wrote a paper which appeared in Crelle's Journal. The paper proposed that there are two different orders of infinity. Cantor showed that the set of real numbers can be put in one-to-one correspondence with the set of natural numbers. However the real numbers cannot be put into one-to-one correspondence with the set of natural numbers. This theory in itself raised no eyebrows and stirred no controversy. The fact was established in this paper that the set of real numbers is larger than the set of natural numbers. Cantor was the first to use nested intervals to proved that the set of real numbers is not countable rather than use his diagonal processes produced in his later work.
Cantor commented on his theory as such, (George Cantor in Rucker, 1995).
Cantor's next paper appeared in 1878 with the central idea of one-to-one correspondence and a number of theorems concerning such correspondences given along with suggestions for classifying sets based on these assumptions. This paper contains the proof that the set of rational numbers is countable. Cantor used the word "power" ("Machtigkeit") for the first time to establish that two sets, which can be put in one-to-one correspondence with each other, have the same power. Cantor discusses in some length sets as having the smallest infinite power and demonstrates that…
Related Documents:
Mathematics
George Cantor
The purpose of the paper is to develop a concept of the connection between mathematics and society from a historical perspective. Specifically, it will discuss the subject, what George Cantor accomplished for mathematics and what that did for society. George Cantor's set theory changed the way mathematicians of the time looked at their science, and he revolutionized the way the world looks at numbers.
George Cantor was a brilliant mathematician |
Meta
Math
Another of my favourite functions if the Gamma function, , the continuous generalization of the factorial. While it grows rapidly for positive reals, it has fun poles for the negative integers and is generally complex. What happens when you iterate it?
First I started by just applying it to different starting points, . The result is a nice fractal, with some domains approaching 1, and others running off to infinity.
Here I color points that go to infinity in green shades on the number of iterations before they become very large, and the points approaching 1 by . Zooming in a bit more reveals neat self-similar patterns with alternating "beans":
In the outside regions we have thin tendrils stretching towards infinity. These are familiar to anybody who has been iterating exponentials or trigonometric functions: the combination of oscillation and (super)exponential growth leads to the pattern.
OK,that was a Julia set (different starting points, same formula). What about a counterpart to the Mandelbrot set? I looked at where c is the control parameter. I start with and iterate:
Zooming in shows the same kind of motif copies of Julia sets as we see in the quadratic Mandelbrot set:
In fact, zooming in as above in the counterpart to the "seahorse valley" shows a remarkable similarity.
During a recent party I got asked the question "Since has an infinite decimal expansion, does that mean the collected works of Shakespeare (suitably encoded) are in it somewhere?"
My first response was to point out that infinite decimal expressions are not enough: obviously is a Shakespeare-free number (unless we have a bizarre encoding of the works in the form of all threes). What really matters is whether the number is suitably random. In mathematics this is known as the question about whether pi is a normal number.
Where are the Shakespearian numbers?
This led to a second issue: what is the distribution of the Shakespeare-containing numbers?
We can encode Shakespeare in many ways. As an ASCII text the works take up 5.3 MB. One can treat this as a sequence of 7-bit characters and the works as 37,100,000 bits, or 11,168,212 decimal digits. A simple code where each pair of digits encode a character would encode 10,600,000 digits. This allows just a 100 character alphabet rather than a 127 character alphabet, but is likely OK for Shakespeare: we can use the ASCII code minus 32, for example.
If we denote the encoded works of Shakespeare by , all numbers of the form are Shakespeare-containing.
They form a rather tiny interval: since the works start with 'The', starts as "527269…" and the interval lies inside the interval , a mere millionth of . The actual interval is even shorter.
But outside that interval there are numbers of the form , where is a digit different from the starting digit of and anything else. So there are 9 such second level intervals, each ten times thinner than the first level interval.
This pattern continues, with the intervals at each level ten times thinner but also 9 times as numerous. This is fairly similar to the Cantor set and gives rise to a fractal. But since the intervals are very tiny it is hard to see.
One way of visualizing this is to assume the weird encoding , so all numbers containing the digit 3 in the decimal expansion are Shakespearian and the rest are Shakespeare-free.
Distribution of Shakespeare-free numbers in the unit interval, assuming Shakespeare's collected works are encoded as the digit "3".
The fractal dimension of this Shakespeare-free set is . This is less than 1: most points are Shakespearian and in one of the intervals, but since they are thin compared to the line the Shakespeare-free set is nearly one dimensional. Like the Cantor set, each Shakespeare-free number is isolated from any other Shakespeare-free number: there is always some Shakespearian numbers between them.
In the case of the full 5.3MB [Shakespeare] the interval length is around . The fractal dimension of the Shakespeare-free set is , for some tiny . It is very nearly an unbroken line… except for that nearly every point actually does contain Shakespeare.
We have been looking at the unit interval. We can of course look at the entire real line too, but the pattern is similar: just magnify the unit interval pattern by 10, 100, 1000, … times. Somewhere around $10^{10,600,000}$ there are the numbers that have an integer part equal to . And above them are the intervals that start with his works followed by something else, a decimal point and then any decimals. And beyond them there are the numbers…
Shakespeare is common
One way of seeing that Shakespearian numbers are the generic case is to imagine choosing a number randomly. It has probability of being in the level 1 interval of Shakespearian numbers. If not, then it will be in one of the 9 intervals 1/10 long that don't start with the correct first digit, where the probability of starting with Shakespeare in the second digit is . If that was all there was, the total probability would be . But the 1/10 interval around the first Shakespearian interval also counts: a number that has the right first digit but wrong second digit can still be Shakespearian. So it will add probability.
Another way of thinking about it is just to look at the initial digits: the probability of starting with is , the probability of starting with in position 2 is (the first factor is the probability of not having Shakespeare first), and so on. So the total probability of finding Shakespeare is . So nearly all numbers are Shakespearian.
This might seem strange, since any number you are likely to mention is very likely Shakespeare-free. But this is just like the case of transcendental, normal or uncomputable numbers: they are actually the generic case in the reals, but most everyday numbers belong to the algebraic, non-normal and computable numbers.
It is also worth remembering that while all normal numbers are (almost surely) Shakespearian, there are non-normal Shakespearian numbers. For example, the fractional number is non-normal but Shakespearian. So is We can throw in arbitrary finite sequences of digits between the Shakespeares, biasing numbers as close or far as we want from normality. There is a number that has the digits of plus Shakespeare. And there is a number that looks like until Graham's number digits, then has a single Shakespeare and then continues. Shakespeare can hide anywhere.
In things of great receipt with case we prove,
Among a number one is reckoned none.
Then in the number let me pass untold,
Though in thy store's account I one must be -Sonnet 136
As iteration formula I choose , where c is a multiplicative constant. Iterating some number like 1 and plotting its fate produces the following "Mandelbrot set" in the c-plane – the colours here do not denote the time until escape to infinity but rather where in the complex plane the point ended up, as a function of c. In a normal Mandelbrot set infinity is an attractive fixed point; here it is just one place in the (extended) complex plane like any other.
"Mandelbrot set" for the hyperbolic tanh function tanh(cz).
The pinkish surroundings of the pattern represent points attracted to the positive solution of . There is of course a corresponding negative solution since tanh is antisymmetric: if z is an attractive fixed point or cycle, so is -z. So the dynamics is always bistable.
Incidentally, the color scheme is achieved by doing a stereographic projection of the complex plane onto a sphere, which is then fitted into the RBG cube. Infinity corresponds to (0.5,0.5,1) and zero to (0.5,0.5,0) – the brownish middle of the Mandelbrot set, where points are attracted towards zero for small c.
Sphere used to stereographically map complex numbers to colors.
Another property of tanh is that the function has singularities wherever for integer . Since Great Picard's Theorem, that means that in the vicinity of those points it takes on nearly all other values in the complex plane. So whatever the pattern of the corresponding Julia set is, it will repeat itself near there (including images of the image, and so on).This means that despite most z points being attracted towards zero for c-values inside the unit circle, there will be a complex stitching of undefined points since they will be mapped to infinity, or are preimages of points that get mapped there.
Zooming into the messy regions shows that they are full of circle-cusp areas where there is a periodic attractor cycle. Between them are the regions where most of the z-plane where the Julia sets live is just pure chaos. Thanks to various classic theorems in the theory of complex iteration we know that if the Julia set has non-empty interior it is the entire complex plane.
Walking around the outside edge of the boring brown circle gives a fun sequence of patterns. At there are two real fixed points and a straight line border along the imaginary axis. This line of course contains the singularity points where things get sent to infinity, and near them the preimages of all the other singularities on the line: dramatic, but visually uninteresting.
Tanh 'Julia set' for c=1.
As we move along the circle towards more imaginary c, there is a twisting of the border since each multiplication by c corresponds to a twist: it is now a fractal spiral covered by little spirals. As the twisting gets stronger, the spirals get bigger and wilder (especially when we are very close to the unit circle, where the dynamics has a lot of intermittency: the iterates almost but not quite gets stuck close to certain points, speed away, and then return to make rather elliptic spirals).
When we advance towards a cuspy border in the c-plane we see the spirals unfold into long twisty tentacles just before touching, turning into borders between chains of periodic domains.
Tanh 'Julia set' for c=1.1*exp(0.6*i).
But then the periodic domains start to snake out, filling the plane wildly.
Tanh 'Julia set' for c=1.1*exp(0.6594*i).
until we get a plane-filling, ergodic Julia set with no discernible structure. For some c-values there are complex tesselations of basins of attraction, and quite often some places are close enough to weakly repelling fixed points to produce small circular false basins of attraction where divergence is slow.
Tanh 'Julia set' for c=1.1*exp(0.66*i).
One way of visualizing this is to make a bifurcation diagram like we do for real iteration. Following a curve we plot where iterates end up projected along some line (for example their real or imaginary part, or some combination). To make structure stand out a bit more I decided to color points after where in the whole plane they are, producing a colorful diagram for r=1.1:
Note how spirals unfold until they touch each other, forming periodic domains or exploding across the entire plane, making a chaotic full-plane attractor… which often blinks into complex patterns of periodic domains only to return to chaos.
I like the idea of a thanksgiving day, leaving out all the Americana turkeys, problematic immigrant-native relations and family logistics: just the moment to consider what really matters to you and why life is good. And giving thanks for intellectual achievements and tools makes eminent sense: This thanksgiving Sean Carroll gave thanks for the Fourier transform.
These days a razor in philosophy denotes a rule of thumb that allows one to eliminate something unnecessary or unlikely. Occam's was the first: William of Ockham (ca. 1285-1349) stated "Pluralitas non est ponenda sine neccesitate" ("plurality should not be posited without necessity.") Today we usually phrase it as "the simplest theory that fits is best".
Principles of parsimony have been suggested for a long time; Aristotle had one, so did Maimonides and various other medieval thinkers. But let's give Bill from Ockham the name in the spirit of Stigler's law of eponymy.
Of course, it is not always easy to use. Is the many worlds interpretation of quantum mechanics possible to shave away? It posits an infinite number of worlds that we cannot interact with… except that it does so by taking the quantum mechanics formalism seriously (each possible world is assigned a probability) and not adding extra things like wavefunction collapse or pilot waves. In many ways it is conceptually simpler: just because there are a lot of worlds doesn't mean they are wildly different. Somebody claiming there is a spirit world is doubling the amount of stuff in the universe, but that there is a lot of ordinary worlds is not too different from the existence of a lot of planets.
Simplicity is actually quite complicated. One can argue about which theory has the fewest and most concise basic principles, but also the number of kinds of entities postulated by the theory. Not to mention why one should go for parsimony at all.
But in day-to-day life Occam works well, especially with a maximum probability principle (you are more likely to see likely things than unlikely; if you see hoofprints in the UK, think horses not zebras). A surprising number of people fall for the salient stories inherent in unlikely scenarios and then choose to ignore Occam (just think of conspiracy theories). If the losses from low-probability risks are great enough one should rationally focus on them, but then one must check one's priors for such risks. Starting out with a possibilistic view that anything is possible (and hence have roughly equal chance) means that one becomes paranoid or frozen with indecision. Occam tells you to look for the simple, robust ways of reasoning about the world. When they turn out to be wrong, shift gears and come up with the next simplest thing.
Simplicity might sometimes be elegant, but that is not why we should choose it. To me it is the robustness that matters: given our biased, flawed thought processes and our limited and noisy data, we should not build too elaborate castles on those foundations.
The minimal example would be if each risk had 50% independent chance of happening: then the observable correlation coefficient would be -0.5 (not -1, since there is 1/3 chance to get neither risk; the possible outcomes are: no event, risk A, and risk B). If the probability of no disaster happening is N/(N+2) and the risks are equal 1/(N+2), then the correlation will be -1/(N+1).
I tried a slightly more elaborate model. Assume X and Y to be independent power-law distributed disasters (say war and pestillence outbreaks), and that if X+Y is larger than seven billion no observers will remain to see the outcome. If we ramp up their size (by multiplying X and Y with some constant) we get the following behaviour (for alpha=3):
(Top) correlation between observed power-law distributed independent variables multiplied by an increasing multiplier, where observation is contingent on their sum being smaller than 7 billion. Each point corresponds to 100,000 trials. (Bottom) Fraction of trials where observers were wiped out.
As the situation gets more deadly the correlation becomes more negative. This also happens when allowing the exponent run from the very fat (alpha=1) to the thinner (alpha=3):
I like the phenomenon: it gives us a way to look for anthropic effects by looking for suspicious anticorrelations. In particular, for the same variable the correlation ought to shift from near zero for small cases to negative for large cases. One prediction might be that periods of high superpower tension would be anticorrelated with mishaps in the nuclear weapon control systems. Of course, getting the data might be another matter. We might start by looking at extant companies with multiple risk factors like insurance companies and see if capital risk becomes anticorrelated with insurance risk at the high end.
If you take positive independent integers from some distribution and generate ratios , then those ratios will have a distribution that is a convolution over the rational numbers:
One can of course do the same for non-independent and different distributions of the integers. Oh, and by the way: this whole thing has little to do with ratio distributions (alias slash distributions), which is what happens in the real case.
The authors found closed form solutions for integers distributed as a power-law with an exponential cut-off and for the uniform distribution; unfortunately the really interesting case, the Poisson distribution, doesn't seem to have a neat closed form solution.
In the case of a uniform distributions on the set they get .
The rational distribution g(a/(a+b))=1/max(a,b) of Trifonov et al.
They note that this is similar to Thomae's function, a somewhat well-known (and multiply named) counterexample in real analysis. That function is defined as f(p/q)=1/q (where the fraction is in lowest terms). In fact, both graphs have the same fractal dimension of 1.5.
It is easy to generate other rational distributions this way. Using a power law as an input produces a sparser pattern, since the integers going into the ratio tend to be small numbers, putting more probability at simple ratios:
If we use exponential distributions the pattern is fairly similar, but we can of course change the exponent to get something that ranges over a lot of numbers, putting more probability at nonsimple ratios where :
The rational distribution of two convolved Exp[0.1] distributions.Not everything has to be neat and symmetric. Taking the ratio of two unequal Poisson distributions can produce a rather appealing pattern:
Rational distribution of ratio between a Poisson[10] and a Poisson[5] variable.Of course, full generality would include ratios of non-positive numbers. Taking ratios of normal variates rounded to the nearest integer produces a fairly sparse distribution since high numerators or denominators are rare.
The rational numbers do tend to induce a fractal recursive structure on things, since most measures on them will tend to put more mass at simple ratios than at complex ratios, but when plotting the value of the ratio everything gets neatly folded together. The lower approximability of numbers near the simple ratios produce moats. Which also suggests a question to ponder further: what role does the über-unapproximable golden ratio have in distributions like these?
In any case, there is something simultaneously ugly and exciting when neat patterns in math just ends for no apparent reason.
Another good example is the story of the Doomsday conjecture. Gwern tells the story well, based on Klarreich: a certain kind of object is found in dimension 2, 6, 14, 30 and 62… aha! They are conjectured to occur in all dimensions. A branch of math was built on this conjecture… and then the pattern failed in dimension 254. Oops.
It is a bit like the opposite case of the number of regular convex polytopes in different dimensions: 1, infinity, 5, 6, 3, 3, 3, 3… Here the series start out crazy, and then becomes very regular.
The volume of a unit sphere increases with dimension until , and then decreases. Leaving the non-intuitiveness of why volumes would shrink aside, the real oddness is that the maximum is for a non-integer dimension. We might argue that the formula is needlessly general and only the integer values count, but many derivations naturally bring in the Gamma function and hence the possibility of non-integer values.
Another association is to this integral problem: given a set of integers , is the integral ? As shown in Moore and Mertens, this is NP-complete. Here the strangeness is that integrals normally are pretty well behaved. It seems absurd that a particular not very scary trigonometric integral should require exponential work to analyze. But in fact, multivariate integrals are NP-hard to approximate, and calculating the volume of a n-dimensional polytope is actually #P-complete.
We tend to assume that mathematics is smoother and more regular than reality. Everything is regular and exceptionless because it is generated by universal rules… except when it isn't. The rules often act as constraints, and when they do not mesh exactly odd things happen. Similarly we may assume that we know what problems are hard or not, but this is an intuition built in our own world rather than the world of mathematics. Finally, some mathematical truths maybe just are. As Gregory Chaitin has argued, some things in math are irreducible; there is no real reason (at least in the sense of a comprehensive explanation) for why they are true.
Mathematical anti-beauty can be very deep. Maybe it is like the insects, rot and other memento mori in classical still life paintings: a deviation from pleasantness and harmony that adds poignancy and a bit of drama. Or perhaps more accurately, it is wabi-sabi. |
Pythagoras (572-492BC), a student of Thales, is now one ofthe most famous mathematicians of all time. According to David E. Smith. "Pythagoras said have discovered the fifth and the octave of anote that can be produced on the same string by stopping at 2/3 and 1/2 of its length, respectively." He based his metaphysic andeverything in a number. He believed all universe was based in numbers; he thought that musical proportion governed the motion of the planets andthat "the heavenly bodies in their motion through space gave out harmonious sounds. He is perhaps best known for the theorem named in hishonor. I think he was really intelligent, but for me he was kind of crazy because no it is true that we need the number to do a lot ofthings, but he numbers are no everything; he sometimes compare them with god so I did not know him. The only thing I know, he was a genius.Pythagoras believed that the good life it he contemplative life, therefore, an odious sinful habit. Today that would be a goodlife-style, a contemplative life, because while you are expecting to learn always new you could succeed in the life.
For me all of thephilosophers left something new for us, and we choose how to use them. For example, the Pythagoras theorem is really usefull for us in geometry |
It occurred to Albert Einstein that gravity could be modeled as a geometric phenomenon. Instead of saying that a particle was deflected from a straight trajectory by the force of gravity, one might say that that gravity affects space and time in such a way as to alter the notion of what a straight trajectory is. To this end Einstein began to adapt the mathematics of Riemann's metric geometry to describe the physical world. One night, Einstein sat in his study, grinding through pages and pages of coordinate transformations. His hand hurt from the writing because each term in an equation had to be preceded by multiple capital-sigma summation signs. "I'm so sick of these these stupid sigmas. I wish I could just leave them out," Einstein thought. He looked back down at the page, paused for a moment, then scratched out all the summation signs. Because the terms only ever involved summation over coordinate indices, and the sums were invariably over those indices that appeared exactly twice, removing the sigmas did nothing to obscure the meaning of the equations, but did make them easier to read and write. "So much better," Einstein said to himself. "Einstein, you're a genius!" |
Thursday, July 12, 2012
Fortune Favors the Brave: Eratosthenes and the Circumference of the Earth
One of the most persistent myths is the
belief that Christopher Columbus sailed to the New World in order to prove that
the earth was round. In fact, learned people had known this for several
centuries before he was born. People had observed that the earth's shadow
during a lunar eclipse is always circular. And they knew that the only object
that casts a circular shadow from every direction is a sphere.
And the natural next question to be
answered was how big that sphere is. Greek scientists proposed many answers
based on various conjectures. Remarkably, one of them actually figured it out.
In the 3rd century B.C.,
Eratosthenes, a native of Alexandria Egypt, realized something as simple as a
shadow could be used to solve this problem. He knew that on the summer
solstice, the sun was directly overhead at the southern Egyptian city of Syene
(modern day Aswan). In Alexandria, at the same time, the sun is at a 1/50th
declination from being directly overhead (which could be accurately measured
using shadows).
Taking just that information, Eratosthenes knew that he could
calculate the circumference of the earth as long as he knew how far apart
Alexandria and Syene are. Estimating that distance based merely on how long it
took him to go from one city to the other on a camel, he arrived at the figure
of 5000 stadia. There is some debate about what Eratosthenes' stadia was equal
to. But if we assume that it was the Egyptian stadion, his calculations arrive
at 25,000 miles for the circumference of the earth. That is less than a hundred
miles off from what scientists today measure as the circumference (24,902).
When you consider that we also today know that the earth is not even a perfect
sphere (it bulges a bit at the equator), Eratosthenes breakthrough is
astounding.
The story of Eratosthenes shows us that
the solution to a problem can sometimes be a matter of approaching it from a
different angle. In this case, it let a man learn the true size of the world
for the first time, all without ever leaving Egypt |
Engaging students: Introducing the number e Nada Al-Ghussain. Her topic, from Precalculus: introducing the number e.
How can this topic be used in your students' future courses in mathematics or science?
Not every student loves math, but almost all students use math in his or her advanced courses. Students in microbiology will use the number e, to calculate the number of bacteria that will grow on a plate during a specific time. Biology or pharmacology students hoping to go into the health field will be able to find the time it takes a drug to lose one-half of its pharmacologic activity. By knowing this they will be able to know when a drug expires. Students going into business and finance will take math classes that rely greatly on the number e. It will help them understand and be able to calculate continuous compound interest when needed. Students who do love the math will get to explore the relation of logarithms and exponentials and how they interrelate. As students move into calculus, they are introduced to derivatives and integrals. The number e is unique, since when the area of a region bounded by a hyperbola y= 1/x, the x-axis, and the vertical lines x=1 and x= e is 1. So a quick introduction to e in any level of studies, reminds the students that it is there to simplify our life!
What interesting things can you say about the people who contributed to the discovery and/or the development of this topic?
In the late 16th century, a Scottish mathematician named John Napier was a great mind that introduced to the world decimal point and Napier's bones, which simplified calculating large numbers. Napier by the early 17th century was finishing 20 years of developing logarithm theory and tables with base 1/e and constant 10^7. In doing this, multiplication computational time was cut tremendously in astronomy and navigation. Other mathematicians built on this to make lives easier (at least mathematically speaking!) and help develop the logarithmic system we use today.
Henry Briggs, an English mathematician saw the benefit of using base 10 instead of Napier's base 1/e. Together Briggs and Napier revised the system to base 10, were Briggs published 30,000 natural numbers to 14 places [those from 1 to 20,000 and from 90,000 to 100,000]! Napier's became known as the "natural logarithm" and Briggs as the "common logarithm". This convinced Johann Kepler of the advantages of logarithms, which led him to discovery of the laws of planetary motions. Kepler's reputation was instrumental in spreading the use of logarithms throughout Europe. Then no other than Isaac Newton used Kepler's laws in discovering the law of gravity.
In the 18th century Swiss mathematician, Leonhard Euler, figured he would have less distraction after becoming blind. Euler's interest in e stemmed from the need to calculate compounded interest on a sum of money. The limit for compounding interest is expressed by the constant e. So if you invest $1 at a rate of interest of 100% a year and in interest is compounded continually, then you will have $2.71828… at the end of the year. Euler helped show us many ways e can be used and in return published the constant e. It didn't stop there but other mathematical symbols we use today like i, f(x), Σ, and the generalized acceptance of π are thanks to Euler.
How can technology be used to effectively engage students with this topic?
Statistics and math used in the same sentence will make most students back hairs stand up! I would engage the students and ask them if they started a new job for one month only, would they rather get 1 million dollars or 1 penny doubled every day for a month? I would give the students a few minutes to contemplate the question, without using any calculators. Then I would take a toll of the number of the students' choices for each one. I would show them a video regarding the question and idea of compound interest. Students will see how quickly a penny gets transformed into millions of dollars in a short time. Money and short time used in the same sentence will make students fully alert! I would then ask them another question, how many times do you need to fold a newspaper to get to the moon? As a class we would decide that the thickness is 0.001cm and the distance from the Earth to the moon would be given. I would give them some time to formulate a number and then take votes around the class, which should be correct. The video is then played which shows how high folding paper can go! This one helps them see the growth and compare it to the world around them. After the engaged, students are introduced to the number e and its roll in mathematics. |
Yesterday, the Edinburgh Mathematical Society met in St Andrews. Thomas Jordan gave a talk on multifractal analysis. I had no idea what this was, and went along without too much expectation of being enlightened. But Thomas began with a well-chosen example (dating from before the term "multifractal analysis" was invented) that gave the flavour very clearly. So here is a summary of the first ten minutes or so of the talk.
Any real number x in [0,1] has a base 2 expansion. The density of 1s in this expansion (the limit of the ratio of the number of 1s in the first n digits to n) need not exist; but, according to the Strong Law of Large Numbers, for almost all x, the limit exists, and is 1/2.
So it might seem perverse to consider the set Xp of numbers for which the limit is equal to p, for arbitrary p. But that is precisely what Besicovich did, and he proved a nice theorem about the Hausdorff dimension of this set.
Briefly, given a subset of [0,1], take a cover of it by intervals of length at most δ, and take the sum of the sth powers of the lengths of the intervals. Now take the infimum of this quantity over all such coverings, and the limit of the result as δ tends to 0. The result is the s-dimensional Hausdorff measure of the set. There is a number s0 such that the measure is ∞ for s < s0, and is 0 for s > s0; for s = s0, it may take any value. This s0 is the Hausdorff dimension of the set.
Besicovich calculated the Hausdorff dimension of the sets Xp defined earlier. It turns out to be equal to the binary entropy function of p; this is H(p) = −p log p−(1−p)log (1−p), where the logarithms are to base 2. The function H(p) takes its maximum value when p = 1/2, when it is equal to 1; and 1-dimensional Hausdorff measure coincides with Lebesgue measure. So Besicovich's result handles this case correctly.
The entropy function suggests a relation to binomial coefficients and Stirling's formula, which is indeed involved in the proof. (The logarithm of the binomial coefficient {n choose pn} is asympototically nH(p), as follows easily from Stirling's approximation for n!.)
All this can be phrased in terms of the dynamics of the map 2x mod 1 on the unit interval (which acts as the left shift on the base 2 expansion), which suggests a good direction for generalisation, and suggests too that that this generalisation will involve concepts from ergodic theory such as entropy and pressure. Most of the lecture was about this.
(Perhaps worth mentioning that all these sets are negligible in the sense of Baire category, according to which almost all real numbers have the property that the lim inf of the density is 0 and the lim sup is 1.) |
Mathematician pair find prime numbers aren't as random as thought
(Phys.org)—A pair of mathematicians with Stanford University has found that the distribution of the last digit of prime numbers are not as random as has been thought, which suggests prime's themselves are not. In their paper uploaded to the preprint server arXiv, Robert Lemke Oliver and Kannan Soundararajan describe their study of the last digit in prime numbers, how they found it to be less than random, and what they believe is a possible explanation for their findings.
Though the idea behind prime numbers is very simple, they still are not fully understood—they cannot be predicted, for example and finding each new one grows increasingly difficult. Also, they have, at least until now, been believed to be completely random. In this new effort, the researchers have found that the last digit of prime number does not repeat randomly. Primes can only end in the numbers 1, 3,7 or 9 (apart from 2 and 5 of course), thus if a given prime number ends in a 1, there should be a 25 percent chance that the next one ends in a 1 as well—but, that is not the case the researchers found. In looking at all the prime numbers up to several trillion, they made some odd discoveries.
For the first several million, for example, prime numbers ending in 1 were followed by another prime ending in 1 just 18.5 percent of the time. Primes ending in a 3 or a 7 were followed by a 1, 30 percent of the time and primes ending in 9 were followed by a 1, 22 percent of the time. These numbers show that the distribution of the final digit of prime numbers is clearly not random, which suggests that prime numbers are not actually random. On the other hand, they also found that the more distant prime numbers became the more random the distribution of their last digit became.
The researchers cannot say for sure why the last digit in prime numbers is not random, but they suspect it has do with how often pairs of primes, triples and even larger grouping of primes appear—as predicted by as the k-tuple conjecture, which frustratingly, has yet to be proven.
More information:
Unexpected biases in the distribution of consecutive primes, arXiv:1603.03720 [math.NT] arxiv.org/abs/1603.03720
Abstract While the sequence of primes is very well distributed in the reduced residue classes (mod q), the distribution of pairs of consecutive primes among the permissible ϕ(q)2 pairs of reduced residue classes (mod q) is surprisingly erratic. This paper proposes a conjectural explanation for this phenomenon, based on the Hardy-Littlewood conjectures. The conjectures are then compared to numerical data, and the observed fit is very good.
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(Phys.org) -- The same freezing which is responsible for transforming liquids into glasses can help to predict some patterns observed in prime numbers, according to a team of scientists from Queen Mary, University of London25 commentsA number of discoveries I have made about primes and trigonometric functions. I have tried submitting reports on these to various mathematics and "science" journals, but been ignored each time. Among Suspecting a connection between trigonometric functions, pi, the ratio of the circumference of a circle to is diameter, and p(i), the i'th prime number, I looked then at (p(i)/pi) – [p(i)/pi] and I found that the scattershot produces repeating parallel collections of straight lines, ten in a bunch, with a space big enough for another line separating the bunches. This pattern does not occur for any other irrational number, like e = 2.71828..., sqrt(10)=3.1622... .I'd expect there could be a difference between prime-base and composite-base systems.
Among
I'm not sure if I understood what exactly you did... if you plot i on x-axis and sin(p(i)) on y-axis, you do indeed get two overlapped waves, but only up to 25th prime. After that, it just appears random. At least that's what I got plotting it in Mathematica It may not have any immediate place in proving the facts about primes and trigonometric functions, but, for any integer n > 2, the product of sin(n*pi/j), where j goes from 2 to n-1 is 0 if and only if n is composite. Also, the sequence of values p(i+1)-p(i), the differences between successive primes, has a choppiness that resembles sin(i) or sin(i^2) or some such. In fact, summing the products of j^(1/pi) times some expression involving the sine function, where j goes from 1 to n, and multiplying the sum by an expression like n^a/(ln(n)^b) can tend to produce nearly constant values that seem to approach a limit.
for any integer n > 2, the product of sin(n*pi/j), where j goes from 2 to n-1 is 0 if and only if n is composite.
this one is simple; the product of sin(n*pi/j) is 0 if sin(n*pi/j) is 0 for at least one j. If n is prime, it isn't divisible by any j, and n/j isn't an integer. However, sin(n*pi/j) is 0 exactly when n/j is an integer. Meanwhile, a composite number is divisible by at least one j smaller than it, so sin(n*pi/j) is 0 for that j Could you plot it yourself and explain what I should be looking at? Did it over first 10000, and looked at various sections of it separately too because it was a mess. Could you plot it yourself and explain what I should be looking at?
As for the product of sines, that's why I looked at that. That was not an accidental discovery. If the physical scale of the vertical axis is made about the same as the horizontal, the appearance can be like that of overlapping sine curves.
LifeBasedLogic
Just because someone doesn't want to take a risk with a website that might be programmed to download more from your computer than you authorize, and especially since JongDan provided URL's for the other images but not the one they supposedly constructed with 1000 primes, does not mean someone is ignoring the request. The image for sin(p(i)) vs. iis at If I find my computer has been compromised by this I will let you know. A point. The correct formula from an earlier comment is a sum of j^(1/pi) times some simple expression involving sin(j), from j=1 to j=i. p(i) divided by this sum, multiplied by ln(i)^a/(i^(1/pi))^b will tend to approach some constant value.
Just because someone doesn't want to take a risk with a website that might be programmed to download more from your computer than you authorize... ...If I find my computer has been compromised by this I will let you know.
You're aware imgur is a widely used image hosting site, right? If they were "downloading more from your computer than you authorize" one of its millions of users would have noticed by now. I've yet to hear of any way to pull any file from your computer other than the one/s you select in the file picker.
That being said, thank you for providing exactly what you're trying to show. It looks interesting but it's also a sine function so it wouldn't be too surprising if it looks sinusoidal. I defer to those who are more knowledgeable of math to delve any meaning from that, though.
My system froze twice and failed to reboot twice, then declared a password was needed to get it out of a locked state after accessing imgur. The fact is, many people don't know how their computer works or whether, in fact, it is being misused. Why do you think so many end up being hacked? As for the shape of the curve, as I said, sin(i), where i is an integer, looks like a chicken wire array of nested hexagons. That is not consistent with a sinusoidal curve. It's more than that the sine is involved that produces the sinusoidal appearance of sin(p(i)).
My system froze twice and failed to reboot twice, then declared a password was needed to get it out of a locked state after accessing imgur
@juli this is far more likely of a problem on your PC than because of imgur, especially due to advertising and other hackable secondary pop-up's and stuff which most people ignore most Antivirus (AV) software will also track threats...
there aren't many viruses that are transmitted via graphics, and any decent AV will be able to protect you
there are free versions of AV you can DL you may want to consider
but again, the problem you had is far more likely to have come from other sites (including PO) than from imgur
(you can also alleviate ads & etc with either ad-block and or a HOST file - i suggest researching a little for what would be best for you and what your AV will best integrate with)
This mentioned was the fact that 1 followed by 1 had only a 18% chance of occurrence whereas a 1 followed by a 3 is a 30% chance. This is obvious when you look at how the prime numbers are formed as all primes greater than 3 can be written as (6n-1) and (6n+1) this means that once a prime number has been discovered say of the form (6n-1) then the next number that would be checked is of the form (6n+1), hence 2 apart. If the prime number was of the form (6n+1) then the next number checked would be of the form (6n-1) hence 4 apart. This is due to the order of the primes being generated and nothing else.
Just to clarify and add to this further the end numbers that should occur more frequently as consecutive numbers are: 1 - 3, 3 - 7, 7 - 9, 7 - 1, 9 - 1, 9 -3. If the probability of the primes are random then the next possible prime from a digit unit of 1 using the form 6n + 1 or 6n - 1 with a gap of 4 and 2 respectively has a chance of 25% (estimate). For 6n + 1 your next option 6n - 1 would end in 5 hence 100% chance of p'. So lets think about stopping when we hit a prime here are the options For 1 start at: 6n + 1 p = 0% (end in 5) p' p = 25% (end in 7) p' p' p = 19% (end in 1) p'p'p'p = 11% (end in 3) p'p'p'p'p = 8% (end in 7) p'p'p'p'p'p = 6% (end in 9) p'p'p'p'p'p'p= 4% (end in 3) p'p'p'p'p'p'p'p= 0% (end in 5) p'p'p'p'p'p'p'p'p= 3% (end in 9) p'p'p'p'p'p'p'p'p'p=2.5% (end in 1)
Using excel I have analysed 99.9% of the probability data this has revealed that starting at 1 the consecutive probabilities are 1 - 17.4% 3 - 30.9% 7 - 32.6% 9 - 19.1% Which conclude that their findings just show that these prime digits are random after all. Not really difficult Maths I might do a short tutorial to explain my findings and this disproof of the statements made in this article.
compose
My research using basic probability disproves the part that says that two consecutive primes ending in 1 have a 25% chance of occurring. This is because you are testing whether the next number of the form 6n + 1 or 6n - 1 is prime. Even if I used Riemanns number system instead of 25% this would not make any difference because it is proportional. If anyone needs to check my excel document for proof you can do altering the 25% would show the same outcome.
If you consider the universe as a mathematical structure, prime numbers are a common if not universal language in nature. It would make sense to use prime numbers to make it less likely to conflict with another. They give the illusion of randomness because we perceive most things as divisible. Do spiral galaxies complete a full rotation as a prime?
Just as a correction to my own comment; Below 10,000,000,000,000 there are 346,065,536,839 primes as all primes above 5 end in 1, 3, 7 and 9. This means that out of the total of numbers 4 in every 10 can have a chance of being prime. Hence we have 346,065,536,839 out of 4,000,000,000,000 giving 8.65% chance of a number ending with 1, 3, 7 and 9 being prime. If we analyse the probability of these using the above method this will give 1 followed by 1 – 22.0%, 1 – 3 – 27.4%, 1 – 7 27.3%, 1 – 9 23.4% This is still nowhere near 25% for 1 and is within the range analysed.
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Numinations — January, 1999
Millenium!
Let's Numinate about next New Year's Eve. Virtually everyone will
celebrate the new millenium when all those zeros are about to roll into
view. That is, on the 31st of December, 1999. There may be a few
purists who will celebrate the new millenium a whole year later, and I
will briefly recite their argument for this. However, there are good
and substantial arguments for properly celebrating at the end of this
year.
The current numbering of years, the one that gives this year as the
year 1999, began with a year numbered one. To celebrate the ending of
a full two thousand years, we have to complete the year 2000. At the
end of the current year, we have completed only 1999 years – not quite
a full millenium. At least, this is the way the reasoning goes. The
root of the problem was in calling that first year the "year one."
Did you ever notice that this is the 20th century, but all its years
begin with 19xx? This is because measuring, unlike counting, begins at
zero. The confusion sets in when we pair the cardinal numbers (one,
two, three) with the ordinal numbers (first, second, third).
Measurement of something continuous, like the measurement of time in
years, starts at zero and only gets to one when a full year has gone
by. Just as the first century had a zero in the hundreds place and the
first millenium a zero in the thousands place, during the first year,
the "year number" should have been "zero point something." The ordinal
number "one" (or first) should have been paired with the cardinal
number "zero." Thus, the mistake is 2000 years in the past. We can
fix it simply by thinking of the year before 1 CE as 0 CE. With that
year added to the count, the current year is the 2000th. And, just
like when the number 2000 first rolls into view on our car's odometer,
marking the completion of 2000 miles, the end of our current year will
mark the end of the second millenium and the beginning of the third.
Why celebrate new years, centuries, and millennia any differently than
we celebrate our own birthdays? We celebrate the big 40, the big 50,
and so forth. It's the appearance of those zeros that's significant.
Your age isn't one until you've had your first birthday. Your 10th
birthday marks a full decade. Likewise, the first instant of the year
2000 should mark two full millennia.
The year One was supposedly the year in which Christ was born. Now, if
his birthday is the 25th of December, isn't that rather close to the
new year anyway? There is no reason to assume that the year was fixed
correctly in the first place, and then one week later came a new year.
Was that first week in year 0 or 1? Should the first full year be
numbered 1 or 2? There's no way to tell at this point, two thousand
years later, too much is clouded by the mists of time.
The Julian calendar, which was the one in use at the time, was still in
the "experimental" stage then, anyway. That calendar was instituted by
the decree of Julius Caesar, about two years before he died (in 44
BCE). It was the first calendar that specified a 365 day year with a
leap year every four years (inserted between the 23rd and 24th of
February). The first mistake with this new calendar was finally
detected in 8 BCE, when the Emperor Augustus had to cancel three leap
days. And in the year 4 CE, the Julian calendar was finally
synchronized with time as we know it.
However, it drifted out of synch over the years, because it had an
annual error of 11 minutes and 14 seconds. On the 24th of February,
1582, Pope Gregory XIII decreed that the day after October 4, 1582
would be October 15, 1582. He further decreed that centennial years
would be leap years only if divisible by 400. Note that the definition
of a centennial year is one that ends with at least two zeros.
Already, we have significance attached to years ending in zeros!
This Gregorian, or "New Style," calendar was recognized almost
immediately by countries of the Catholic persuasion. Protestant
Germany adopted it in 1700, Britain in 1752, and Sweden in 1753. Greek
Orthodox countries in Eastern Europe adopted it in 1912-1917, the USSR
adopted it in 1918, and Greece didn't adopt it until 1923.
In any case, you can see how much of this is completely arbitrary. Now
that we use the letters CE and BCE, instead of BC and AD, to denote a
separation of "eras" instead of the birth of Christ, this arbitrary
numbering may slowly gain acceptance by other religions as well. This
new terminology has evolved in the past twenty years as evidenced by
dictionaries and encyclopedias from my own shelves. My 1975 Webster's
doesn't list CE and BCE. My 1993 Britannica refers you to an article
on Judaism in which the definition of "Common Era" is embedded. But,
my 1993 Random House Dictionary refers you to "Christian Era" for the
definition of "Common Era."
So, here we sit at one end of a 2000 year yardstick (yearstick?) whose
other end is pretty much clouded by the mists of time. Why not simply
say that the other end begins at zero? We really can't distinguish
otherwise. So what if the year 1 BCE was the year just before the year
1 CE? Measurement demands a zero point. Ordinality demands that we
complete a year, or move one year away from that zero point, before we
get to the number one. So, the year 1 BCE is simply a synonym for the
year 0 CE. Now, it's done. Instead of waiting another year for the
millenium, our counting will simply start (properly) one year sooner at
the other end of the "yearstick."
Let's celebrate the millenium the way we'd celebrate our own decade or
century mark – when the zeros roll over. And, rest assured without a
doubt, your position is completely defensible. After all, you've given
thorough Numination to this whole matter |
You are here
John Tynan — Mathematics
Every class that Dr. John Tynan teaches starts the same way: with a joke.
"Mondays are OK jokes, Wednesdays are bad joke days and Fridays are good joke days," he says. "The students in Tuesday-Thursday classes are lucky because they get to skip bad joke day."
Dr. Tynan is personable, but there is nothing to kid about with the subjects he teaches. He has been teaching in the Mathematics and Computer Science Department since 2001. His favorite course is Abstract Algebra. For Dr. Tynan, the jokes serve a purpose. "They loosen the kids up, get them to laugh or pay attention—some of them actually look forward to my jokes. The routine is: I tell them a bad joke and then I ask if they have homework questions. It warms up the room."
Hazel Brogdon '12 (Chugiak, Alaska) recently graduated with a degree in Biochemistry and a minor in Math. She knew Dr. Tynan before she ever sat in one of his classes. "I met him through the Foster Parent Program," she says. "He and Tracy (his wife) are my foster parents through softball. Their daughters (Fredley and Lynncoln) are like my little sisters."
Coming from Alaska, she knew she would need a little extra support. "In between home games there is time for players to visit with their families and friends. My family is in Alaska so it was really nice that I had a family there supporting me. They even went to Florida over spring break this year to watch some of my games."
Developing a strong relationship with the Tynan family has meant a great deal to her. "I never expected to have this home environment offered to me."
For Dr. Tynan, that's just a part of being a faculty member at a small, liberal arts college. "Our job is to get them ready for what that next step is," he says.
Since he started at Marietta, there have been 60 students graduate with degrees in Mathematics. More than one-third of those students continued on to graduate school and nine are pursuing a doctorate degree. Of the nine, four are pursuing a doctorate in mathematics while the others focus on Computer Science, Industrial Engineering, Veterinary Medicine, Physics and Electrical Engineering.
"What math teaches people is problem solving," Dr. Tynan says. "The joke is: 'What can you do with a major in math?' The answer: 'Anything.' " |
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Introduction to Probability with Texas Hold'em Examples illustrates both standard and advanced probability topics using the popular poker game of Texas Hold'em, rather than the typical balls in urns. The author uses students' natural interest in poker to teach important concepts in probability....
Quarks, Leptons and The Big Bang, Third Edition, is a clear, readable and self-contained introduction to particle physics and related areas of cosmology. It bridges the gap between non-technical popular accounts and textbooks for advanced students. The book concentrates on presenting the subject...
Easily Create Origami with Curved Folds and Surfaces
Origami—making shapes only through folding—reveals a fascinating area of geometry woven with a variety of representations. The world of origami has progressed dramatically since the advent of computer programs to perform the necessary |
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Letters to A Young Mathematician
Leading research mathematician Ian Stewart offers insights into mathematics for aspiring student mathematicians. Describes the importance and beauty of mathematics, the relationship between logic and proof, and the peculiarities of the mathematical community. |
Toothpaste, Custard and Chocolate: Maths gets messy
Abstract
This talk will look at mathematical modelling of real, complex fluids in flow situations – some with serious commercial applications, and some just for fun. We'll spend most of the time looking at the chocolate fountain. We'll experience one of the key day-to-day tools of an applied mathematician: scaling analysis; and we'll answer the question: why doesn't the chocolate fall straight down? |
"A Mathemusician's Journey" – A Talk by Dr. Sudeshna Basu
Math and music are usually organized into two separate categories, without obvious overlap. It tends to be that people are good at math and science or art and music, as if the two elements could not be placed together logically. In actuality, math and music are indeed related and we commonly use numbers and math to describe and teach music.
The Mathematics Society of St. Stephen's College wrapped up its string of events for the present academic session with a talk by Dr. Sudeshna Basu named "A Mathemusician's Journey".
Dr. Sudeshna Basu grew up in Kolkata, India. She started her early musical training under the tutelage of famous Hindustani classical vocalist Smt. Meera Bandyopadhyay of Patiala gharana. After a rigorous training in classical music Dr. Basu started taking Rabindrasangeet lessons under the maestro Sri Ashoketaru Bandyopadhyay.Recently, she started taking lessons on Dhurpad from the famous Gundecha Brothers. She obtained her Ph. D from Indian Statistical Institute, Kolkata. She then went to United States to pursue her academic goals. After teaching at universities in the US, she quit her full time job to dedicate herself whole heartedly to music. She now divides her time between mathematics and music.
Her talk mainly focused on the sub-conscious usage of mathematics in Indian Classical music. Without the boundaries of rhythmic structure – a fundamental equal and regular arrangement of pulse repetition, accent, phrase and duration – music would not be possible.
She illustrated the above with the example of a tihai, which is a rhythmic variation that marks the end of a melody or rhythmic composition, creating a transition to another section of the music. The basic internal format of the tihai is 3 equal repetitions of a rhythmic pattern), interspersed with 2 (usually) equal rests.
For example,
If the phrase is 16 beats long,
like in the rhythmic cycle called Teental,
the outline of a Anagat Tihai might look like 4 2 4 2 4.
Here, each "4" represents a rhythmic pattern that is 4 beats long,
and each "2" represents a rest that is 2 beats long.
(4+2+4+2+4 = 6+6+4 = 12+4 = 16).The start of the next phrase fall exactly on the downbeat.
The mathematician/musician spoke about the influence of music in the lives of various mathematicians in the past including that of Albert Einstein.
He once said that had he not been a scientist, he would have been a musician. "Life without playing music is inconceivable for me," he declared. "I live my daydreams in music. I see my life in terms of music…I get most joy in life out of music."
Being a mathematician as well as a musician both Tagore and Einstein had great influence on her. Dr. Basu ended the talk with a slideshow of photographs from her visit to Einstein's home in Princeton, Mercer County, New Jersey, United States, which included those of Rabindra Nath Tagore's visit to Einstein's home in 1930.
To sum it up, the talk was a great experience for math and music enthusiasts.
Dr. Sudeshna Basu's credit lies in how she juxtaposes her life of a Mathematics teacher at a university in the US and her passion for singing Tagore songs. |
Geometric solid(s) – The cube and its division
CHAPTER "C" FOR KIDS
Geometric solid(s) – The cube and its division
In this Chapter we will get acquainted with the construction of the CUBE in the ancient way with a compass and an unmarked straightedge. This solid is the element that is, or should be, the fundament of human activity. In the beginning we will divide it into its simplest parts, and thereafter in the forthcoming chapters into all of its other parts. Why is it the fundament of human activity? Because the proper construction of the cube will lead us to a series of "unknowns" that have been disregarded and which would bring us to the recognition of the natural harmony between man and his activity in nature. Yet so far we have not made much of an effort to explore this relationship and teach it to ourselves and our descendants, but on the contrary, exultant with our technological orientation, natural harmony was omitted, thus creating a chaos of unnatural figures and forms of whatever comes to mind, all the way to the crazy conviction that with technology i.e. techno-gadgets we can do anything we want. How erroneous such a conviction was best felt by a Japanese company who wanted to erect a replica of the pyramid by techno-technological appliances, and the whole venture was a flop. On the other hand the cube is a code system without which ancient enigmas cannot be solved, such as doublings of squares and cubes, the relation of measurements because they solve the progressions of halves, thirds, fifths, sevenths by geometrical and congruent progressions. We should mention that ancient peoples had no knowledge of decimals. And what is a decimal? A fraction. And a fraction? A part of a whole. So, a part can be drawn, without words, by the ancient means of division. Division of what? The cube, and this can be the diametrical square or cubical – only with a compass and an unmarked straightedge. Therefore we won't waste much of our time on present-day "discursive" constructions of the cube, but will deak with the way that ancient geometry teaches us.
* * *
Although they have the same length of edges (even this isn't taught today), we get some arbitrary parallels by which a shape of three dimensions is produced to create the impression of the geometrical solid of a cube. Then isn't it a pity to waste our time on this?
* * *
The cube – in the ancient manner (compass & unmarked straightedge) as shown in chapter "B" on the construction of the square. Therefore, the given length is the radius. The same goes for the cube. Describe a circle of arbitrary length of the cube's edges, and divide it with circles of the same radius into six parts.
* * *
- Connect poles with lengths. We acquire the circles of a hexagon.
– Connect the subtended poles. Cube (the eighth edge of the cube is hidden behind the front edge). Now we divide it.
* * *
The centerlines of the sides are already inscribed when we partition the circle with circles of same radius. Those are their exterior intersections and among other things they serve as control points for correct drawing. Therefore, subtended intersections are connected by straight lines (directions). ( These directions pass through the center).
* * *
The intersections of the straight lines and the cube's edges are the points that we connect in the direction of the edges. We divided the cube on the cube of 3, i.e. 23 – 2 x 2 x 2. This is one of the ways. Now we can go further on with the progressions of 2.
* * *
Another way, albeit shortened, is used only when an enlargement of the presentation is desired. Thus, the inscribed hexagonal circle of radius of the arbitrary value of the cube's edges – cube, star-shaped hexagonal polygon (every other pole). Thereby we create the conditions to divide the cube 2 x 2 x 2 = 23 .. Its further division is logical.
* * *
But with the hexagonal star-shaped polygon we also created conditions to divide the cube on 3 x 3 x 3 = 33 and onward. But we will stop here and show the enlarged (shortened) form. Hence, circle of radius length of arbitrary edges – hexagon – cube – star-shaped hexagonal polygon.
* * *
Now we connect the intersections of the star-shaped polygon – first left and right from the diameter (parallel with the diameter) – vertically.
* * *
- The other two intersections are parallel with the other diameter since the hexagon has three diameters. Those lengths should go to the edges of the cube…
* * *
- … and the third two parallel with the third diameter of the hexagon…
* * *
… and now we just connect the acquired points on the edges. We divided the cube of 3 as follows: 3 x 3 x 3 = 33 and its radius (radius of the circle) into 3 parts, and the diameter into 6 parts (thirds and sixths). That would be enough for now, since we still need two important basics (the fifths and sevenths). |
1
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Right. Yeah well origami has a surprisingly rich mathematics and geometry to it.
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It's I originally got interested in origami because it just posed a lot of interesting mathematical questions you have
3
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this sheet of material and you have very simple rules you can't stretch it and you can't tear it
4
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and what can you do just by a reconfiguration just by folding and so it's very kind of a simple set up
5
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but the answers turn out to be surprisingly complicated and you need to use a lot of powerful geometry
6
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and algorithms to figure out what you can fault in many senses of you can really fold anything out of a sheet of paper
7
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and you can prove that mathematically. And that's sort of where we got started.
8
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It's very exciting
9
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and finding more interesting ways to make structure the fold between different shapes also has a lot of practical
10
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applications in science and medicine
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and engineering where you want to build some kind of structure that can change transform it shape from one thing to
12
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another so maybe you want to fold it down to some small size for storage or transportation.
13
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Like if you want to put something inside the body.
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Maybe you need to transport through small blood vessels and so you need to make it very compact
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or you want to deploy something into space.
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You want to fold it small so it fits inside your space shuttle and then going to unfold it when it gets there.
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I think it's even more exciting is you imagine like buildings or gadgets
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or things that can transform from one shape to another and serve different functions depending on what you need.
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Maybe your house a room in your house can transform from a kitchen to a bedroom and this kind of thing.
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One of a couple of the areas that we were exploring are things like printing. Now to robots.
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So there are a lot of rapid prototyping machines that are designed to make flat sheets of material.
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And how can you use them to make three D.
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Robots and other structures
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and folding is a good way to do that you can transform your two dimensional sheets into some cool three D.
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Structures so we one of our goals in this printable robot project is to make robots that can for like ten
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or twenty dollars of materials you can cut them and make them within a couple of hours.
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So everyone can make their own robot and customize their robot to do whatever they want.
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Another fun application is in making nano scale structures so we have out of the whole computer chip fabrication
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technology we have really good ways to pattern two dimensional surfaces at with nano scale features like nanometer
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resolution. But we're not so good at making three D. Structures at that scale and so folding offers another way.
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That's more in process and experimental but. An exciting possibility for folding is for you something that is.
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What makes your position as a professor sort of roses. Yeah.
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Yes I know lots of different players interested in different aspects of folding maybe more practical side I'm more on
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the theoretical side and developing new mathematics
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and tools to show to help those sort of kind of underlying technology for people to build on to make useful things.
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So we especially like to prove what we call universe ality results where we say in this kind of regime of origami
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design or folding design.
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You can make anything you want and we give you a computer algorithm to do that
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and so you can come in with your specifications like oh I'd like something that looks like this and it
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and the algorithm will give you how to fold exam. Actually that thing.
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And you know we get very general results they're not always the most practical because often we don't take into
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consideration things like the thickness of the material or other kind of structural issues
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and something we're trying to get to but by kind of simplifying looking at the core geometry we can get very general
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and powerful results and then that those can be adapted to more practical scenarios where you are going for something.
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You would think that there's a limitation but I don't know every year it.
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I'm amazed at what origami artists come up with there's new people with new ideas
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and it seems like almost limitless possibilities
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and especially if you start with a large enough sheet of paper you can really fold really really complicated things
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and there's still aspects we don't understand.
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For example area we look at a lot is curved crease folding so most are Grammies made with straight creases curved
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creases are a lot harder to understand and analyze and we're starting to make progress on the mathematics
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but there's still a lot we don't know we don't have any good design algorithms to say Oh I'd like to fold something
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that looks like this. Here's the curve creases you need to do that.
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So instead we've been experimenting a lot with just playing around trying different curve crease patterns
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and see what they produce and trying to be able to model that mathematically.
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And that led my father
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and I into the sculptural side of paper folding so most of the sculpture we made is make is around curved crease
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folding initially we're just experimenting trying to figure out what's possible and what what can be done
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but we kept making all these beautiful forms and so I started to embrace that as a purely sculptural.
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Endeavor as well but there's a lot of back and forth.
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I think that will do something sculpture really that will inspire new mathematics.
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We discuss we understand something better about curved creases mathematically that inspires new sculpture
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and so it's a lot of fun to go back and forth between the two cards right between art and science I think in general.
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That's a big appeal to why people like to explore origami
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and mathematics together because you have this sort of scientific purpose maybe an engineering application
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or the beauty of the mathematics but then one of the applications is also to make sculpture.
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So it's really exciting to see these kinds of collaboration is a lot of engineering teams are bringing on origami
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artists at to help design new folding structures so artists have a lot of practical experience of how to make
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interesting folding structures and then they know the the literature which is a lot of people folding stuff
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and then that can inspire and inform new scientific discoveries. So your work your age. These are.
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Yeah that's where we like to live is right on the edge of knowledge where we we have a lot of tools
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but there's still something we don't understand.
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And so we try to push push that frontier of what's what's known on the scientific side
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and we use sculpture to kind of help explore that area more tentatively we can we can often make things that we don't
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yet fully understand.
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And so that lets us go a little beyond the frontier and sort of explore what's out there and see what's possible
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and then.
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Hopefully eventually understand that part mathematically where you where your science part formed you well what was it
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like for you just what was there. Yeah well in general we're looking at unsolved problems.
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I mean part of some sense one of the hardest parts is to figure out what the right question is. So you might want.
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There.
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Two types of questions about folding structures one is I give you a structure
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and I want to understand its properties and sort of analyze what it does how good it is what it folds into
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and the other side is the design side so you have some more high level specification of what you'd like to fold
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and then you want to automate the design of a structure that folds with those parameters designs maybe the more
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exciting side
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and there's many different ways you might formulate what you want to fold sort of the classic origami design problem is
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is shaped design. I say I give you a three dimensional shape. I want to fold that thing.
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What's a good way to fold that thing.
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And we're still we're still finding good algorithms for that we have some general procedures that work
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but they may not be so efficient as one of the standard measures of efficiency is if I have a square of a particular
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size material. How large of a version of that shape can I fold.
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I don't want to fold a really tiny thing because that means I'm kind of wasting a lot of my material five a big square
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filled with little microscopic things not very efficient material usage.
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So how can we optimize that scale factor we still don't know the best way to do that.
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There's a sense in which we can't know exactly how good we can do that
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but we can hope to approximate the best solution.
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So that's something we're still actively working on for example our current.
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Favorite technique is called organizer and it's it's also free software
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and it's an algorithm we've been analyzing over the last several years to give an arbitrary three D.
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Shape that gives you a way to fold exactly that shape.
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It seems to be a good method but we don't know it's the best method
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and then there are many other questions based on other types of goals you might want like maybe you want to have a
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folding structure that can make two different shapes.
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We don't know much at all about that question or you want to make a folding that actually.
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Works with really thick material because you're making out of sheet metal
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or you want to make a practical mechanical structure.
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We're still understanding that we've made some progress on
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but there's still a lot of questions we don't know the best way to deal with with these kinds of practical issues
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and so that as becoming really relevant these days because a lot of people are trying to build these structures
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and sometimes it works sometimes it doesn't like to understand that threshold then
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and ideally automatically design structures that always work really well in practice.
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Yeah I mean I think we like to build objects and we.
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And it's even cooler when those objects can change shape so almost anywhere you imagine a gadget of some sort.
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I think folding could offer some interesting perspectives on on reconfigure ability.
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C one one area we haven't talked about is protein folding which is a kind of origami it's a little bit different
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but it's kind of essential to how just understanding how life works and also potentially drug design.
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So every living thing that we know of in this world is built up out of lots of little proteins kind of making life
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happen and proteins are centrally one dimensional pieces of paper that quite a lot into complicated three D.
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Structures in that three D.
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Structure kind of determines how it interacts with other proteins and what what its function is.
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And we don't really understand that process of folding kind of a one dimensional strip of paper into these three D.
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Structures how nature does it how we could do it how we could design proteins that fold into geometries that we want to
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like combat. You can imagine some disease comes along new disease.
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You could design a protein to fight specifically that disease but we don't know how to design.
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Proteins that folded the way we want to.
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And so we're trying to understand how proteins fold in order to sort of just understand how biology is functioning
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but also so that we can kind of control it in useful ways to kill viruses and things like that.
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So that's that's an exciting but difficult interaction.
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I would really like a sort of universal programmable gadget you know like we have lots of gadgets where you can
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download software updates like your smartphone you can download software updates and it does new things
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but we don't yet have a gadget where we can download new shapes
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or new geometries you can imagine a kind of universal gadget that can take on any shape I mean it has to preserve mass
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that you could imagine it unfolding and becoming a large thing at folding into a more compact.
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Structure changing shape maybe it's a chair or one one moment and it becomes a bicycle the next moment
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or I mean anything in principle is possible.
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It's we need to figure out what the practical regimes are
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but instead of having a separate gadget that does different functions
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or separate separate furniture that does different things you could imagine having fewer objects that are more
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reconfigurable so that that excites me like I really like gadgets.
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But I can have a gadget that can do more different things or be more customizable I think that's really exciting.
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I owe a lot of people in the field.
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Got into folding because they've been folding since they were kids and doing origami
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and then they learn about mathematics
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and think oh oh maybe we should combine these two I came from the other side so I was a beginning graduate student at
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University of Waterloo and I was just curious. I was looking for interesting problems to solve.
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I knew that I really like geometry and algorithms and. My father remembered an old.
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Unsolved problems that he had read about
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when he years ago from a column by Martin Gardner who used to write for Scientific American about mathematical games
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and it's a problem that comes from the magic community and the concept is you take a piece of paper you fold it flat
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and make one complete straight cut and then unfold the pieces
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and magicians like Houdini could produce a five pointed star lots of different simple shapes
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and Martin Gardner I was wondering you know what are the limits can you make anything by this process
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or what can you do
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and so that's the problem we started working on like OK I've got geometry in algorithms now seems like a cool unsolved
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problem to work on and it turned out to be fairly challenging took us a year or two to solve
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but it also was very exciting that we got our first universality result
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and we showed that you can make any poly gun any sheet metal straight sides
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or actually you can make several shapes all at once. Just by a one straight cut after folding. So those very exciting.
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It's fun problem motivated by magic.
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It turns out to have some practical applications also there are some designs for airbag folding collapsing airbags flat
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that are based on the same kind of algorithm that we didn't intend that at the time
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and it really got us excited about this world of folding where it seems to have very rich
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and complicated mathematics but also those kind of fun and visual
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and you can you can demonstrate these things you know you can fold a piece of paper make a kite and make a swan
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or whatever shape you want to so it has an attendee ability that everyone can kind of appreciate even if they're not a
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mathematician you can say hey look we solved this magic problem that's cool.
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You know something for you where it was well before right. Well. Yeah it's going to start.
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I guess my father became a single parent when I was two years old
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and so we've been close for a long time especially from when I was ages seven to eleven
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and we started traveling together and visited many different places.
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Mostly east coast of United States and just traveling for fun.
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There was no particular reason other than seeing different cultures within the United States
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and exploring which was really fun and throughout that time my dad treated me as a peer. So we would.
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Jointly decide where we're going to go next to how long to stay in a place some places we'd stay just for a few days
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other places we'd stay for years and that was a really fun and bonding experience for us.
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Growing up and also because we're traveling a lot.
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We try to at home school and home school turned out to work really well for us.
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I would spend only like an hour a day doing sort of the breadth of regular school
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and so that I have many other hours during the day to explore things
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and very quickly for me exploring was computer programming.
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I got really excited about that essentially how to video games I played a lot of video games I was curious how they
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were made and my dad knew a little bit about computer programming to get us started
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and then we'd go to the library to learn more.
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This was all before the Internet and so I was sort of racially learning about computer programming
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and having a lot of fun there and then when school got out I would go and play with kids and things like that.
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So that was a really great time for me growing up and I went very fast in the computer science
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and eventually mathematics side of things right. So right. Yeah I asked over that when I was.
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Five or so and six years old my dad and I had our first collaboration we like to say with the Eric
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and dad puzzle company.
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So I helped design wire take apart puzzles and my dad would make them bending wire
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and then we sold to twenty stores across Canada and we split the income fifty fifty and it was a lot of fun.
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That was definitely the beginning of my interest in puzzles which is still to this day something I had like a lot
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and probably also the beginning of my interest in mathematics and geometry
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and things like that although that came much later as we were twelve. Yes yes. So after we ended this travel.
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I wanted to learn more about computing in computer science I learned was a thing
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and you have to go to university to learn about it.
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So there was some complication but I started undergraduate at twelve
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and took lots of classes because I at that age you can really soak in a lot of material and so I ended up finishing
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when I was fourteen and then went to graduate school and got a master's and Ph D.
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By the time I was twenty and then went on the job market and became a professor here at MIT. You're right. Yes.
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Yeah it's really. We really value. Having fun and enjoying the work that we do.
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There's a very there's essentially no line between the work that we do and the things we do for pleasure.
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So it's all mixed together just with different kinds of outcomes maybe becomes a math paper maybe it becomes a
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sculpture maybe it's there's no outcome we're just doing it for fun but.
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It's all for fun and the philosophy is that if we do work that we enjoy
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and find pleasurable then we'll do it very well excel at it and that has been a useful guiding principle
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and I would encourage everyone to do the same it's definitely it may seem risky at times
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and certainly there was a worry that the work that we do is to recreational like you know we're studying the
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mathematics of a magic trick how could that be useful for anything but it turned out to be unexpectedly.
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But I think a lot of specially in mathematics there are just a lot of basic questions that are very curious
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and you want to know the answer to
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and if they're basic enough the sort of very simple set up like paper folding is a very simple set up a very few rules
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about what's what you're allowed to do and yet it's very complicated to understand it's a nice context for.
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I think basic research tends to become useful eventually even though you may not see the applications ahead of time
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and so mathematicians tend to be attracted to like very simple questions that have complicated answers.
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Those tend to be also useful questions to answer always but if you solve enough of them.
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Many of them will become practical and so even though you do it for fun.
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It tends to have useful applications as well so that you might be worried by a lack of applications
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but turns out to be OK. It's for real life. It's really sweet.
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I mean we have pretty much ideal set ups where we can work on what we enjoy and get paid for it
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and have fun doing it and have all the resources to do it. We're very lucky.
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Yes So the glass blowing interest comes from.
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My dad's background which is more on the visual arts side so before I was born.
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In the late sixty's early seventy's he had the first glass studio in Canada.
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It's called the father of Canadian glass
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and so he had to have a studio made lots of glass work it was it was the early days in the studio movement of glass
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blowing in North America
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and so he was experimenting exploring what's possible trying different recipes to make glasses and glass colors
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and things and then he didn't blow glass for many years until
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and I never really saw him blood last until we came to MIT fifteen years ago
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and we discovered he'd MIT has a glass blowing studio called The Glass lab and so my dad got curious to try
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and glass flying again.
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And so he started teaching there became one of the instructors and started blowing glass again
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and I got to see him blow glass and watched him make things and it's so beautiful and amazing to watch
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and then eventually like maybe maybe I should try clasp like that said Yeah you know you should at least see what it's
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like but be careful. It's addictive.
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So I quickly got into a glass blowing and now we blow glass together and make things together and it's a lot of fun.
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It's a little more difficult because there's a lot of physics going on with glass blowing which is not exactly my forte
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but we're always looking for interesting math and connections between mathematics and glassblowing and we've found.
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We've found some interesting books there I think there's still a lot more to be explored.
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I would love to have algorithms to automatically design interesting because this sort of operations you can do are
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glassed in glass blowing a very simple. You know you can.
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You're turning your piece you can swing it around you can play with sort of gravity in this way you can heat different
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parts and cool other parts and that totally changes the shape that you produce.
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But it's a very complicated relationship and so it's hard to model all of that mathematically.
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But we've found some interesting regimes where it's simple enough that it's mostly geometric what's going on
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and so we can use computers to help design new patterns in glass.
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So we have some free software called virtual glass that we've been developing where you can design what are called
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Glass came patterns very simple.
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Conceptually simple but hard to visualize where you set up some essentially straight lines of color and glass
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and then twist them.
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And so you get some really cool twisty patterns they've been used in glass flying for for centuries. But.
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Pretty much everyone who makes glass cane follows one of standard set of patterns
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and so we were curious whether there were more patterns for Glass can that were possible in the software lets you
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explore those patterns and lets you try new things and sometimes you try a new thing
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and it looks kind of like an old thing.
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So it's not really interesting but other times you try a new pattern and it looks amazing in the software
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and that tells you here.
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This is something we should spend the time to actually learn how to make in real life software doesn't tell you exactly
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how to make it but it gives you a kind of schematic and then you have to do the glass blowing hard work
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but at least you know that the thing you're trying to make is really beautiful and so it's worth working towards.
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So you can rapidly try lots of different designs and software to find the one you want and then go physically make it.
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So it's really hard.
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Yeah I think it's I mean I think in general working on the boundary between two different fields you find interesting
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areas that.
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People tend to specialize in just one area and so they miss the things that the boundaries
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and so we've had a lot of fun exploring these boundaries
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and I think it comes partly from our different backgrounds my dad with the art background me with the more math
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and science. Background and we're always talking to each other and so we see we see the connections.
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When I started graduate school I was doing this sort of more theoretical mathematical work my dad's side
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and saying looks. That's interesting.
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This kind of creativity you're going through
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and solving unsolved mathematical problems is very much like the kind of thing that I go through in designing new
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sculptures or thinking about new art to build
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and so we started working together then he got he I taught him to become a mathematician.
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And he taught me to become an artist and so now we work on both together
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and it's really it's a lot of fun for us to collaborate in that way but also leads to really interesting questions
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and inspirations where instead of just thinking OK the math is the serious stuff
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and everything else is just you know side project we think of everything is like main projects
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and they inspire each other in ways that we couldn't predict. So I'm just working for years. We're definitely.
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Yeah I mean you could say frontiers of science and art maybe.
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Or that interplay but you know we're always as scientists we're always excited about the unknown
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and I mean that's as soon as we understand something fully it becomes almost boring
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and we want to move on to the next thing I mean we write down what we know and publish it
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and share it with the world so they can build on top of it
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but then we're always excited about the next question which we don't understand that's that's really what drives us is
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the price that we don't quite understand or like that seems a little strange. And we're curious about and.
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Yeah that's that's where we explore next. You know years at. Wells you might. Yeah it's a good question.
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I think in the in the folding regime. I work in many different areas but in the folding world.
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I think the biggest challenges right now are taking the nice mathematical geometric design algorithms that we have
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and adapting them to to real world materials.
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So we're starting to look at how does the thickness of the material affects what we can fold.
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How does the rigidity of material affect what we can fold off and you're making things out of plates and hinges.
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So you can really only fold at the creases whereas on paper it's more flexible.
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Between the creases So this is a world called rigid origami still trying to understand how to design within that space
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but it's very practical and exciting and for us it's nice
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and challenging because we don't know that's what that's what we don't know how to do
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and so that's where we're attracted So I think in the next couple of years we'll make a lot of progress in that kind of
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trying to take the rich and very general mathematics and adapting it
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or how to deal with the parameters of real world materials where you know other areas where you
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or where you're working well yeah I see there are a lot of so I mean like the traditional origami set up is you have a
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square paper and all you can do is fold and it's really interesting to see what you can do just by folding
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but there are a lot of practical setups like in our printable robots project where it's.
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Also find to cut the material I mean why not. It's folding is very powerful it's a good way to go from two to three D.
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but We don't have to start from a square of material probably we're starting from some kind of rectangle of sheet
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material and why not also cut it in two dimensions before you fold
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and that's exciting because it can lead to much more efficient foldings potentially use all of the material now
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and you can make structures you couldn't make just by folding or you can you can make them much more efficiently
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and in different ways it's also a little it's tricky from a mathematical perspective because now we have so much more
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freedom we can cut and fold in some sense it's more freedom than we know what to do with
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and so that's that's kind of a new direction of folding where we also are cutting because why not.
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It's a practical thing you can do
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and maybe there are some settings where you want to add lots of cut some settings where you want to add fewer cuts we
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don't know the right balance between us and I think that's a new frontier we're still exploring
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and trying to understand but potentially leads to much better ways of folding structures.
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And going back to your you're very free for. Case don't have this space. If you go.
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I think it played a big role I mean it's hard to know exactly
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but I think growing up with so much free time unstructured time where I could just explore what interested me really
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gave me a big edge. Instead of sort of wasting time which a lot of schools do just filling that time.
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So as a kind of child care. It's set up.
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There's social aspects which are good too but a lot of time I feel like is wasted in school
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and so having the home school opened up this window where I could explore what interested me and
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and really dive in deeply and that let me go far ahead in the computer science world
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and I think in general could let students go really far ahead in the thing that excites them the most you still have to
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add in the breadth and socialize with other kids and so on but really
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and then going to university at a young age I think really gave me another edge whereas you can learn so much at a
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young age and so when you get to university.
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Suddenly there's really interesting things you're learning and it's really exciting
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and I still remember the things that I learned back then.
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So that's really powerful as a way to to get started and I think a lot of people could do it.
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There's also a more general sense of.
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Because we were improvising if we want a long travelling around we would talk to our neighbors learn about what they
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knew about and if they knew some interesting topic they would teach me and teach my dad.
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So I learned different aspects about the magic that way.
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I learned different kinds of cooking that way
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and that was a fun way to say it to appreciate different people of different backgrounds
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and different knowledge sets and I think in directly that influenced me
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and my dad to think a lot about collaboration
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and in current day we we collaborate with a lot of different mathematicians
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and different papers I've written papers I think over four hundred people at this point
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and on the art side we're also looking for collaborators interesting ways to combine different ideas from different
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minds we collaborate a lot with each other. Of course but also looking for outside inspiration. I thing.
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When you combine multiple people together you can really you can solve problems that could not be solved individually
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on the mathematical side this is because there's just so many areas of mathematics. You can't really know all of them.
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But some problems require lots of different tools to solve and so you can either go
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and learn about that tool it takes a long time
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or you could just collaborate with the person who ARE THE knows the tool
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and they can solve that piece of the problem really. Well you can solve your piece you combine the right people.
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You can solve big problems relatively easily
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and on the art side you get inspiration things that no one person could make because they have the creative voice from
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multiple people you have to be willing to let go of your own ego to do this and I think that probably for my dad
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and I came from this period where we're just kind of exploring together and being open to the people that we meet
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and learning from them. Not that I know of. And it's an interesting challenge to try to model.
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Fun or humor or surprise. Mathematically I've heard.
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I know I have some friends who are trying to answer that question but I don't know of one sort of I usually go by.
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You know it when you see it kind of definition. So it was like. Each Other. Large I see here.
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Yeah it's certainly a fascinating topic to think sort of at a high level of like mathematics for example has a kind of
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a branch mathematical logic where tries to understand where we try to understand mathematically what mathematics is
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and why it works or when it works when it doesn't work. But.
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Of course the mathematics we practice in real life is a kind of a social dynamic you know do you believe
362
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but someone claims they have a proof written down there
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but to really check the proof you have to check it very carefully and it's humans are perfect
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and so it's there's a social dynamic to the.
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The body of research we we create and in some ways it makes it more fascinating
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and colorful that that kind of mind share of what we know
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or what we think we know is always kind of changing usually we're adding things we think are true
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or that we claim are true.
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Sometimes we take them back away we look at an old theorem and people have been building on
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and realize oh actually that proof is wrong
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and then there's this flurry of activity trying to fix the proof make a new proof so that the results that are built on
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it are the result may still be true but sometimes we need to find a new way to prove it.
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Sometimes the results and being false. That's that's more. It's occasionally scary but it's exciting.
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Always trying to discover new things but also make sure they're really correct.
375
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And definitely to me one of the appeals of mathematics is that you there is at least a sense of real truth of ultimate
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truth that in principle if you're doing it correctly
377
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and you prove something you really know that it is without a doubt true.
378
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There's no other area where you can be as certain but still even then
379
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or stuff like quite certain because humans make mistakes all the time. Yeah.
380
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So I mean certainly we can see a lot that we can kind of build up lots of evidence that something is true by
381
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constructing lots of examples and sculpture and and more practical engineering structures and so on
382
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but to know that it's always true is a little bit different to know that it's usually true. This start.
383
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Well you're right.
384
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You have puzzles of remained an active interest
385
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and in some sense all the mathematics we do is a kind of puzzle we have some set up of like what you're allowed to do
386
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say with paper folding are some other simple mathematical structure
387
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and the puzzle is you know what's possible what can you make these are kind of met a puzzle set sense
388
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but even puzzles themselves like the kind of board game puzzles you get or the like sliding blocks
389
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or these kinds of things are actually really interesting to study mathematically as well.
390
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And so my dad and I and many collaborators like to explore the mathematics of games and puzzles
391
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and we do it for video games like we studied Tetris and Super Mario Brothers
392
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and other Nintendo games that I grew up playing now I can study them mathematically
393
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and the sorts of things that we prove are that it's really hard to play these games perfectly.
394
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So if you if I give you a level of Super Mario Brothers and say can you get from start to finish.
395
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That's actually computationally difficult problem
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and you can prove that solving that problem is really hard for a computer to do
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and my philosophy is that humans are essentially a kind of computer
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and so that tells you that it's also really hard for humans to play these games perfectly
399
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or to solve these puzzles to play a Tetris game optimally
400
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or to slide blocks around to get one block out of the box all these problems are really really hard
401
00:39:37,36 --> 00:39:40,83
and I think it helps explain for humans.
402
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Why we enjoy them because humans like a challenge things should be challenging but not too difficult
403
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and proving these problems are computationally difficult.
404
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They're still solvable given enough time
405
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but in general you need an amount of time that grows exponentially with the size of the puzzle and so.
406
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It means it's beyond a certain size it really becomes intractable and even in a small size it's a challenge
407
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but still feasible.
408
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I think that's why it's fun to have this kind of mathematical justification for why we like playing games
409
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and puzzles and it's also a fun way to explore puzzles and games that I grew up with or know or love
410
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and be able to study studying them from a mathematical perspective lets me essentially play the game
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but in a more interesting way in some ways.
412
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Usually we have to design new levels new puzzles within a design space in order to show that.
413
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Oh we can build like logic gates and we can essentially build a computer within this game or puzzle.
414
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And that's how you show that it's hard for a computer to play because computers are not it's really hard for a computer
415
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to simulate a computer sexually.
416
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That's sort of the hardest thing that they can do
417
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and so we get to have fun by playing became by designing new levels and so on.
418
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In order to prove these kind of interesting mathematical results that actually this game is really challenging.
419
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And difficult.
420
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So it's a bit of both Certainly I also just like playing games playing board games playing video games
421
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and that's a lot of fun. Just as. Mediums to explore human experience I guess.
422
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And I like the the role playing aspects I like the having fun with friends aspect
423
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or exploring a world that's I mean these days.
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Video games tell really powerful stories and so it becomes a new medium for storytelling.
425
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So lots of more personal and just sort of fun aspects like that as I would be as a as a kid playing again.
426
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And there's definitely a lot of nostalgia playing even playing old games as.
427
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New Again they still hold up as being very exciting. But there's always even when I'm playing just for fun.
428
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There's always in the back of my mind thinking I wonder if we can set this up as a clean mathematical problem
429
00:42:13,68 --> 00:42:15,87
and analyze the complexity of this game
430
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and some games are more amenable to this kind of mathematical analysis some of them require some adaptation to be a lot
431
00:42:24,24 --> 00:42:29,17
of games have a lot of different elements it's really complicated mathematics is really good at getting at the core of
432
00:42:29,17 --> 00:42:32,8
a problem. So it's a lot better when you have set up a simplified version.
433
00:42:33,13 --> 00:42:40,17
Maybe you say oh let's just focus in on this one particular aspect of the game which if that's harder than the whole
434
00:42:40,17 --> 00:42:45,24
thing is of course also even harder and so you can kind of isolate out the different parts
435
00:42:45,24 --> 00:42:52,01
and tease out an interesting mathematical problem out of a real game or puzzle and then analyze that. So I mean.
436
00:42:52,52 --> 00:42:56,32
It's it all fits together so as I'm playing a game. I'm always thinking about.
437
00:42:56,51 --> 00:43:00,84
I wonder what I can tease out of this game and as I'm playing and having fun.
438
00:43:01,08 --> 00:43:05,13
I'm also trying to think about that that mathematical formulation so.
439
00:43:05,41 --> 00:43:17,6
It's good because then you get inspiration for new problems to solve just by having fun all day so.
440
00:43:17,61 --> 00:43:19,16
Yeah I should get one.
441
00:43:19,31 --> 00:43:28,35
But we made these wire take apart puzzles so each It's multiple pieces each piece is just made out of a piece of metal
442
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wire that my dad would bend with pliers into shape and so one shade might be trouble clef
443
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or some some recognizable shape or house I remember designing that one and then there be other pieces attached to it.
444
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Everything's made out of wire.
445
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Maybe some metal rings also and so these pieces appear to be interlocked and the challenge is to separate them.
446
00:43:52,63 --> 00:43:59,86
And so while they look interlock there's actually some complicated procedure for pulling one piece at a.
447
00:44:00,00 --> 00:44:04,6
Yeah there and you've solved the puzzle of that you have to put it back in and give it to someone else to solve.
448
00:44:04,72 --> 00:44:13,16
So these are challenging the kind of a mix of geometry and apology in their design and.
449
00:44:14,28 --> 00:44:16,67
Yeah there are a lot of fun you can be quite hard.
450
00:44:17,56 --> 00:44:27,00
Some of them require hundreds of moves to solve some of them are easy if you know how but yeah.
451
00:44:27,2 --> 00:44:33,61
So I think that the challenge of a human playing video games comes from different elements
452
00:44:34,34 --> 00:44:39,83
and some parts are easy for computers to solve and other parts we sure are difficult for a computer to solve.
453
00:44:39,89 --> 00:44:48,32
So if you imagine like solving a level of Super Mario Brothers there's there's kind of the physics of like the physical
454
00:44:48,32 --> 00:44:51,73
aspect of pushing the buttons at the right time either pressing them really quickly
455
00:44:51,73 --> 00:44:55,18
or exactly the right moment just before you fall off the ledge you jump
456
00:44:55,18 --> 00:44:59,03
and land in the right place that sort of thing that computers are actually really good at doing
457
00:44:59,03 --> 00:45:08,65
and there are people who exploit that in the tool assisted plays plays of games where they use computers to like slow
458
00:45:08,65 --> 00:45:10,04
everything down and.
459
00:45:10,52 --> 00:45:16,3
Time exactly the right moment to make a jump and things like that the computers can do some aspects really well
460
00:45:16,3 --> 00:45:19,87
but there's kind of a there's a broader medal level in solving a puzzle
461
00:45:19,87 --> 00:45:26,26
or solving a level in a video game where you need to plan out I should do this thing first
462
00:45:26,26 --> 00:45:28,79
and then I'll go do this thing there's a time limit
463
00:45:28,79 --> 00:45:34,79
and so it's really sensitive to how I should plan out the overall execution of the level executing it may be hard for
464
00:45:34,79 --> 00:45:41,01
human easy for a computer but the planning part for a sufficiently complicated game is usually really difficult
465
00:45:41,95 --> 00:45:46,41
and you can prove that that's computationally challenging now real world levels
466
00:45:46,41 --> 00:45:51,37
and puzzles are usually designed to be right at the edge where you have to try several different options
467
00:45:51,37 --> 00:45:55,23
but it's not impossible but in some sense
468
00:45:55,23 --> 00:45:59,67
and that challenge comes out of this broader setting if you have a really large.
469
00:46:00,00 --> 00:46:05,46
Well you can encode really hard problem inside that that puzzle and solving it.
470
00:46:05,56 --> 00:46:08,61
You can show us how hard even for a computer to play.
471
00:46:09,72 --> 00:46:16,96
So another example is like Tetris you have Tetris The usual the practical challenge is that you have this time limit.
472
00:46:16,99 --> 00:46:21,85
You know the pieces falling you have to decide where to put it really fast computers are good at doing things really
473
00:46:21,85 --> 00:46:25,17
fast but deciding where to put is actually really hard
474
00:46:26,00 --> 00:46:31,86
and you can you can show that sort of long term planning of where you should put your pieces so that you won't run out
475
00:46:31,86 --> 00:46:33,21
of space in your Tetris board.
476
00:46:33,38 --> 00:46:40,21
That's actually competition intractable also so it's interesting I think real world games you have this interesting
477
00:46:40,21 --> 00:46:45,43
mixture of making it hard for a human by giving time limits or physical execution can be challenging
478
00:46:45,43 --> 00:46:49,35
and then you have this mastery aspect which very appealing to gamers.
479
00:46:50,5 --> 00:46:53,91
But then usually there's this underlying computational difficulty that.
480
00:46:53,99 --> 00:47:15,79
So you're solving this hard problem under time constraints that it's exciting for people. It's research for your site.
481
00:47:16,66 --> 00:47:21,2
There definitely are some consequences I have a lot of games
482
00:47:21,21 --> 00:47:26,75
or in some sense a like video games are often an abstraction of a real world problem.
483
00:47:26,77 --> 00:47:29,15
Typical example is motion planning.
484
00:47:29,34 --> 00:47:32,17
So you're either you have a bunch of robots
485
00:47:32,17 --> 00:47:38,5
or you're a bunch of people trying to execute some goal you have a lot of objects you want to rearrange them into a
486
00:47:38,5 --> 00:47:39,16
particular pattern
487
00:47:39,16 --> 00:47:44,24
or maybe you're in a warehouse moving products around the every every product has a place it needs to go.
488
00:47:44,39 --> 00:47:47,72
What's the optimal way for moving all these parts around.
489
00:47:48,07 --> 00:47:55,54
That's those kinds of problems end up in a lot of video games also usually in a somewhat abstracted simplified form.
490
00:47:55,56 --> 00:47:59,83
So proving those problems are hard shows also that.
491
00:48:00,00 --> 00:48:04,27
These kinds of more real world problems are difficult as well maybe you can
492
00:48:04,27 --> 00:48:11,48
or it helps you maybe try to isolate what are the what is special about the real world instances maybe your warehouse
493
00:48:11,48 --> 00:48:14,36
is mostly two dimensional because you don't stack lots of things
494
00:48:14,36 --> 00:48:22,33
or what are the it's speciality is of the real world instance that make them easier than the videogame So there's
495
00:48:22,33 --> 00:48:24,07
definitely that kind of interplay
496
00:48:25,21 --> 00:48:31,65
but I think a lot of people study the Myself included study the complexities of these games
497
00:48:31,65 --> 00:48:35,47
and puzzles because it's fun and it's kind of a recreational pursuit.
498
00:48:35,61 --> 00:48:42,33
So it's a little bit less serious than some areas of mathematics computer science but still we enjoy it
499
00:48:42,33 --> 00:48:43,46
and it's kind of a fun.
500
00:48:43,57 --> 00:48:50,85
I use it a lot as a way to get students excited about research because most people come in with their own background of
501
00:48:50,85 --> 00:48:53,5
like what are fun games and puzzles that they grew up playing
502
00:48:54,41 --> 00:48:59,17
and those inspired new mathematical problems either directly about those games
503
00:48:59,17 --> 00:49:01,21
or about sort of the underlying principles
504
00:49:02,02 --> 00:49:09,61
and these kinds of hardness personally call them to show these games are competition intractable are a nice way to get
505
00:49:09,61 --> 00:49:15,09
started in research because you get to play with the game you get to use the expertise you have from having grown up
506
00:49:15,09 --> 00:49:15,73
playing this game.
507
00:49:15,9 --> 00:49:17,69
You probably spent way too many hours playing them
508
00:49:17,69 --> 00:49:21,69
and that expertise is actually really helpful for solving the underlying math problem
509
00:49:21,69 --> 00:49:26,99
and it can get people excited about. Oh this is this is computer science research I want to do more of this.
510
00:49:28,35 --> 00:49:35,86
There's also the I think the broad appeal of you know there's some mathematical results that are hard for.
511
00:49:36,23 --> 00:49:37,94
The general public to appreciate.
512
00:49:38,01 --> 00:49:43,86
But you analyze a game or a puzzle that everyone has played or big segment the population is played
513
00:49:43,86 --> 00:49:46,97
and they can appreciate like oh yeah I remember that being really hard.
514
00:49:46,99 --> 00:49:48,77
Oh you can prove that mathematically Oh that's interesting.
515
00:49:49,14 --> 00:49:55,08
I wonder how they do that and that can inspire people to enter the field or at least get a curiosity or
516
00:49:55,08 --> 00:49:59,7
and learn about fields that they're not necessarily working in and appreciate that.
517
00:50:00,00 --> 00:50:06,22
There's interesting things you can do about problems I happen to care about because most people like games
518
00:50:06,98 --> 00:50:21,91
and so this isn't a nice kind of broad appeal connection where our years ten years.
519
00:50:21,92 --> 00:50:30,33
It's hard to know exactly where my research will take me I definitely like MIT as a base because it's I mean there are
520
00:50:30,33 --> 00:50:34,33
mazing students here amazing people doing all sorts of great and crazy things
521
00:50:34,33 --> 00:50:40,17
and just a lot of flexibility to essentially do what we want and explore whatever we find interesting.
522
00:50:40,5 --> 00:50:49,08
So what will be most interesting to us in ten years is hard to guess but this definitely is a nice.
523
00:50:49,1 --> 00:50:54,24
Powerful base to do it from so different enjoying my time here. |
Wednesday, May 02, 2007
Life Is Math
I'll go even deeper: the best comedy works because it's mathematically sound.
As any comedian will tell you, comedy is all about timing... and timing is math. Sure, relatability is the humane filter that determines whether or not we develop an emotional response to something (like humor), but whether or not a joke works boils down to simple mathematics.
The same goes for music, painting, photography, theatre, dance, writing, architecture, film... Everything we consider to be aesthetically engaging and unspeakably profound -- even sex -- can be broken down to mathematics.
Why?
Composition, color scheme, rhythm, tone... each element that affects the "finished product" of art is mathematically based. Change any element, even by a fraction, and you have a wholly different end result -- one which society may react to in completely different ways.
The same goes for life beyond art. Great feats in sports are merely the result of body mechanics. The difference between a great golf swing and one that slices into the woods off every tee might be one-sixteenth of an inch.
Likewise, traffic patterns are entirely mathematical -- humans operating machines that are simultaneously computing millions of equations, all the while being influenced by external data (weather, temperature, distance, congestion).
The entire concept that life is math occurred to me in college, and I recall being profoundly depressed at that observation. Who, at the age of 20, wants to believe that his or her entire life will be dictated by the uncaring objectivity of numbers?
But Chris's post highlighted the element that mitigates the objectivity of math: we're all subjective creatures. Life might be entirely mathematical, but individuality is our one-of-a-kind way of interpreting the exact same set of data. |
Foolproof, and Other Mathematical Meditations
Overview
Brian Hayes wants to convince us that mathematics is too important and too much fun to be left to the mathematicians. Foolproof, and Other Mathematical Meditations is his entertaining and accessible exploration of mathematical terrain both far-flung and nearby, bringing readers tidings of mathematical topics from Markov chains to Sudoku. Hayes, a non-mathematician, argues that mathematics is not only an essential tool for understanding the world but also a world unto itself, filled with objects and patterns that transcend earthly reality. In a series of essays, Hayes sets off to explore this exotic terrain, and takes the reader with him.
Math has a bad reputation: dull, difficult, detached from daily life. As a talking Barbie doll opined, "Math class is tough." But Hayes makes math seem fun. Whether he's tracing the genealogy of a well-worn anecdote about a famous mathematical prodigy, or speculating about what would happen to a lost ball in the nth dimension, or explaining that there are such things as quasirandom numbers, Hayes wants readers to share his enthusiasm. That's why he imagines a cinematic treatment of the discovery of the Riemann zeta function ("The year: 1972. The scene: Afternoon tea in Fuld Hall at the Institute for Advanced Study in Princeton, New Jersey"), explains that there is math in Sudoku after all, and describes better-than-average averages. Even when some of these essays involve a hike up the learning curve, the view from the top is worth it.
About the Author
Brian Hayes is Senior Contributing Writer at American Scientist. His writing has appeared in Scientific American, The Sciences, Wired, the New York Times Book Review, the New Republic, and other publications.
Endorsements
"Each of these essays brings unexpected twists of perception and presentation; what a fine imagination Hayes has! I enjoyed the book enormously." —Nick Trefethen FRS, Professor of Numerical Analysis, University of Oxford; creator of Chebfun; author of Trefethen's Index Cards
"Brian Hayes takes us with him as he roams far and wide across the mathematical landscape. Whether he's braving the borderlands of the latest research or poking around in some forgotten corner of history, his chronicles of what he finds there are consistently captivating and revelatory." —Steven Strogatz, Jacob Gould Schurman Professor of Applied Mathematics, Cornell University; author of The Joy of x
"With a journalist's instinct for story, a mathematician's concern for accuracy, and a storyteller's sense of narrative, Brian Hayes lets the general reader in on a secret mathematicians already know: math is fun! His vignettes are like the snapshots of a returned traveler, showing us exotic lands and the marvelous creatures that live there. Foolproof shows that the mathematical enterprise is one of high adventure." —James Propp, Professor, University of Massachusetts Lowell |
Mathematics
Mathematics
Mathematics is an universal language. It is also known as language of science.Developing interest towards maths depends on your industry in the initial stages and later on your intelligence. It is one of the few subjects, wherein you can score hundred percent marks to enhance your overall percentage.
Many students don't like maths and some students are really scared of maths.But it's an easy and interesting subject,once you understand the basics.be thorough in them.Without the basic knowledge about fundamental formulas,techniques,multiplication tables,laws and theorems you cannot expect to be good at maths.
Unlike other subjects, each lesson in maths is made on the previous one. Falling a day behind puts you in confused position. never hesitate to ask questions. A little uncleared doubt now,leads to a huge roadblock in future.
Reasoning is the backbone of mathematics.Failure in not getting correct solution teaches you how to look back to locate your mistake.This applies to real life also.Convert your life problem into real mathematics.draw a formula.Locate the main crisis.Once you are clear about the cause of your problem,automatically you would know the method of approach to sort it |
Briony Thomas
Lecturer in Design Theory, School of Design, University of Leeds, UK
"After studying the geometry of repeating patterns as a student of
textile design, I became interested in the possibilities of patterns
repeating in three-dimensions, around the faces of mathematical solids.
This interest led to an investigation into which of the 17 plane classes
can regularly repeat around the Platonic solids. The successful
application of a pattern to repeat across the faces of a polyhedron, is
determined by the pattern's underlying lattice structure and its
inherent symmetry operations. The project resulted in the creation of a
collection of pattern designs and regularly patterned polyhedra,
inspired by geometric tilings found at the Alhambra Palace in Granada,
Spain."
"Polyhedra de los Leones "
2007, Laser-etched wood composite, 56 cm x 20 cm x 20 cm
Polyhedra de los Leones features the tetrahedron patterned with p6m,
the octahedron patterned with p3m1 and the icosahedron patterned with
p6. These patterns are constructed on a hexagonal lattice, where the
unit cell comprises two equilateral triangles. The rotational symmetries
of the tetrahedron and icosahedron are maintained by pattern classes p6
and p6m due to the higher order of symmetry within the plane pattern.
Centres of six-fold rotation, characteristic of the plane pattern,
become axes of three-fold rotation at each vertex on the tetrahedron.
Axes of two- and three-fold rotation are also preserved in the solid,
with six reflection planes evident. Only pattern classes containing
six-fold rotation are applicable to regularly patterning icosahedron.
Centres of six-fold rotation in the pattern become axes of five-fold
rotation at each vertex and all other rotational symmetries are
preserved. The two units, created by each half of the pattern class p3m1
unit cell, permit repetition across the eight octahedral faces and
result in two-fold rotation at each vertex. Although no rotation is
present at the solid's edges, six planes of reflection are evident. |
Beauties
This is a collection of mathematical results that I consider as gems based on their beauty and elegance alone. Neither the depth nor the width of its application is criteria for consideration in my collection. It is only the beauty that matters here. Hence the title 'Mathematical Gems. Again as they say, beauty lies in simplicity, simplicity of the statement of the result is therefore one of my criteria. Also since number theory is my area of interest, a majority of my gems belong to number theory. For each gem, I have given a one line explanation why I saw a gem in it. But I believe that for these gems, the best explanation is in the theorem itself.
Gem 1. An all in one formula
This asymptotic formula is one of my personal favorites because it unites most of the important topics and constants of analytical number theory under one single formula: |
Today From Bedtime Math: Trevi Treasure Trove
The Trevi Fountain in Rome, a giant statue of marble men and horses swimming in rushing water, is the biggest fountain in the famous city. This isn't just a little angel statue: the fountain stretches 161 feet across and stands 86 feet tall! The legend is that people who throw coins into the fountain will have a safe return to Rome again in the future – but you have to throw the coin over your left shoulder using your right hand. People seem happy to do this, since the fountain collects over $3,000 in coins every day! As with most fountains, the coins are collected and used to buy food and clothing for people in need.
Now see if you and your kids can find out how much money those marble men and mares round up each year.
Wee ones: If you and 3 friends each throw a coin in the fountain, how many coins do you throw?
Little kids: If the marble statues include 3 men and 2 horses, how many statues are in the fountain? Bonus: How many legs do they all have together?
Big kids: If the fountain caught exactly $3,000 of money per day, how much would it catch in a week? (Hint if needed: you can add and multiply "thousands" the way you add apples, cookies or anything else.) Bonus: About how long does it take the fountain to catch $100,000 — a month, a year, longer?
Answers: Wee ones: 4 coins. Little kids: 5 statues. Bonus: 14 legs (6 on the people, 8 on the horses). Big kids: $21,000. Bonus: About a month — it will take just over 33 days, which would be $99,000.
Resources
Activities |
Computational mathematics
A black and white rendition of the Yale Babylonian Collection's Tablet YBC 7289 (c. 1800–1600 BCE), showing a Babylonian approximation to the square root of 2 (1 24 51 10 w: sexagesimal) in the context of Pythagoras' Theorem for an isosceles triangle. The tablet also gives an example where one side of the square is 30, and the resulting diagonal is 42 25 35 or 42.4263888.
Computational mathematics involves mathematical research in areas of science where computing plays a central and essential role, emphasizing algorithms, numerical methods, and symbolic computations. Computation in research is prominent.[1] Computational mathematics emerged as a distinct part of applied mathematics by the early 1950s. Currently, computational mathematics can refer to or include: |
This old diagram could serve as a spinner for allowing luck to determine how much ice cream to serve oneself. It's from a wholly different context: Raising P. V. Squabs for Profit by John S. Trecartin, 1920. |
5. Do some independent research on fractals: where they occur in nature, how fractals are used in the movies, how are they used in the stock market, how are fractals connected to chaos.
6. Check out Bob Devaney's (NCTM speaker) page
7. Make a fractal card from Shephali chokshi-Fox (NCTM speaker) |
From early in life, humans have access to an approximate number system (ANS) that supports an intuitive sense of numerical quantity. Previous work in both children and adults suggests that individual differences in the precision of ANS representations correlate with symbolic math performance. However, this work has been almost entirely correlational in… (More)
Nonhuman animals, human infants, and human adults all share an Approximate Number System (ANS) that allows them to imprecisely represent number without counting. Among humans, people differ in the precision of their ANS representations, and these individual differences have been shown to correlate with symbolic mathematics performance in both children and… (More) |
Data science and programming
Thu, 18 Jan 2018 15:29:39 +0000enhourly1 science and programming
Bayes' Theorem is the FizzBuzz of Data Science
20 Dec 2017 12:00:53 +0000 other day, I did a technical interview that involved applying Bayes' Theorem to a simple example. It stumped me. And it left me feeling empathy for folks who have had trouble with the FizzBuzz interview question.
It doesn't reflect day-to-day work
FizzBuzz depends on understanding a few concepts, like conditional execution, the modulus operator, divisibility of numbers, and common denominators.Every programmer should be familiar with the modulus operator and it's relationship to divisibility, but knowing about it doesn't mean it's part of your bread and butter. The day-to-day of software engineering usually takes place at th higher level of understanding good design patterns, parsing requirements, and using APIs for their team's platform or framework. Diving down to a lower level to reproduce divisibility from simple mathematics is a shift in perspective and takes several mental cycles to get right if it's out of practice.
It's stressful to problem solve on the spot
Unless you're made of steel, you'll probably get some form of jitters during an interview and this can hurt the way you solve problems.
In 2005, Beilock & Carr published a paper on performance of math problems between high working-memory and low working-memory undergraduate students. The HWM group, which performed very well on simple mathematical problems at baseline, low-stress conditions performed significantly poorly in a high stress. (Incidentally, Beilock & Carr's experiment used a modular arithmetic task in their experiment, the same concept that is integral to FizzBuzz.)
Add the stress of trying to explain your solution to an interviewer and this is a recipe for a meltdown.
Experience teases out edge cases, but experience fades with time
One of the really tricky parts about FizzBuzz is that is involves an edge case that can trip people up. Namely, when the number is 15, it's possible to print out all three of "Fizz," "Buzz," and "FizzBuzz" if you're not careful.
The problem with this is that edge cases aren't over-arching principles or theoretical concepts, they're anomalies. And people learn how to deal with anomalies by experiencing them. At work, edge cases would get discovered in a handful of ways, but most likely being a) a developer on the team has come across it before and b) there's a deliberate, likely time-consuming effort to enumerate possible cases to test them out. As I've mentioned above, case (a) isn't very likely because the edge cases of FizzBuzz aren't top-of-mind, and case (b) presents problems because of the stress of interviewing.
Bayes' Theorem
Bayes' Theorem (or Bayes' Rule) is just tricky enough behave like FizzBuzz in these situations. It's certainly something that every data scientist should know, but it isn't something he or she would use every day. It's complicated enough to be relegated to the class of abstract math problems, which take a hit during stressful situations. And finally, for edge cases in applying Bayes' Theorem (like whether two events are independent), it's difficult and unlikely for individuals to come up with suitable examples to test immediately.
]]> Deep Learning is Easier Than You Think
06 Jul 2016 13:00:15 +0000 came across a great article on using the Deep Learning Python package tflearn to perform inference on some classic datasets in Machine Learning like the MNIST dataset and the CIFAR-10 dataset. As it turns out, these types of models have been around for quite a while for various tasks in image recognition. The particular case of the CIFAR-10 dataset was solved by a neural network very similar to the one from the mentioned post. The general idea of using convolutional neural networks dates back to Yann LeCun's paper from 1998 in digit recognition.
Since I have a lot of experience working with data but not a lot of experience working with deep learning algorithms, I wondered how easy it would be to adapt these methods to a new, somewhat related, problem: going through thousands of my photos to identify pictures of my cat. Turns out, it was easier than I thought.
This is definitely a cat photo
Training a similar neural network on my own visual data just amounted to connecting up the inputs and the outputs properly. In particular, any input image has to be re-scaled down (or up) to 32×32 pixels. Similarly, your output must be binary and should represent membership of either of the two classes.
The main difficulty involves creating your dataset. This really just means going through your images and classifying a subset of them by hand. For my own run at this, all I did was create a directory like:
images/
cat/
not_cat/
I put any cat photos I found into the cat directory while putting any non-cat photographs in the other folder. I tried to keep the same number of images in both directories to try to avoid any class imbalance problems. Then again, this wasn't as much of a concern since roughly half my photos are cat photos anyway.
From there, tflearn has a helper method that lets you create an HDF5 dataset from your directory of images with a simple function. The X & Y values from that data structure can be used as the inputs to the deep learning model.
By using around 400 images (roughly 200 for each class), my classifier achieved about an 85% accuracy rate on a validation set of data. For my purposes, namely just automatically tagging potential photos of my cat, this was accurate enough. Any effort to increase the accuracy of this would probably involve some combination of:
adding more training data by putting images into my class folders
changing the shape of the network by adding more layers or more nodes per layer
That's all it really takes. If you know a bit of Python and can sort a few of your photos into folders based on their categories, you can get started using sophisticated deep learning algorithms on your own images.
You can find the code for this on my account at Github. If you want to chat or reach out at all, follow me on Twitter @mathcass.
]]> 09.58.28.jpgCommunicate about your work
28 Apr 2016 12:53:27 +0000 gave a lightning talk earlier this month to the PyData Atlanta Meetup. I've given hour-long talks on technical subjects before, but I hadn't done anything quite that concise before. This fact freaked me out quite a bit. I wanted to reflect a bit on why it's always a good idea to communicate more what you do.
No matter how mundane or "been done before" you believe your work is, there's value in showing it to others because some people will learn from it. In machine learning, some methods are designed to try all possible permutations of a set of options to choose the one with the best performance. As the complexity of a model about your data grows, inevitably this tree search method breaks down and you need to apply some heuristics to the problem. What people don't mention is that these heuristics can come from anywhere, whether it's a research paper, a book, a mentor, or even a five minute talk you saw on a Thursday night.
Like any good person who over-prepares for things, I read up a bit on it which helped me come to this conclusion (and that helped me think through public speaking in general). Here are some resources:
Some helpful tips from Time.com (including a quote from Woodrow Wilson that is one of my favorites)
Oh yeah. I think my talk went pretty well. Here's a link to my Google Drive slides or a PDF copy. If you'd like to chat about data or about my work, feel free to reach out to me via email or on Twitter.
]]>
25 Apr 2016 12:45:48 +0000 a long time, I've been interested in with web technology. In high school, I read Jesse Liberty's Complete Idiot`s Guide to a Career in Computer Programming learning about Perl, CGI (common gateway interface), HTML, and other technologies. It wasn't until I finished a degree in mathematics that I really started learning the basics, namely HTML, CSS, and JavaScript.
At that point, folks were just starting to come out of the dark ages of table-base layouts and experimenting with separating content (HTML) from presentation (CSS) from behavior (JavaScript). A popular discussion was over what the best type of layout was. I remember reading discussions of left vs right handed vs centered pages. (The latter is still a pain to implement but the web has still come a long way since then.) Those discussions stuck with me and motivates the study I'm running right now.
In mathematics, people have been using fundamental matrix algebra to help them accomplish some very interesting things. One application is in image decomposition, basically, breaking images into simpler, more basic components. A space of vectors (or coordinates that can represent data points) have "eigenvectors," which are vectors that you can use to reconstruct other vectors that you have in front of you. In facial recognition, applying this technique to images of faces yields Eigenfaces, patterns that are common to many of the images.
Coming back to the idea of website layouts, I reasoned there must be a way to see this in real web data somehow. If we took images of the most popular websites on the web, how would their "eigenlayouts" look overall. Would certain layouts (like left or right or center) just pop out of the data somehow. Well, to answer that question, we need some data, we need to analyze it, and then we need to interpret it.
Data Retrieval
To run the analysis on these websites, we need to turn them into images. For that, I turned to PhantomJS, a headless browser geared toward rendering websites. For the sake of having a portable solution (able to run just about anywhere without needing too much bootstrapping), I decided to use a community-submitted Docker image that I found that that did that job nicely. Basic usage has you specifying the website you want to render as well as the output filename for the image. You can additionally pass in arguments for your webpage resolution. I went with 800px by 1200px because it's a sensible minimum that people are creating websites for.
When I need to run a lot of commands and don't need a who programming language, I typically turn to GNU make for defining a pipeline of work and for parallelizing it out.
The gist of this is that first you run make mkdomains to create a directory domains/ filled with dummy targets of each domain you want to look up (10,000 take up about 80MB of space). Then, running make will use that seeded directory of domains to pull each one down and drop the image file in the images/ directory. You can file all of this under "stupid Makefile hacks."
Next Steps
So far, this covers a very small portion of the 80% of data science that's cleaning and munging data. The next blog post will focus on loading and analyzing this image data in Python using the scikit-image library.
If you liked this post, please consider subscribing to updates and following me on Twitter.
Data Scientists often need to sharpen their tools. If you use Python for analyzing data or running predictive models, here's a tool to help you avoid those dreaded out-of-memory issues that tend to come up with large datasets.
Enter memory_profiler for Python
This memory profile was designed to assess the memory usage of Python programs. It's cross platform and should work on any modern Python version (2.7 and up).
To use it, you'll need to install it (using pip is the preferred way).
pip install memory_profiler
Once it's installed, you'll be able to use it as a Python module to profile memory usage in your program. To hook into it, you'll need to do a few things.
First you'll need to decorate the methods for which you want a memory profile. If you're not familiar with a decorator, it's essentially a way to wrap a function you define within another function. In the case of memory_profiler, you'll wrap your functions in the @profile decorator to get deeper information on their memory usage.
If your function looked like this before:
def my_function():
"""Runs my function"""
return None
then the @profile decorated version would look like:
@profile
def my_function():
"""Runs my function"""
return None
It works because your program runs within a special context, so it can measure and store relevant statistics. To invoke it, run your command with the flag -m memory_profiler. That looks like:
python -m memory_profiler <your-program>
Profiling results
To see what the results look like, I produced some sample code snippets that show you some examples.
While these examples are contrived, they illustrate how tracing memory usage in a program can help you debug problems in your code.
Above we have a pretty obvious logical error, namely we're loading a file into memory and repeatedly appending its data onto another data structure. However, the point here is that you'll get a summary of usage even if your program dies because of an out-of-memory exception.
When should you think about profiling?
Premature optimization is the root of all evil – Donald Knuth
It's easy to get carried away with optimization. Honestly, it's best not to start off by immediately profiling your code. It's often better to wait for an occasion when you need help. Most of the time, I follow this workflow:
First, try to solve the problem as best as you can on a smaller sample of the actual dataset (the key here is to use a small enough dataset so that you have seconds between when it starts and finishes, rather than minutes or hours).
Then, include your entire dataset to see how that runs. At this point, based on your sample runs you should have a) an idea of how long the full dataset should take to run and b) an idea of how much memory it will use. Keep that in mind.
Next, you have to monitor it running, so that could mean three possible outcomes (for simplicity)
It finishes successfully
It runs out of memory
It's taking too long to run
You should start thinking about profiling your code if you encounter either of the latter cases. In the case of overuse of memory, it will help to run the memory profiler to see which objects are taking up more memory than you expect.
From there, you can take a look at whether you need to encode your variables differently. For example, maybe you're interpreting a numeric variable as a string and thus using more RAM. Or it could be time to offload your work to a larger server with enough space.
If the algorithm is taking too long, there are a number of options to try out, which I'll cover in a later post.
Concluding remarks
You just saw how to run some basic memory profiling in your Python programs. Out-of-memory while analyzing a particular dataset is one of the primary hurdles that people encounter in practice. The memory_profiler package isn't the only one available so check out some of the others in the Further Reading section below.
If you liked this post, please share it on Twitter or Facebook and follow me @mathcass.
As a data scientist, it really helps to have a powerful computer nearby when you need it. Even with an i7 laptop with 16GB of RAM in it, you'll sometimes find yourself needing more power. Whether your task is compute or memory constrained, though, you'll find yourself looking to the cloud for more resources. Today I'll outline how to be more effective when you have to compute remotely.
I like to refer folks to this great article on setting up SSH configs. Not only will a good SSH configuration file simplify the way you access servers, it can also help you streamline the way you work on them.
I find Jupyter to be a superb resource for writing reports and displaying graphics of data. It essentially lets you run code in your web browser. However, one issue with using it on a remote machine is that you may not be able to access the interface because the server is blocking the necessary port to see it on the web(this is a great thing for security and prevents others from seeing your work). There's a way to work through this by using SSH's ability to forward ports.
To do that, first you'll need to log into your remote machine:
ssh -L 8888:127.0.0.1:8888 <remote host>
That means you're connecting to your remote host, except any time you want to access port 8888 on your local machine (127.0.0.1), it will forward it to the remote machine's port 8888.
Then, you'll need to start Jupyter:
cd <project>
jupyter notebook
Finally, head to the url to find yourself accessing the remotely running copy of your notebook.
Here's a screenshot of what that should look like.
Note the highlight line on the right. There isn't a web browser installed on my remote machine but I was still able to access this notebook by using my local computer.
Whether you want to load a 50GB data frame into Pandas or use jobs=-1 in Scikit Learn, you should find yourself more able to do your work.
Today we're going to talk about what a Bloom filter is and discuss some of the applications in data science. In a later post, we'll build a simple implementation with the goal of learning more about how they work.
What is a Bloom Filter?
A Bloom filter is a probabilistic data structure. Let's break that term down. Any time you hear the word "probabilistic" the first thing that should come to mind is "error." That is, it sometimes has errors. When you hear "data structure" you should think about "space," more specifically storage space or memory.
Bloom filters are designed to answer questions of set membership, that is, "is this item one of X?" Here are some simple questions you might be working with if you were considering them:
Have we seen this email address sign up to our site recently?
Is this a product the user has bought before?
Are the registrations we're seeing from this IP address on a whitelist of IPs that we can trust?
Basically, it compresses simple set membership information into a smaller memory footprint at the cost of a bit of error.
By design, Bloom filters only implement two types of operations, add and contains. So, once you've added a member to it, you wouldn't be able to remove it. Additionally, you wouldn't be able to query for a list of elements in it.
So, why would you be okay with error in data in exchange for using less memory? We can answer that question (as well as better understand the use-case of set membership) with a few useful examples.
How are Bloom filters used?
It would be hard not to mention this Medium article on the subject because it clarifies how they apply to a data science task. At Medium, they have some version of a distributed data store. Now, distributing data helps you scale out information across servers, but it also has a chance of increasing your variance in expected response times (read: more requests might take longer to finish running). Most web companies focus on making the user experience as pleasant as possible, so they value response time. In this case, one particular request that was important was the set of articles the particular user had read before.
At risk of retelling the already well-told story, suffice it to say that they used a Bloom filter to prevent recommending to the user articles he or she had read before. This was a case where they used a data science model (the store recommendation engine) and they augmented it by including a component that could use compressed information (the Bloom filter) to prevent the user from needing to see the same article twice. Moreover, even though there's potential for error in that filter, that error is negligible (in the sense that the user's experience isn't hinder by it). To summarize, because they could efficiently represent the set of "articles this user had read before" and since it had a defined error rate, they could improve their user experience.
As another example, think about buying items on Amazon.com. Amazon has almost any item imaginable and probably also distributes their data for scale. They're still able to tell you right on the product page whether you've bought an item before and when. I don't have any insight into what's going on behind the scenes but this is another perfect place for a Bloom filter by using one at the user level (holding the set of products someone has bought) or at the product level (holding a set of all the users who have bought the product). A negative match (which will be correct 100% of the time) means operationally you don't need to perform that database lookup to see if someone bought this item. A positive match (which will probably be rare most of the time) will be the only time when you confirm and go see that transaction data.
Finally, I wanted to point out a use-case that I found by perusing various implementations of the tool. Bloom filters can also be used to track time-dependent information (or various forms of time series data). One thing you could do is store aggregate level information (like whether someone bought a particular product in a given time horizon like the last 30 or 90 days) in a Bloom filter. Then, based on that information, you can make modeling decisions like what sort of ads you show this person.
I hope this post helped you learn a little bit about Bloom filters. In a later post, I'll go into some detail on how they're implemented with a focus on pedagogy. In the meantime, follow me on WordPress or Twitter for updates.
Additional links to check out
A Python package from Moz on using Redis as a backend (they were using it to ensure they didn't crawl the same websites multiple times)
A very detailed article of several other probabilistic data structures
A great bitly post on the subject as well as their own implementation (which also supports removing set members)
]]> useful books for learning Data Science
15 Jan 2016 12:45:35 +0000 was listening to an old episode of Partially Derivative, a podcast on data science and the news. One of the hosts mentioned that we're now living in the "golden age of data science instruction" and learning materials. I couldn't agree more with this statement. Each month, most publishers seem to have another book on the subject and people are writing exciting blog posts about what they're learning and doing.I wanted to outline a few of the books that helped me along the way, in the order I approached them. Hopefully, you can use them to gain a broader perspective of the field and perhaps as a resource to pass on to others trying to learn.
I first found Learning from Data through Caltech's course on the subject. I still think it's an excellent text but I'm not sure if I would recommend it to the absolute beginner. (To someone who is just coming to the subject, I would probably recommend the next choice down on the list.)
However, I have a Master's degree in mathematics so I was familiar with the background material in linear algebra and probability as well as the notation used. Learning from Data taught me that there was actual mathematical theory behind a lot of the algorithms employed in data science.
Most algorithms are chosen for their pragmatic application, but they also have features in and of themselves (such as how they bound the space of possible hypotheses about the data) that can help determine their effectiveness on data. There's also a general theory for how to approach the analysis of these algorithms. At the time of reading, a lot of it was still a bit over my head, but it got me incredibly curious about the field itself.
Now, understanding a few things about the theory is great, but most of the time, people want to know what it can actually do.
I'll admit to only having had a cursory understanding of what was possible before I read Data Mining Techniques. I knew that the most widely used algorithms were used for assessing risk, like credit scores. However, I didn't know much about how you could make gains in the world of marketing using data science techniques.
I appreciated that the authors have a lot of experience in the field, especially experience that predates most of the growth in big data these days. This book makes it clear that many of the most useful algorithms have been around and in use for decades. The authors also offer some explanations from the direct marketing case (print magazines and physical mail) that I hadn't considered, such as ranking algorithms, which were originally used to prioritize a list of people to contact because of the high costs of mailing paper to people.
More than anything, I liked the breadth of the topics, since they cover just about every form of marketing algorithm and do a great job of giving you a high level view of why they matter.
You won't walk away from this book knowing how to implementing everything they talk about, but you will get a sense for which algorithms are suited for particular tasks.
This book gave me a better way to think through the initial phases of a project, but I still needed some help in learning how to communicate about data and how to fit it directly into the business context.
I read through this one while I was on vacation (yes, I know, I'm that type of geek). That didn't stop me from soaking up a lot of information from it about how data science applies to a company trying to use these models. Most of the book is focused on helping you think through how to operationalize the process of running and managing a data science project and what outcomes you might expect from the effort.
Beyond that, I think it taught me how to communicate better about data at a company. Being able to talk about the many months it will take to bring a project into fruition and weigh it against alternatives is the bread and butter of working at a company that wants to make money. Moreover, if you believe that a particular project is the right choice, you need to be able to back up that choice by communicating about the benefits.
I want to say that this is a very "bottom-line" type of book, but that's okay to hear about some of the time. Data science doesn't always have to be about the hottest technique or the biggest technology if your priorities include keeping your costs below your revenue. However, I still didn't learn much about getting my hands dirty with the data on a day-to-day basis. For that, I had to rely on the final book I present.
This is a book on predictive modeling in R and on using a package that the author developed for doing that. This isn't simply about someone tooting their own horn because caret is a quality piece of software. Overall, I think that even if you don't end up using R as your go-to tool for analyzing data, you'll still learn a lot from this book. It thoroughly demonstrates the power caret can offer you in a project, to the point that you'll seek the same functionality in your tool of choice (or hopefully build its equivalent for us).
Caret is a package that offers a consistent interface for just about any predictive task (classification or regression) that you could ask for. One issue some people have with R packages is that the interface for algorithms isn't very consistent. Learning how to use one package won't always lead to the same understanding in a completely different package. Caret addresses that by giving you the same way to set up a modeling task for many different algorithms. Moreover, it also automates several tasks like:
Data splitting into training and test sets
Data transformations like normalization or power transforms
Modeling tuning and parameter selection
Essentially, it makes working in R a lot like using Scikit Learn (an excellent library itself) but with many more options and model implementations.
So that's all you need, right? Just read a couple of books and you're on your way? Not quite. You'll actually have to apply some of this and learn from it. Perhaps next time you're in a meeting discussing priorities for your company, you will need to frame the conversation about your next data project and directing the data effort toward your business goals (Data Science for Business). When you're brainstorming possible things that you could try to predict and use in a marketing campaign, you will need to outline the possible techniques and what they could offer you (Data Mining Techniques). If you're evaluating candidate algorithms for their ability to perform the task accurately, you will need to gauge their effectiveness from a theoretical (Learning from Data) and practical (Applied Predictive Modeling) standpoint.
I hope this helps you apply data science at work and gives you perspective in the field. Also, if you're not a follower on Twitter, please follow me @mathcass.
I've recently been reading a great book on how people make decisions and what organizations can do to help folks make better choices. That book is Nudge.
What is a nudge?
The authors describe a nudge as anything that can influence the way we make decisions. Take the primacy affect, for instance, namely the idea that order matters in a series of items. We're more likely to recall the first or last option in a list of items simply because of their positions. This would be a nudge if you later chose the first movie from a list that a friend had recommended mostly because it was the first one to come to mind in the store.
The fact that humans have these biases is in indicator that we don't always act rationally. In cases where we haven't had enough experience to learn from our decisions, we need a bit of help finding the most appropriate option for our needs. Most people only decide what type of health care plan they need or at what rate to contribute to their retirements plans a few times in their life, so there isn't much opportunity to learn at all.
All in all, the book is a great read, and much of it is an explanation of how proper nudges have excellent applications in areas like health care and making financial decisions.
How does Data Science fit in?
So, why bring in Data Science? Well, lately companies have been looking to the field of Machine Learning and Statistics to determine how to make better business decisions and these methods can play an important role in helping define the right nudges to use.
The authors emphasize that proper nudges should a) offer a default option that is stacked in the favor of most people and b) make it easy to stray from the default option as needed.
When I think about those two, a few things come to mind. In Machine Learning, a mathematical optimization takes data about outcomes and selects the best set of choices. And recommender systems are designed to, when given a few hints, offer up suggestions of similar or like items.
In the case of deciding on the most favorable default option, that decision should be made based off of the available data. The authors talk about health-care and Medicare Part D and the fact that the government randomly assigned plans, thereby leaving most people in a sub-optimal situation. An approach to solve this problem given the available data would have been to make a survey of citizens and their prescription needs, and then selected a default plan from every option in a way that minimize some variable, such as the median cost per participant.
Additionally, the authors describe a tool for Medicare Part D that allowed someone to input their prescriptions and assigned someone a plan to choose from. One of the difficulties with this system was that it rarely gave the same answer, even with the same inputs, because the plans would change over time. This gave people a false sense of which plan was good for them. A better approach would have been to give recommendations of appropriate plans, by taking the drug information and matching it to available plans. When presented with 100s of options, people have a difficult time making a choice that will work, but if those 100 could be winnowed down to the 3-5 most appropriate ones, people will have an easier time weighing the pros and cons.
Obviously, there is still plenty of constructive work to be done in supporting any nudge. And I believe that the tools that Data Scientists use day-to-day are valuable to keep in mind in these efforts.
]]> Power of Perspective
14 May 2015 01:40:31 +0000 used to be a person who would get jealous at others, namely their technical ability. If I thought that the person I was working with were better at math or programming compared to me, it'd cause a drive in me to get better at both of those. I'd pour myself into books on the relevant subjects to try to enhance my ability. I'd work on projects to try to get familiar with these advanced techniques. I'd be lying if I said this didn't help me become a better programmer or analyst, but I definitely it increased my stress levels more than I needed.
I think I was missing the point the entire time. I lost sight of two factors that hadn't occurred to me. First, I hadn't even known that my perspective was incredibly full of worth. Secondly, I had forgotten that the people I was jealous of weren't even really doing work that I truly wanted to to. Think about that for a moment.
"There will always be people who are better than you at something." At least that's what I keep hearing people say when it comes to life, work, and career progress. So, if that's the case, then how did your boss get hired? Or the CEO of a public company? Couldn't they have just found someone better than they are to do the job? I'm almost sure of it, but I'm betting it's not because of raw ability in any particular skill but rather it's because of the perspective they bring to the table. Your viewpoint is an extremely valuable asset. How you think about a situation or problem is more unique than you think it is and if your boss isn't using your perspective to enhance his or her own view, both of you are losing out.
On the other point, you have to ask yourself if you're really doing work that you want to do. Will mastering the skills you're working on get you to the job that you want. Additionally, I've been in situations where a colleague is working on the project that I want to work on. The project. However, every time this has happened, it's because I never voiced my interest in working on it. And then half the time, the person assigned the work didn't want to do it nearly as much as I did.
In essence, don't ever overlook some neglected assets, especially when it comes to sharing how you see the world (and your work) and your unique desire to persue a particular kind of work. |
Study Tips
Learning Physics
The Nature of Physics
In the evolution of western, scientific human thought the step out of
the "ocean" onto the "shore" was the very revolutionary idea that we
should seek some sort of understanding of phenomena by looking at the
numbers generated by (freely designed) measuring devices. I think
this began with Boyle, et al, in the 1600's. Prior to this, such a
mode of understanding was not even a part of human thinking. Since
then it has become the essential feature of every human invention
which goes by the name of quantitative science .
Physics is a logical structure composed of quantitative statements,
and the language it is written in is mathematics. Historically
mathematical expressions have often proven to be applicable far beyond the
phenomena that first gave rise to the expression. As a result I believe
that the mathematical nature of physics is a reflection of reality
itself being mathematical. However, pure mathematics is not physics;
physics must correspond to observable and measurable events. In
physics there are always pictures behind the mathematics. If there
are no pictures there is no physics, and if you don't understand the
pictures you don't understand the physics. In essence, mathematics is
most useful as a compact description of the pictures.
Physics has levels of understanding, and in fact physics is more deep
than broad. If the mathematics at the first level is properly
understood, that mathematics acts as one of the pictures for the next
level. Thus full understanding at deeper levels cannot be achieved without
first having obtained reasonable understanding of the mathematics of
simpler levels. At each stage you must translate the pictures into a
mathematical understanding that you can use to construct your
qualitative understanding of the next stage. The interplay between
qualitative understanding and quantitative calculation is vital.
Neither alone is physics.
The Nature of Physics Knowledge
Psychologists have studied the question of how physics knowledge is
stored and used in the human brain. I am most familiar with the work
of Jill Larkin, although certainly many others have contributed.
Larkin's work involved having beginners and experts work standard
physics problems aloud.The records of this work were analyzed for
patterns of thought, and especially for differences in the patterns of
thought of beginners and experts. I will come back to this work in
talking about how to solve physics problems, but for now what I am
interested in is the way people store physics information
mentally. The most compact form of physics information is of course
equations, and equations are stored mentally.But they are not stored
alone.Two features of the storage stand out:
Equations are stored in related groups: ask an expert for a particular
equation and he/she also recalls several other equations automatically.
Pictures of various types and English explanations always accompany the
equations.
An example of a group of equations that are always stored together are
the equations for motion under constant acceleration. Incidentally,
experts always store a label of this type with the equations. The
label ensures that the equations are not misused in situations to
which they don't apply. The equations for constant acceleration are
x = x0 + v0t + (1/2) a t2
v = v0 + a t
The definitions of the quantities are recalled whenever an expert
recalls the equations themselves:
x is the position of a moving object
v is the velocity of the object
a is the acceleration of the object (a constant, remember)
t is the time at which the position and velocity are measured
A subscript of 0 means the value of that quantity when t=0.
There are some additional things I have observed in myself and other
trained physicists. I carry a picture of the graph of the
equations in my head which I recall whenever I see an equation of
reasonably simple form. For complicated equations I generate a mental
graph of individual terms or even of factors within the terms. Given
v = (1/2) a t2, my mind reacts with "parabolic in t, it
will rise quickly." However, the reaction is conceptual, not in terms
of a statement in English.
Another form of visualization is providing interpretations for the
symbols. I read "x", I say "x", I write
"x", but my mind always reacts "position." I may even translate an
equation mentally, just as I would with a difficult phrase in a
article in French. Hence
v - v0 = a t
is read mentally as "the change in velocity is given by the
acceleration times the elapsed time."
Experts always keep the physical meaning of equations in mind. The
equation itself is just a sequence of nonsense syllables. What is
important is what the symbols mean. I have a mental image of my moving
around a fixed starting point, changing velocity or not according to
the situation. Sometimes I am the starting point and it is a
car moving. Often I visualize a checkerboard pattern on the ground as
a kind of graph paper. I do similar things with other equations,
whenever possible.
How to Learn Physics
Keep up with the homework. In a recent class I was led to calculate
the correlation between number of homework problems attempted and the
overall semester grade for each student. I found that
87% of the variation of the semester scores of
students in the class could be predicted using only the percentage of
homework problems they attempted. A couple of extraneous things raise
this correlation: All of these students were honors students quite
capable of learning physics, so there was little competing talent
effect on the correlation, and some of this correlation comes about
because the same personal qualities that lead to getting homework done
on time are good for other aspects of learning physics. Nonetheless,
the correlation is too large to be ignored. Do the homework.
Physics must be "understood." You will hear this from me and other
physics teachers again and again, only typically we don't explain what
we mean. "Understanding" refers to making the connections between
phenomena we know about and the logical/mathematical descriptions of
the phenomena that make up physics theory. For each principle and
equation, find something you are familiar with or at least have seen
demonstrated that is an application of the principle. Figure out how
changes in the principle that seem otherwise plausible to you would
conflict with the familiar events. Use these conflicts to
"understand" why the principle is expressed the way it is rather than
in some alternate form. Practice making these connections. Ideally,
after you are done, you find it difficult to understand how a piece of
physics could be any different than it is. If you have to memorize
physics by rote, you don't understand it yet.
I do not mean to say that memorization is unnecessary. Definitions
are essentially arbitrary, and at least some will have to be
memorized. Moreover, memorization provides fast recall of equations
for tests, is a familiar process you already know how to do, and by
now you have a pretty good feel for when you have something reliably
memorized. It is also a boring, time-consuming task, and memorized
information is prone to disappearance in the middle of a test when you
need it most. So you want to memorize judiciously and have backup
ways of recalling information if necessary. If you know you can
derive or reconstruct an equation, you are far less likely to forget
it anyway.
So here are some tips on what to memorize and how to organize your
memory work:
The structure of physics can be used to help you learn equations.
Prepare study sheets (sample) with equations in
logical groups and in logical order within the groups. Arrange the
groups themselves in a logical order. It is easier to remember
structured information than isolated facts.
You have to know the meaning of equations to use them, so
memorize the meaning in English and practice translating the English
into equations. Or if you prefer, make sure you practice translating
the equations into English in order to retrieve the meaning.
On study sheets, always include diagrams with your equations to
show the meaning of the variables in the equations. When possible
include a graph that shows the behavior of the equation.
Use units to help construct equations.
Make use of the fact that most quantities come into simple
equations to the first power only. Use qualitative arguments to
figure out whether the quantity is in the numerator or denominator.
Take special note of quantities that are not present to the
first power, and if possible figure out why they occur the way that
they do. This technique is especially effective in checking that
an equation you are not sure of is in fact correct. |
This Mind-Boggling Map Explains How Everything in Mathematics Is Connected
Advertisement
Unless you
were a total pro at mathematics in high school, you probably only have a vague
recollection of things like geometry, algebra, and some guy called Isosceles (what a great name) and
that sucks, because mathematics is one of the most fascinating languages
humanity has ever devised, but without university-level expertise, you're going
to have a very bad time trying to figure out how things like chaos
theory and fractal
geometry tie in with machine learning and all those crazy
prime numbers we keep finding.
"The
mathematics we learn in school doesn't quite do the field of mathematics
justice - we only get a glimpse of one corner of it, but mathematics as a whole
is a huge, and wonderfully diverse subject," Walliman says in the video
below.
To navigate
this complex and busting Map of Mathematics, the best place to start is in the
middle, where the orangey brown circle depicts the origins of human interest
into how numbers explain our Universe:
We've then
got two main sections that represent the two major fields in mathematics today
- Pure Mathematics (an appreciation of the language of numbers itself) and
Applied Mathematics (how that language can be used to solve real-world
problems). You can mess around with and download a high-res, zoomable version here, and
print it on a throw pillow here,
because we all need something to look at on the couch when Taboo is getting
a little too weird.
To fully
appreciate Walliman's Map of Mathematics, you should definitely watch the video below to
get the proper walkthrough. All those names of things - topology, complex
analysis, and differential geometry - might not sound like much to you now, but
you'll soon learn that they're really just describing the shapes of things in
our Universe, and the way those shapes change in time and space are explained
by things like calculus and chaos theory.
Now that
you've made it through the trickiest theoretical stuff, it's on to applied
Mathematics, which applies to the disciplines of physics, chemistry, and
biology, where number systems are integral to understanding how the Universe
and everything in it behaves. You've also got engineering, economics, and
game theory, and probability, cryptography, and computer science - all of which
simply wouldn't exist had our very cluey ancestors not laid the foundations of
number-sleuthing for us centuries ago.
What's that?
Mathematics literally applies to everything in life and the
Universe? [Internal cheering by maths teachers intensifies]
If all of
this sounds all too basic for you, don't worry, there's more to this map than
just pure and applied mathematics. It even covers what could be the biggest
mystery of the entire discipline - how researchers examining the foundations of
maths have failed to find a complete set of fundamental rules, called axioms,
that are provably consistent across every little nook and cranny of the
mathematical universe. |
The Meaning Of Numbers: 5 / Numerology | Andrea's Number
In this video, I explain the meaning of the number 5. Footage was taken in Times Square, New York City several years ago when I was a travelling person |
How Does A Calculator Work?
In religion we hear a lot about faith. I've seen it described as "belief without evidence." Something that is difficult to comprehend, or explain, but we have complete trust. It is as if we are saying, we are not sure how it works, but we don't doubt that it works.
This is kind of how I would describe my relationship with a calculator. I have no clue how it works…but I know it does. Sometimes I will test the calculator and see if I can catch it having a bad day. I will enter 99 x 99 and think that maybe this will be the time is spouts out 9999 instead of 9801.
Guess what? It says 9801, which is correct. It is always correct. Never once have I gotten a wrong answer from a calculator (although, if I need a calculator I wouldn't know what the right answer is, so I wouldn't know if the calculator's answer is wrong).
How does it do that? Time to ditch the faith and start to learn some answers in today's edition of Wonder Why Wednesday…
How Does A Calculator Work?
Fun fact…the first calculators were not small. In fact, they were so big they had to be built into a desk. According to Wonderopolis, the Casio Computer Company released the Model 14-A in 1957, thus creating the world's first all-electric compact calculator.
Four years later the British Bell Punch/Sumlock Comptometer ANITA reduced the size to a trim and lean 33 pounds.
With the advancements in technology, calculator size and cost kept going down and down until the 1980s when the devices were small enough to fit in your pocket and cheap enough to be common in many schools.
All that is great, but it doesn't answer the question of how it works.
Calculators, like Mexican restaurants rely heavily on chips. These chips, known as integrated circuits contain transistors that can be turned on and off with electricity to perform mathematical calculations.
They do this by processing the information in binary form. Like a kite, binary form relies heavily on string. Binary uses two digits to do the work: 0 and 1. With the help of chips, our calculator takes the numbers we enter (99 x 99 in my example above) and converts them into binary strings of 0s and 1s.
The chips use those strings to turn transistors on and off with electricity to perform the desired calculations. Confused? Me too. So I will turn to Wonderopolis who says,
Since there are only two options in a binary system (0 or 1), these can easily be represented by turning transistors on and off, since on and off easily represent the binary options (on = 0 and off = 1 or vice versa). Once a calculation has been completed, the answer in binary form is then converted back to our normal base-ten system and displayed on the calculator's display screen. Most calculator displays use inexpensive technologies common today, such as liquid crystal displays (LCD) or light-emitting diodes (LED)."
Got all that? So now I am picturing that if I were to open up a calculator I would find a bunch of chips and string. Throw in a random penny or two and you would have the exact same thing I find when looking between my couch cushions.
I think I might just continue to rely on that who faith thing when dealing with calculators. |
Why would anyone investigate your hallucination? It would cost money and we have no reason to think it would tell them anything they didn't know. Also, once again, you are ignoring the fact that the experiment has been done. It showed that pi was constant.
A fixed number for a pi is boring. Variable number to pi, it is dramatic and intriguing. Therefore, I have no doubt that a scientific institution will conduct the experiment, and will merit fame. I suggest you wait patiently, the historical experiment will come. I have a lot of patience.
A small square is perfectly similar to a large square. Lines of squares will overlap perfectly with each other A small circle is not exactly like a large circle. Lines of circles will never overlap. Hence the idea of changing pi
That's a good start. Now draw a a circle in the square so that it just touches all 4 sides. The circle has a diameter that is the same as the length of the side of the square Then start to fill the gaps between the circle and the square with smaller squares. Using smaller and smaller squares you can fill the whole of the gap as closely as you like. Here's a very rough sketch. squares.jpg (16.81 kB . 362x346 - viewed 623 times)
Then do the same, but starting with a bigger square.
You can find the circumference of the circle by adding up the sides of the squares next to it. The two sets of squares and circles are the same. So pi is the same. |
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Number A number is a mathematical object used to count, label, and measure. In mathematics, the definition of number has been extended over the years to include such numbers as zero, negative numbers, rational numbers, irrational numbers, and complex numbers. [Wikipedia (En)] |
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Read Sir Cumference and the Dragon of Pi with your kids. We found a copy at our local library. It's not only a great story, it's a clever way to help kids understand Pi. In the story, Sir Cumference drinks a potion that accidentally turns him into a dragon. His son, Radius, must find a cure. Along the way, he meets Geo and Sym Metry, brothers who make wheels and helps his cousin, Lady Fingers bake pies and solves the mystery of Pi.
Now have the kids try what Radius did in the story. Take a piece of string and measure the diameter (d) of one of their favorite round snacks. We used an orange.
In the story, Radius took equal lengths of pie crust to create a wheel. The spokes reached across the pie fine, but when he tried to create the outside circle of the wheel, three wouldn't fit. There was a gap.
Now, have the kids carefully triple the length of string. Essentially, they're multiplying the diameter by three. Cut the string.
Try to wrap the string around the outside of the orange (or whatever food you used). It won't reach all the way around; there's a gap. That gap is why we need Pi. It's that .14159265359… that we need for our string to reach all the way around.
So this Pi day, celebrate Pi along with all things round and irrational, like pie for breakfast. Okay, maybe that's not exactly irrational, but it is delicious. Happy March 14!
P.S>. Did you know that March 14 is also Albert Einstein's birthday? Happy Birthday Al!
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Jennifer Cooper is the blogger behind Classic-Play.com, an online resource for creative families. Her favorite past times include: dancing around her living room, watching the Pink Panther with her kids and daydreaming. She lives in Baltimore, MD with her husband, photographer Dave Cooper, and two children. |
The geometry of weird-shaped dice
skullsinthestars
10 months ago
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I've been enjoying a bit of reminiscing about my childhood lately, hunting down old copies of role-playing games I enjoyed in my youth as well as exploring newer games that have come out since then. One thing that has changed dramatically since my gaming days is the proliferation of types of dice. Most human beings never go beyond ordinary 6-sided dice, which we in the gaming world call a "d6." Classic Dungeons & Dragons players, however, are familiar with the d4, d6, d8, d10, d12, and d20.
But these days, there are even more imaginative varieties! I've starting collecting dice of every shape and size, and my current collection is shown below, in order¹:
This is a really amazing variety of dice! This is my "special dice" collection in its entirety, and includes some duplicates (don't ask how I got 4 identical d60s), but in some cases, such as the d7, d12 and d24, there are varieties in shapes even with the same number of faces!
This variety got me wondering: how does one design dice with a weird number of faces? What mathematical strategies does one use to make them? What other types of dice are possible? And, perhaps most important: are these dice "fair"?
I thought it would be fun to answer these questions with a blog post, in which we discuss the geometry of dice!
We will start with the most familiar types of dice, and work ourselves gradually into strange and unfamiliar territory. Before we begin, we should note that a major consideration for any type of die is that it be "fair": that is, every number on the die should be equally likely to be rolled. The most obvious way to do that is to make the die have a lot of symmetry in its shape, which brings us to our first category…
The Platonic solids. The most symmetric shapes, as their name implies, formally date back to the Greek philosopher Plato (428-348 BCE) , though most of them were recognized, or at least crafted, even earlier. The Platonic solids include the d4, d6, d8, d12, and d20, as shown below.
The Platonic solids, taken from an old set of Dungeons & Dragons dice.
The Platonic solids are the only polyhedra (multi-sided objects) which are convex (have no concavities) and regular. A "regular" polyhedron is one for which not only are all faces equivalent to one another, but so are all edges and all vertices (points). To put it another way: a Platonic solid can be rotated to make any edge, vertex or face look exactly like any other one. Obviously, this is ideal for making a fair die — there is no preferred edge, vertex or face on the solid.
The Platonic solids are somewhat profound, in that there are a small number of them, and no (obvious) reason why they have the number of faces that they do. This profundity captured the imagination of Plato, who gave the solids a fundamental role in his dialogue Timaeus (c. 360 BCE), assigning each of them to one of the elements. From Timaeus,
To earth, then, let us assign the cubical form; for earth is the most immoveable of the four and the most plastic of all bodies, and that which has the most stable bases must of necessity be of such a nature. Now, of the triangles which we assumed at first, that which has two equal sides is by nature more firmly based than that which has unequal sides; and of the compound figures which are formed out of either, the plane equilateral quadrangle has necessarily, a more stable basis than the equilateral triangle, both in the whole and in the parts. Wherefore, in assigning this figure to earth, we adhere to probability; and to water we assign that one of the remaining forms which is the least moveable; and the most moveable of them to fire; and to air that which is intermediate. Also we assign the smallest body to fire, and the greatest to water, and the intermediate in size to air; and, again, the acutest body to fire, and the next in acuteness to, air, and the third to water. Of all these elements, that which has the fewest bases must necessarily be the most moveable, for it must be the acutest and most penetrating in every way, and also the lightest as being composed of the smallest number of similar particles : and the second body has similar properties in a second degree, and the third body in the third degree. Let it be agreed, then, both according to strict reason and according to probability, that the pyramid is the solid which is the original element and seed of fire; and let us assign the element which was next in the order of generation to air, and the third to water. We must imagine all these to be so small that no single particle of any of the four kinds is seen by us on account of their smallness: but when many of them are collected together their aggregates are seen. And the ratios of their numbers, motions, and other properties, everywhere God, as far as necessity allowed or gave consent, has exactly perfected, and harmonised in due proportion.
So fire is a d4, earth is a d6, air is a d8, and water is a d20. Of the 5th Platonic solid, the d12 dodecahedron, Plato vaguely says, "There was yet a fifth combination which God used in the delineation of the universe. "
Later thinkers also thought the Platonic solids played a fundamental role in the cosmos. In 1596, the German astronomer Johannes Kepler published the book Mysterium Cosmographicum, in which he speculated that the positions of the known planets corresponded to the Platonic solids inscribed within one another, as illustrated below.
Obviously, this model did not work out, but it illustrates how the set of 5 solids captured the imagination of philosophers trying to make sense of the universe.
There is a big problem with the d4 when being used as a die, though: it doesn't roll very well! One solution in recent years has been to truncate the tips of the tetrahedron and place the actual numbers on the tiny flat surfaces. Such dice are easier to roll and to read the result. A traditional d4 and a newer one are shown below.
The only downside of the newer d4 is the extremely slim chance that it could land right on one of the small flat surfaces. That is really, really unlikely, though.
There is a curious illusion associated with the d8 that is worth noting before moving on. You may note that the d8 can be created by taking two square-base pyramids made of equilateral triangles and fixing their bases together. Looking at such a die, it seems like it is mirror-symmetric across the plane of the bases, but not so symmetric that every edge and vertex is the same!
Two d8s side by side, in the "upright" position.
I interpret this as an illusion due to the way the numbers are put on the die. They are all placed with their bottoms towards a single base plane, which makes it look like this is the only symmetry. However, if you look at the die from other edges, you can see that it looks the same! There are 3 different base planes.
Two d8s side by side, but tilted along a different axis for each die.
If this isn't obvious from looking at actual dice, it is much clearer looking at a transparent model. See if you can trace the 3 squares formed by the edges in the image below.
An octahedron, via Wikipedia. There are 3 square "base planes": the one that is horizontal, the one that is vertical and facing us more or less edge on, and the one that is vertical and lies in the plane of the picture.
Mirror-symmetric dice. But the idea of mirror symmetry provides a way to make dice of any large even number of sides! We can make a "pyramid" of sorts with any number of faces and a base and glue two of them base to base; as noted, a d8 can be thought of as gluing two 4-faced pyramids together. In this way, we can construct the final member of the original Dungeons & Dragons dice, the d10. We can also, however, make a different form of d12, as well as a d14, d16, or higher! Such a structure is formally known as a trapezohedron. My d10 and d16 are shown below.
A d10 and a d16. Note that the "10" on a d10 is labeled "0" because it is used in percentile calculations.
You may notice a slight difference between the d10 and d16: the faces of the d10 are shifted off-center across the mirror plane, while the faces of the d16 are aligned across this plane. This shift of the d10 is necessary because there are 5 faces on either side of the mirror plane. If the faces across the plane were lined up, then the die would come to rest on one face, but an edge would be facing upward! The shift means that a face is upward when the die comes to rest. This is necessary for any mirror-symmetric die with an odd number of faces on either side of the mirror plane.
One can take the idea of mirror symmetry to a ridiculous extreme. Below is pictured a d50, which uses the same principle. Note that it also has its faces shifted, because there are an odd number of faces on either side of the edge!
A 50-face trapezohedron, via Wikipedia. I think this die is ridiculous, so it's one of the few I don't own.
Face doubling. So far, all the dice we've considered have had an even number of faces. The trapezohedrons, however, suggest a very straightforward, if unimaginative, way to make dice with an odd number of faces: simply double up on the lower numbers! In this way, a d6 becomes a d3, a d10 becomes a d5, and a d14 becomes a d7, as shown below.
D&D players were doing this for years before actual dice were made: players simply said, for instance "a roll of 1-2 on a d6 is 1, 3-4 is 2, and 5-6 is 3." We were certainly able to handle that, but it is just nice to have dice explicitly marked for this strategy.
My favorite of this type is the d2: the 2-sided die! As can be seen below, it is in essence a die for which 3 sides in a "U"-shape represent 1 and the other 3 sides in a "U" represent 2. Each trio of sides has been rounded together to make the die have, effectively, 2 rounded sides!
A 2-sided die, with some runic bling drawn along the edge.
Lots of people will say, "Why do you need a 2-sided die? Can't you just flip a coin?" You could, but gamers such as myself love to roll dice. Flipping a coin just doesn't have the same feeling to it!
Catalan solids. We will have more to say about odd-numbered dice in a few moments, but first let's return back to dice with an even number of faces. Is there any way, other than trapezohedron, to create more varieties?
In fact, we can, by removing some of the symmetries of the Platonic solids. The Platonics, if you recall, were symmetric in faces, edges, and vertices: that is, every face, edge and vertex was equivalent to every other one. But, for a fair die, it would seem that we really only need every face of the die to be equivalent: the Platonic solids have, in a sense, more symmetry than we need. If we throw out edge and vertex symmetry, this leads us to the d24, d48, d30, d60 and d120!
These objects are members of a group of solids known as Catalan solids, after the Belgian mathematician Eugène Charles Catalan who described them in 1865. The complicated names of those pictures above are, respectively, the tetrakis hexahedron, the disdyakis dodecahedron, the rhombic triacontahedron, the deltoidal hexecontahedron, and the disdyakis triacontahedron.
That's a lot of wordage, but we can understand most of these Catalan solids as extensions of the Platonic solids. For example: suppose we take each face of a cube, and "pull" it outward to make it a narrow pyramid, as shown below.
We have now created a tetrakis hexahedron, as there are 4 sides per pyramid and six faces to the cube! 4 times 6 = 24, obviously! It should be noted that the vertices are no longer all the same: the peak of each pyramid is not equivalent to one of the corners of the former cube.
We can similarly think of the d48, the disdyakis dodecahedron, as making an 8-sided pyramid out of each face of a cube. Alternatively, we can think of it as making a 6-sided pyramid out of each face of a d8 (octahedron).
There is another Catalan solid that is a d24, as well; it is known as the deltoidal icositetrahedron, and is shown below.
This 24-sided die can be viewed as making a triangular pyramid out of each face of an octahedron.
The d60 and d120, the deltoidal hexecontahedron, and the disdyakis triacontahedron, may similarly be thought of as modifications of a d20. For the d60, we make a 3-sided pyramid out of each face of the d20, and for the d120, we make a 6-sided pyramid. I highlight these pyramids in the figure below, if it isn't clear.
Some Catalan solids cannot be simply fashioned from the Platonic solids, however. The d30 is one of those, as is a different version of a d12, a shape known as a rhombic dodecahedron. These two are shown below.
The rhombic dodecahedron has the curious property that its shape can be used to perfectly fill a volume in three-dimensional space. That is: if you had enough of these d12s of the same size, you could stack them together to fill a region of space without any gaps, just as you could with a bunch of d6s.
There are a few other tricks we can do with these two. If we take the rhombus that forms the faces of the d30 and pyramid it into four equal sections, we end up with a d120! If we take the rhombic d12 and pyramid up each of the faces right in the middle, we end up with the d24. These divisions are illustrated below.
It should be noted that the d60 and d120 end up pushing the limits of what might be considered a useful shape for a die. They are both quite spherical (the d120 is larger than a golf ball), and will roll for a long, long time before coming to rest on a number. Furthermore, the numbers are so closely packed together that it can be somewhat tricky, at a glance, to figure out what the actual rolled result is! It may be possible to make dice which have an even larger range of numbers, but they wouldn't be terribly practical.
Crystal-shaped dice. Let's now return to dice with an odd number of sides. Dice makers are not, of course, limited to making dice that conform to some standard of geometric perfection! Inspired by the mirror symmetry dice described earlier, one can simply remove the edge across the mirror plane, effectively reducing the number of sides in half. If the faces are to remain flat, this means that the edges have to be rounded, resulting in something like the d3 pictured below.
If the image doesn't make it perfectly clear, picture it as a filled tube with a triangular cross-section, and the edges are rounded. Provided the edges are rounded in the same way, it is still a fair die, because each of the 3 flat sides is the same.
This same idea has been used to design what are often referred to as "crystal dice," which look like amethyst crystals or something similar. For instance, here's a lovely set of crystal d4s, via Gamemaster Dice.
Here the edges aren't rounded, but beveled to a peak, so that the die cannot rest stably on those flat edges. Some manufacturers have been particularly imaginative with the tube shape of these dice, making them look, for instance, like a rocket ship.
The cylindrical d5. As we carry on to even more varieties of dice, we begin to enter truly strange territory. For instance, consider the d5 pictured below.
We can view this as being similar to the d3 discussed above, except that the ends of the triangular cylinder have been made flat and turned into additional faces instead of being rounded. But now we have a problem: the faces of the sides of the cylinder are of a different size and shape than the faces of the ends of the cylinder: how can we possibly know if this die is fair or not?
Arguably one can do some calculations to prove it, or do brute force testing of dice of various lengths, but we will simply argue that there must be a size of die which is fair; presumably the dice-maker figured out what that size is! Imagine first a die which is made of a very, very long cylinder, for instance with the same length to thickness proportions as a pencil. Such a die will be almost certain to land on its length, on one of the three sides around it. If we now imagine a die which is made of a very, very short cylinder, like a coin, it is almost certain to land on one of its ends.
Somewhere between the long die which will land on its length and the short die which will land on its end, there must be a die of a length which will land 2/5ths of the time on an end, and 3/5ths of the time on a side. Because each of the ends are the same, and each of the sides are the same, this die must be fair! This idea is illustrated below.
In this case, we have again taken advantage of the symmetry of the triangular cylinder: because each of the ends is the same and each of the sides along the cylinder length is the same, there is only one free parameter to tune — the cylinder length — in order to get a fair die.
This design is based on spacing points as equally as possible on a sphere and then cutting planar slices perpendicular to those directions.
In short: the plan is just to make the die as symmetric as possible, slicing faces at points that are roughly equally spaced on a sphere. It turns out that it is not even possible, except in the Platonic cases we've already considered, to space points equally on the sphere, and there are a number of different ways to define and calculate equal spacings; a bit of mathematical description is given here.
Are these dice fair? Odds are against it, but there's one way to find out: roll it a lot of times and see what happens! On one lonely night in a hotel room during a recent work trip, I did just that with a d7. (I only tested the d7 because bigger dice would require many more rolls to get good statistics.)
The tabulated results are as follows:
1: 63
2: 57
3: 72
4: 113
5: 75
6: 81
7: 39
There were 500 total rolls; if the die was fair, I would expect about 71 rolls of each number. Clearly there is a bias towards rolling a 4 and a bias against rolling a 7! The other numbers are relatively balanced, though evidently not perfectly.
So the die is not fair. But we can ask: does it matter? As noted at the beginning of this post, such dice are usually employed in role-playing games, where there is usually a large amount of flexibility in the rules anyway. "Mildly biased" dice aren't really an issue where the rules are often made up on the fly!
Other stuff. In this post, I've focused on the geometry of individual dice, but there's even more interesting stuff that can be said about the mathematics of combinations of dice! There are the so-called Sicherman dice, which are differently numbered pairs of d6s that produce the same sums as two ordinary d6s! There are also non-transitive dice, in which each die of a set of 4 can always be beaten, on average, by another member of the set! I blogged about these a few years back, if you're interested in learning more.
If you've read this far, congratulations! You've read 3500 words on the geometry of dice! Hopefully I've gotten across the point that the design of dice is not child's play. |
dace.co.uk : mathematics :
The Quadratic equation formula of Al Khwarizmi
Why, when the original proof (which I shall reveal to you) is simple and beautiful, are schoolchildren expected to take an important piece of mathematics on trust? I mean, the formula enabling the solution of equations in x² (x-squared) of the form:
For me, and perhaps for you also, up until the famous quadratic equation formula, school algebra was logical and transparent, even if at times trickier than Latin. Then suddenly, like a bolt from heaven, came a horrible-looking formula, the solution to the above equation, that we were supposed to take on faith alone:
Do not misunderstand me: I would not criticise the teaching I received many years ago at school. Our teachers were dedicated to teaching, and loved their subjects. In those days also they had the freedom to teach a few things that were off the syllabus. We lived in a time when professionals were respected a little more than they are today, and were trusted to know their job and how to do it. Thus, as students we were not subjected to time-wasting and soul-destroying examinations merely for the sake of allowing semi-educated politicians to entertain the illusion that they could measure the qualities of experts (end of rant).
Even so, our teachers were under some pressure of time, because of examinations, and to the best of my memory this is why we were told to memorise the quadratic equation formula instead of being led gently through the proof.
Enough preamble! Let us see how Al Khwarizmi leads us through this knot in the Islamic garden of medieval mathematics, and notice on the way how geometry forms the paths in the garden, leading us through the dancing flowers of symbols, signs and formulae.
We start with the equation as Al Khwarizmi worked with it:
x² + bx = c
where c is the area of the whole rectangle.
Next Al Khwarizmi chops the bx term in half, resulting in two rectangles (b/2)x, which he then cunningly rearranges along the edges of the square of side x (x²):
x² + 2(b/2)x = c
If we add a little blue square, top right in the diagram, we can make one big square.
The area of the little blue square is (b/2)². The area of the big square is (x+b/2)². The area of the big square is also equal to (the little blue square + c), which = ((b/2)² + c).
That is to say: (x+b/2)² = ((b/2)² + c)
Therefore (x+b/2) = ±√((b/2)² + c)
Therefore x = -b/2 ±√((b/2)² + c)
= -b/2 ±√(b²/4 + c)
= -b/2 ±√(b²/4 + 4c/4)
= -b/2 ±√(b² + 4c)/2
Therefore x = (-b±√(b² + 4c))/2
This is, as you can see, very close to the quadratic equation formula quoted above, except that we have no 'a' term and we have a plus instead of a minus in the (b² + 4c) part of the formula.
The reason for the plus instead of the minus is that Al Khwarizmi starts with the equation in the form:
x² + bx = c
Whereas we start with:
ax² + bx + c = 0
The interested reader may now obtain the modern formula from Al Khwarizmi's formula by dividing the starting equation (ax² + bx - c = 0) through by a and then substituting the resulting coefficients (the a, b and c terms) into Al Khwarizmi's formula. |
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Dan Asimov: The Bongles of Bingle-Bangle
By Gary Antonick April 27, 2015 12:00 pmApril 27, 2015 12:00 pm
Photo
Bangles by Oscar and Nancy.Credit Tony Cenicola/The New York Times
Our puzzle this week is an original by Dan Asimov, a mathematician and computer scientist who has held research positions at Harvard and at the Institute for Advanced Study, and who has taught at Stanford and U.C. Berkeley. He is also the inventor of the "grand tour" method of animating statistical data. Let's join him as he takes us out of this world with —
The Bongles of Bingle-Bangle
As everyone knows, the planet Bingle-Bangle is the first planet that our astronomers have found to be inhabited by humanoids, who are known as Bongles. Making use of relativistic yotta-photonics, our exosociologists have established rudimentary communication with the Bongles. We know four things about them so far:
1. The Bongles use the word bangle to mean a perfect circle, anywhere in space, having radius equal to 1. All bangles are identical.
2. Every adult Bongle wears a tringle — a triple of separate bangles — to signify their family status. (Our exosociologists have not yet figured out how this works in the very complex Bongle society.)
3. The Bongles consider two such tringles to be the same if one tringle can be moved to become the other tringle, through a series of in-between tringles. Since everything in between must be a tringle, at no point may any two of the bangles come in contact with each other. (For instance, one kind of tringle is represented by any three bangles that are not linked with each other. Another kind of tringle has two of the bangles linked and the third one far away. There is no way to move between these two different types of tringles without two bangles' passing through each other — but this is taboo.)
4. The Bongles refuse to communicate further until we can prove to them that we know what all the different kinds of tringles are. Our exosociologists have tried to reach the world's most eminent topologist and geometer, T.W. Paul, to learn what all the distinct tringles are. But she has been impossible to reach because she is in isolation, putting the finishing touches on her solution to the longstanding unsolved math problem known as the Four-Dimensional Smooth Poincaré Conjecture.
Puzzle: Can you find all the different kinds of tringles? Help us communicate with the Bongles!
Note that because all bangles in any tringle are interchangeable, as long as two tringles look the same, they represent the same tringle.
I asked Dr. Asimov to say a bit about what influenced him to become a mathematician, his favorite topics in the field, and any advice for those considering mathematics as a career. He sent the following reply by email:
When I was five, my Uncle Harold told me about magic squares, and I ate it up. Later I would read Martin Gardner's wonderful "Mathematical Games" column in Scientific American. My first epiphany came at age 14 when I stayed up all night graphing strange formulas like y = 2x – x2. I was also lucky enough to have some great teachers — especially my thesis adviser, M.W. Hirsch.
Math combines mystery with the search for truth and extraordinary beauty. My mother, who is an artist, inspired my special fondness for the areas of geometry and topology. But many other areas of math are equally fascinating.
For any young people getting interested in math: Keep checking out this column. Google to find out more about any math topic of interest. Definitely read Martin Gardner's recreational math books. Talk with your math teacher and with mathematicians. Attend a Math Circle or a summer program providing math enrichment. Above all: If some math stuff intrigues you, ask questions about it and then try to answer them! Dan Asimov:
There are just 6 different kinds of tringles.
Here are representatives for each one:
1) All bangles unlinked with any other
2) Two bangles linked, the third far away.
3) A straight chain of three links
4) and 5) A circular chain of three links. These come in two distinct handednesses, each the mirror image of the other.
6) Two bangles linked, the third one trapped around the region where the link occurs.
Dr. W. came closest to posting the correct solution — I would say he was a hair's breadth (a hare's breath?) away since he mentioned all of them, only was not sure whether the circular chain of three links came in two distinct handednesses. Others came very close.
This kind of puzzle might best be called exploratory mathematics, since rigorous proofs that there are 6 distinct tringles, and no others, would take us beyond he scope of Numberplay.
It's clear that among the 6 different types mentioned above, types 1), 2), and 3) are clearly distinct. Then things get tricky for three reasons:
a) Is a chain of 3 bangles the same tringle as its mirror image? It turns out that a circular chain of 3 bangles can be arranged so that each bangle lies on a perfect torus of revolution in space, as long as its big radius is also equal to 1, no matter what its little radius is. And this can occur in two distinct mirror-image ways.
b) Is the Borromean rings a possible tringle? (And if so, is it the same as its mirror image?) There is a beautiful but advanced proof that 3 perfect flat circles — no matter what their radii — cannot form the Borromean rings.
c) Are there any other tringles that depend on more than ordinary linking?
And the only possibility here is 2 linked bangles with the 3rd one unlinked with the others yet "trapped" around the waist of the link. This was the last of the 6 tringles that I found, and I agree with Seth that this is rather fantastic and unexpected.
Kudos to Mark (96/100) and Dr W (99/100) for coming ever so close to perfection |
ten
Ten is the base of our familiar number
system, which stems directly from the fact that we have ten fingers
on which to count. Ten is the only triangular
number that is a sum of consecutive odd squares (10 = 12 + 32) and the only composite integer such that all of its positive
integer divisors other than 1 are of the form x2 + 1 (2
= 12 + 1, 5 = 22 + 1, 10 = 32 + 1). Strange
but true: the lifespan of a taste bud is ten days. |
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Interesting. Actual maths ability in a pure sense is, nowadays, considered less important than ability to do well in exams. For which a good memory and ability to work at high speed and to "follow recipes" and jump through hoops, are needed. Problem-solving capacity is very much neglected. Which is sad, because this is really the only thing worthwhile.
Yeah, always hated exams, the older I got the more they seemed to be about learning for the test rather than for the future (combined naturally with added pressure), it's one of those short-term gain long-term loss type things.
Got through most of them well enough but still the odd slip-up, even when you remembered pretty much all the information in the study guides you'd still lose marks for not answering the question like the examiner wanted you to... which in itself throws you off a bit on really easy questions/tests because you start quadruple-guessing if the simple question has any hidden meaning.
I used to get asked to perform multiplications while in the street at the back of my house. Those were the days. In general, where I grew up was quite rough, and there weren't many clever people around. Also, I could have just said any answer, as the people asking didn't have a calculator, so had no way of checking my answers were correct...
I often find that with mathematics, and computing and other such technical fields, ability tends to rise on an exponential scale.
I think of myself as being "better than average" at arithmetic, simply because many people can't do arithmetic at all. It's not hard (in this country at least) to be better than average. But then these kids are about 20,000 times better than what I would consider good. They have gotten to the stage where it seems like they don't actually need to think.
With more physical tasks, at least the scale is within reach. For example, doing the 100m in 10 seconds is considered phenomenal, but probably most averagely fit people under 40 could do it in 20 secondsKiribati has estimated that the costs to get them to the World Cup (minus accommodation and food, which the organisers will cover) will be £50,000 (about 56,000 Euros or 68,000 US Dollars). At least half of this will be for the return airfares. For a group of 20 players/officials, this would amount to 3,400 US dollars for each person, which would be impossible for them to pay themselves, as this would be several months' wages for some people. Sponsorship needs to be found quite quickly.
There is "Yorkshire" vs Ellan Vannin this month, and "Surrey" vs Barawa in May. And a small tournament with "Yorkshire" is planned in May but details haven't been released. As well as fixtures vs club teams etc.Stone paper scissors? Guessing how many pennies are in a jar?
I guess with some newly created sports the range may be narrow. In rollball, which was invented only in 2005, India is seen as the "elite", as it is played in schools there and they have played it the longest, and indeed invented it. However, in the first World Cup in 2011, India, with 6 years of practice, were beaten in the final by Denmark, who were not even a member of the Rollball Federation, and who only got together for the tournament, being drawn from a skater hockey club (skater hockey itself has only around 10 active countries). The club had won 2nd place in the 2010 European Skater Hockey Championship for clubs.
Belarus also formed a team for the tournament in a similar fashion (drawn from other sports), and reached the semi-final, losing to Denmark. In general, Belarus and Denmark thrashed all of their opponents despite not having 'real' rollball players (Belarus won all their games by between 5 and 14 goals, except the loss to Denmark, while Denmark won all their games by between 4 and 18 goals, except for the final in which they beat India 3-2). Some of the teams they beat (Bangladesh, Nepal) had a similar length of playing experience to India, but that doesn't seem to have made any difference at all.
The sport of rollball is a combination of handball/basketball on roller skates, and it seems that skating ability is very important, despite the method of scoring goals being more reliant on shooting ability, as demonstrated by a team of proficient skaters with no background in handball or basketball winning the World Cup.
I guess if "betting" were classed as a sport, this could also be the case. It's possible to win a bet on your first bet without knowing anything about the sport you are betting on, while "professional" gamblers may rely too much on their "proven strategy", which may only work 80% of the time, and may fail while the novice gets lucky.
nfm24 wrote:While on the subject of the eligibility criteria, why aren't they available on the ConIFA website? As you know, the constitution requires it: "The Constitution, the Internal Regulations, the decisions and announcements of CONIFA shall be published on the official website in the official language."
website
nfm24 wrote: website |
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Beautiful and Useful Digests
Abel ("Nobel") Prize for Fermat's Last Theorem Proof
Fermat's last Theorem was first conjectured by French mathematician Pierre de Fermat in 1937. However, he did not provide the proof. (apparently he had omitted the proof as there was not enough margin space to write it down!)
We all know we can break down a squared number (25, that is 5^2) into sum of two squared numbers (16, that is 4^2 and 9, that is 3^2). This confirms to Pythogaras' theorem.
What Fermat saw was that it is impossible to do that with any number raised to a power greater than 2. Put differently, the formula, x^n + y^n = z^n, has no whole number solution when n is greater than 2.
After more than 3 centuries, Andrew Wiles solved this problem in 1994 by way of the modularity conjecture for semistable ellipic curves.
He is offered the Abel Prize ("the Nobel of mathematics) for his stunning proof. [We believe that such effort should be recognized even more and should carry a financial reward more than what is currently offered.] |
Karl Friedrich Gauss
Gauss was a German mathematician who lived in the late 1700's through the mid-1800's. Stories picturing him as a child mathematical prodigy abound, though some are probably not true. One such story has him correcting his father's arithmetic on some business accounts when Gauss was only three.
Gauss also attended a one-room schoolhouse as a child. The schoolmaster was a rather harsh individual who believed in using a whipping cane as a motivational device. One day, this teacher assigned his students the task of adding the integers 1-100, an early example of a teacher assigning "busy work". Gauss almost immediately turned in his slate. The schoolmaster, thinking Gauss was mocking him, waited until all slates were turned in to begin grading. After grading all the slates, only Gauss had the correct answer. What did he notice? 1+100 = 101, 2+99 = 101, 3+98=101, 4+97=101…There are 50 pairs of numbers that sum to 101, and 50 x 101 = 5050.
Happily, the schoolmaster then recognized that Gauss was rather gifted, and began to train him in mathematics. |
Chaos Theory
Where chaos begins, classical science
stops." James Gleick "The century's third great revolution in physical sciences." Gleick
Chaos Theory:
The study of complex nonlinear dynamic systems and forever changing
complex systems based on mathematical concepts of recursion. Known as the
irregular side of science- its discontinuities and erratic sides. Deals
with connection between different types of irregularity.
"The
chaos theory predicts that complex nonlinear systems are inherently unpredictable-
but, at the same time, chaos theory also insures that often, the way to
express such an unpredictable system lies not in exact equations, but in
representation of the behavior of a system- in plots of strange attractors
or in fractals." Gleick
Chaos is not about disorder! The word chaos stands for the very essence
of order!
History:
The chaos theory, first discovered by Edward Lorenz in the 1960's, essentially
states that simple systems may actually produce complex behavior. On the
other hand, it has also been proven that complex systems actually have
a simple underlying order. Fractals are an important part of chaos, exemplifying
the beauty and symmetrical magic of the chaos theory. Fractals
in geometry may be the single most perfect way to express the chaos
theory and its principles. Although at first one may think "chaos"
is unpredictable and random, in actuality, chaos is the very essence of
order. Although chaos deals with the erratic side of science and math,
it produces a concrete order in every seemingly random iteration or problem.
Relation to Arcadia:
As Valentine explains to Hannah on page 47, " The unpredictable
and predetermined unfold together to make everything the way it is. It's
how nature creates itself, on every scale, the snowflake and the snowstorm."
They argue about the possibilities of chaos as they attempt to figure out
what Thomasina was really on to in her discoveries. They soon realize she
had gotten a grip on the concept
of iterated algorithms during her experiment with the leaf and its
relation to fractal geometry. At this point in the play, the reader becomes
aware of Thomasina's brilliance. She has already begun to study and understand
the complex concept of the chaos theory. This also links the two time periods
of the play together when Valentine is expressing his knowledge of the
topic. The characters of the present learn that Thomasina was already onto
this subject at her extremely young age.
The chaos theory is also present throughout the play during discussions
of Valentine's grouse project. He speaks of the importance of iterated
algorithms in calculating the grouse population in a given year. In the
situation of determining the growth of a certain species over a period
of time, x is the number of animals and y is the result. The equation summarizes
the effect of the environment on the number of animals each year and their
fluctuations in population.
Arcadia's explanation of noise also relates to the chaos theory. When
working on the grouse project, Valentine refers to noise as the impracticalities
and unpredictable variables of the equation. He refers to the extremely
small factors in nature that, in result, come to have a major impact on
the outcome. Thank you for visiting the "Chaos in Arcadia" web
page!!!!
An outstanding site for the examination of these mathematical concepts
and their relation to the play is Chaos,
Fractals and Arcadia which you may link by clicking the title.
If you are interested in a more in depth discussion and every day life
application of chaos theory, you might want to
visit this page |
Artículos con la etiqueta 'lenguaje de la ciencia'
The vast dimensions of the learning process necessitate the choice of a focussed enough theme — more especially the Sciences-for a purposeful analysis. And Mathematics is the most natural language for such study, as it provides a quantitative basis for articulating the huge dimensions of Science. The purpose of this essay is to bring out the unique role of Mathematics in providing a base to the diverse sciences which conform to its rigid structure. Of these the physical and economic sciences are so intimately linked with mathematics, that they have become almost a part of its structure under the generic title of Applied Mathematics. But with the progress of time, more and more branches of Science are getting quantified and coming under its ambit. And once a branch of science gets articulated into a mathematical structure, the process goes beyond mere classification and arrangement, and becomes eligible as a candidate for enjoying its predictive powers! Indeed it is this single property of Mathematics which gives it the capacity to predict the nature of evolution in time of the said branch of science. This has been well verified in the domain of physical sciences, but now even biological sciences are slowly feeling its strength, and the list is expanding |
"Kurt Gödel's (1906-1978) monumental theorem of incompleteness demonstrated that in every formal system of arithmetic there are true statements that nevertheless cannot be proved. The result was an upheaval that spread far beyond mathematics, challenging conceptions of the nature of the mind."
"Is the core of cognition and animacy essentially only self-representation and self-reference (as in Bach, in our DNA and elsewhere)? Is it essential incompleteness (as in Gödel's Theorem and elsewhere)? Is it strange loops and tangled hierarchies (as in Escher, in Hofstadter's own book, I Am a Strange Loop, and elsewhere)? Is it in the patterns, puzzles, paradoxes, puns, poetry, and programming that we see throughout Hofstadter's work? Or is it elsewhere?
Elsewhere.... Perhaps it is precisely in analogy that we find the common thread of all these cognitive and creative phenomena, and thus the common element in the endeavors that make us human, and thus the core of our humanity..."
"The Year of Mathemagical Thinking"
Lev Grossman, TIME Magazine, March 15, 2007
"I Am a Strange Loop" sets out to probe the essence of the soul—in a philosophical, cognitive sense... Consciousness, soul, and "a light on inside" are all terms referring to the essential "I" which somehow composes an individual human self...
...is very self-referential, and that any explanation of the concept bends back onto the same concept again. The resulting loop, though, isn't like most loops caused by self-reference, since there's no feedback as in... an infinite corridor of TV screens on videotape. So consciousness isn't a regular loop; it's a strange loop.
..while presenting arguments of logic, clever bits of analogy here and there add up to reveal that the book itself is more than just a friendly essay: everywhere you turn, "Strange Loop" is drawing back on itself, too. For example, the book's arguments are made almost entirely through symbols, analogies, and tales of personal experience. Appropriately, Hofstadter devotes much discussion to the reasons that symbols, analogies, and empathy (or, as he calls it, "Varying Degrees of Being Another") actually work. This book is a work of art, unabashedly self-referential on every level..."
Abstract: "...And yet there are some basic ideas that we should not lose track of, and that should help to keep us from confusing wild speculation with grounded reality. In my talk, I will attempt to chart out a way of looking at the "singularity scenario" with one's feet on the ground, and I will try to give, using my moderate familiarity with a number of different scientific disciplines, a personal appraisal of what I see as the likelihood of our being eclipsed by (or absorbed into) a vast computational network of superminds, in the course of the next few decades."
"The So-called Singularity: An Onrushing Tsunami, or Another Y2K?" [MP3] Artificial Life X: Tenth International Conference on the Simulation and Synthesis of Living Systems
Bloomington Campus, Indiana University, June 3-7, 2006
Abstract: "In the past few years, a number of futurologists, extrapolating on the basis of many interrelated exponential curves such as Moore's Law, have come to the conclusion that computer intelligence is rising so swiftly that quite soon, it will inevitably reach and then surpass human intelligence, and that at that monumental juncture in the history of this planet, humanity will be eclipsed and replaced by its own creations. Within a few decades, these cyberprophets proclaim, we humans will be living among superintelligent entities that are just as incomprehensible to us as we are incomprehensible to bacteria, and the upward spiral will continue from there on without limit, resulting in entities "who" are literally billions of times more intelligent than today's humans are, and "who" will soon commandeer stars and then whole galaxies, finally turning the entire universe into one single inconceivably intelligent self-reflective organism akin to the Omega Point of the mystic Jesuit philosopher Teilhard de Chardin..." |
Hyperbolic Crochet Coral Reef Many organisms in the coral reef have a very particular structure; the frilly crenelation seen in coral, kelp, nudibranch, seashells… is a form of geometry known as.
Margaret Wertheim leads a project to re-create the creatures of the coral reefs using a crochet technique invented by a mathematician -- celebrating the amazements of the reef, and deep-diving into the hyperbolic geometry underlying coral creation.
Conformal Models of Hyperbolic Geometry
Conformal Models of Hyperbolic Geometry
Part X. Extra-terrestrial physics as shown in crops: could there be another fifth dimension accessible from our three dimensions of space and one of time, which is spinning, hyperbolic, and has the symmetry of a Mobius strip? And could this be the �vector space� of quantum theory? By Dr. Horace R. Drew |
Three guys go into a hotel, each with $10 in his pocket. They book one room at $30 a night. A short while later a fax from the headquarters directs the hotel to charge $25 a night. So the receptionist gives the bellhop $5 to take to the three guys sharing the room. Since the bellhop never got a tip from them and because he can't split $5 three ways, he decides to pocket $2 and give them each one dollar back. So each of the three guys has now spent $9 and the bellhop has $2 for a total of $29. Where's the extra dollar?"
Woo-hoo! (Yes, I can be a nerd.)
Ok, I'm throwing this in for Joseph who mentioned it. It's the case of the terminator 0, 1/2 and 1:
We have thus proven that when dealing with infinity, S can be any number.
What we have here is a beautiful mathematical explanation to the philosophical and metaphysical presumption that ultimately, we are all one and the same, and that, just as we are part of the universe, the universe is within us. |
Euclidean Fun for Kids
Twenty-three centuries after Euclid of Alexandria composed his Elements of Geometry, some of his favorite shapes – including the triangle, square, and circle – were re-released as repositionable wall graphics today. The sturdy wall graphics substrate, which was not available in Euclid's era, has the potential to make geometry interesting again, and even to change the worldview of primary school children.
"Kids learn with their hands," says experimental philosopher Jonathon Keats, who curated the collection for the wall graphics company Walls 360. "Children figure out how the world works by climbing rocks and falling off bicycles. Repositionable wall graphics extend geometry into the realm of physical experience."
The wall graphics in this new collection include both simple shapes and complex combinations arranged to show the ways that shapes can interrelate. All are repurposed from Oliver Byrne's classic Elements of Euclid, first published in 1847, which illustrated the basics of Euclidian geometry in strikingly modern combinations of bright primary colors. "The images in Byrne's Euclid are remarkably intuitive," Keats says. "You could work out Euclid's system without even looking at the text. In fact, you're better off without all the verbiage, as any child will understand (though parents may not get it)."
Keats, whose ballet for honeybees was featured on Big Think, believes that the future of textbook-free adhesive geometry is enormous, and argues that the greatest impact will be on children who find ways to break Euclid's rules. "The world needs more non-Euclidian thinking," he says. "When kids start trying to stick these wall graphics onto spheres and Calabi-Yau manifolds, they'll begin questioning all our ideas and beliefs."
Big Think Edge helps organizations by catalyzing conversation around the topics most critical to 21st century business success. Led by the world's foremost experts, our dynamic learning programs are short-form, mobile, and immediately actionable. |
I'd live badly if I didn't write and write badly if I didn't live
Fractal
A fractal is a mathematical intrigue; quite simply a geometric figure each part of which has the same statistical character as the large, whole piece. This property is known as self – similarity.
mandelbrot set
The term fractal is a derivative from the Latinfrāctus meaning "broken" or "fractured."
Albert Klein Blue Fractal
Fractals can be used to describe partly random or chaotic phenomena through mathematical formulae. However, to me Fractals have an aesthetic quality which is greatly appealing. Art need not always be free flowing abstract curves, sometimes precision and accuracy, combines with mathematical acumen can produce something truly magnificent. |
Maths Ideas You Really Need to Know
By Tony Crilly
£11.99
Who invented zero? Why 60 seconds in a minute? How big is infinity? Where do parallel lines meet? And can a butterfly's wings really cause a storm on the far side of the world?
In 50 Maths Ideas You Really Need to Know, Professor Tony Crilly explains in 50 clear and concise essays the mathematical concepts - ancient and modern, theoretical and practical, everyday and esoteric - that allow us to understand and shape the world around us.
Packed with diagrams, examples and anecdotes, this book is the perfect overview of this often daunting but always essential subject. For once, mathematics couldn't be simpler.
Contents include: Origins of mathematics, from Egyptian fractions to Roman numerals; Pi and primes, Fibonacci numbers and the golden ratio; What calculus, statistics and algebra can actually do; The very real uses of imaginary numbers; The Big Ideas of relativity, Chaos theory, Fractals, Genetics and hyperspace; The reasoning behind Sudoku and code cracking, Lotteries and gambling, Money management and compound interest; Solving of Fermat's last theorem and the million-dollar question of the Riemann hypothesis.
Biographical Notes
Tony Crilly is a Reader in Mathematical Sciences at Middlesex University, having previously taught at the University of Michigan, the City University in Hong Kong and the Open University. His principal research interest is the history of mathematics, and he has written and edited many works on fractals, chaos and computing. He is the author of the acclaimed biography of the English mathematician Arthur Cayley.
Children of the Ghetto
Elias KhouryA World on Fire
James Heneage Genius Test: Maths
Julia Collins Eternal City
Domenica De Rosa |
Math Philosophy-- Why does 1/∞ not equal 0, and for that matter, what is ∞?Ok, you want an answer...here.
1/infinity = small
If you want to use a concept as part of an equation...I can use a concept as the answer.
Or hell...we are living in the realm of fantasy...so I'll change my answer.
I don't mind if infinity is not considered a number as long as 0 is also treated as such. However this isn't the case in the mainstream.
As I've already shown using reciprocals, 0 is just as distant from 1 as it is from infinity.
0 is infinitely small, infinity is infinitely big. Together they define the radial dimension. To ignore one is to say that the other side of the
coin does not exist.
Note that I'm not defining infinity as a number with an infinite amount of digits, I'm defining it as the upper limit, the farthest possible
distance from 0.
The numbers with the most amount of digits reside in what I call the "awareness barrier". If you count from infinity and 0 at the same time, they
will approach the awareness barrier but never reach each other. In this sense, a certain aspect or understanding of infinity is deferred. You
would think that the number just below infinity would have a ridiculous (and undefinable) amount of digits, but with the advent of shadow numbers (any
number that uses infinity as the origin) this is no longer a problem. Infinity can now be understood as well as 0, but we still can't comprehend the
numbers in the awareness barrier (because we're not aware of them yet and we can't be aware of them all).
Interestingly, I didn't even ask the question "what is ∞/∞" until I saw the answer staring me in the face. I too thought the idea was
preposterous, but geometry doesn't lie. Using the root grid calc you can see how the
fractional representation gets closer and closer to ∞/∞ as it approaches 1. For example, 1/2, 2/3, 3/4, etc. Eventually you'll have really
large numbers like 10000/10001.
However, no matter how big these finite numbers get, they will not equal 1. So what is that point at the top of the grid? What's the
fractional representation of it if it's not ∞/∞? Should we create a new number to define it? To say it doesn't exist is to leave 1 (and -1)
out of the picture.
And don't give me that "undefined" non-answer crap. That's defeatism and the reason mathematics is currently stagnating in schools. Too few
attempt to discover and instead only learn to regurgitate the knowledge of those who came before.
2/∞ must equal zero.
Now, look at 1/0. Does that equal infinity? It may be the only true way to describe ∞, and the most basic of ways. What is(1/1)/(1/0)? (0/1)
What about 3/0? What does that equal? Also ∞, which is situational to this case because if we had this equation:
(3/1)/(3/0),
what do you get? 1/0... ∞
To say that these infinities are interchangeable enough to equal each other and that 1=2... that is nonsensical.
But here lays the difference between you and I. The "fundamental" basis of math that you keep telling me I have no grasp of, appears to be up for
debate. Unless you can prove otherwise, which you have tried to do and cannot.
Ok, you want an answer...here. 1/infinity = small If you want to use a concept as part of an equation...I can use a concept as the answer. Or
hell...we are living in the realm of fantasy...so I'll change my answer. 1/infinity = dragons
Now that is pathetic. You are wonderful at ruining a concept that holds some element of reality. You cannot ruin by disproving me logically, so now
you need to play the funny card. Ooo look, you even have a star! The masses must love you
else through your demeaning rhetoric.
Can't wait to find the day that actually brings you down from the clouds.
And hey, maybe I am a dreamer myself. But I am man enough to admit it... I hope to make things bigger and better. You hope to crush things, making
everything else but yourself smaller. 2 very different approaches toward the same goal.OK - I see what you are saying, and I've re-read your original post. I originally answered because I too love maths, and I see that you want to go
on to a higher level, That's great! And a big part of that is questionning and pushing boundaries as that is how we learn at higher levels.
Just a few things though: It's not what "some guy with a textbook" told me; I do actually have a maths degree, so I AM the guy with the textbook!
:-p
And more seriously; there are some set definitions which give us the rules of maths, which in turn, give us the rules of physics. We cannot answer
your question as it re-defines infinity. It would be similar to re-defining the value of pi to 3. All of a sudden, circles would not work! When you
go on to study more advanced physics, are you going to challenge "some guy with a textbook" telling you that speed is distance/time because you want
to change the definition of it for a hypothetical question?
And just so you know - my question that irritated my physics lecturer was "on a magnet, if this end is North and that end is South, what is in the
middle?" and I wouldn't let that one lie! Haha!
Now that is pathetic. You are wonderful at ruining a concept that holds some element of reality. You cannot ruin by disproving me logically, so
now you need to play the funny card. Ooo look, you even have a star! The masses must love you.
Many people have proved you wrong logically...I just don't think you understand logic.
You wanted to misuse a concept, you wanted to use a concept as a number...I and others tried nicely to show you how your thinking is wrong...you
don't listen. In fact you go further and say that WE are the ones that are confused.
The only thing left to do is to mock you to show how ridiculous your claim is else
through your demeaning rhetoric.
No, clearly you aren't humble at all...and don't try to kid anyone...you do think you are that person.
Can't wait to find the day that actually brings you down from the clouds.
I'm often wrong, I admit it and learn from it.
So don't hold your breath...because I have no problem being wrong.
And hey, maybe I am a dreamer myself. But I am man enough to admit it... I hope to make things bigger and better. You hope to crush things,
making everything else but yourself smaller. 2 very different approaches toward the same goal.
I only live my life by logic...and my only hope is that others would too.
Oh so you have heard of yourself as well? Odd, my notes do not match up to yours. Well, dividing by 0, you can deny it but then you would be denying
the number 0. Is that a dragon as well? Is that how you rationalize things that cannot be completely explained? I am doing the best I can, and
unfortunately you are too which is making you look like a fool.
You do not have to accept everything I say, but you are starting to point out things that you cannot, with any amount of logic deny. OOO someone
stepped out of the box and divided by the very real 0 thingy. Shoot em! It cannot be, mother said so.
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Don't trust that algorithm
Harvard Ph.D. warns of big data's dark side in 'Weapons of Math Destruction'
Whether we know it or not, complex algorithms make decisions that affect nearly every aspect of our lives, determining whether we can borrow money or get hired, how much we pay for goods online, our TV and music choices, and how closely our neighborhood is policed. Thanks to the technological advances of big data, businesses tout such algorithms as tools that optimize our experiences, providing better predictive accuracy about customer needs and greater efficiency in the delivery of goods and services. And they do so, the explanation goes, without the distortion of human prejudice because they're calculations based solely on numbers, which makes them inherently trustworthy. Sounds good, but it's simply not true, says Harvard-trained mathematician Cathy O'Neil, Ph.D. '99. In her new book, "Weapons of Math Destruction: How Big Data Increases Inequality and Threatens Democracy," the data scientist argues that the mathematical models underpinning these algorithms aren't just flawed, they are encoded opinions and biases disguised as empirical fact, silently introducing and enforcing inequities that inflict harm right under our noses. The Gazette spoke with O'Neil, who once worked as a quantitative analyst and now runs the popular Mathbabe blog, about what she calls the "lie" of mathematics and her push to get data scientists to provide more transparency for an often too-trusting public. GAZETTE: How did your work as a hedge fund quant prompt you to start thinking about how math is being used today? Had you given it thought before then? O'NEIL: It absolutely had not occurred to me before I was a quant. I was a very naive, apolitical person going into finance. I thought of mathematics as this powerful tool for clarity and then I was utterly disillusioned and really ashamed of the mortgage-backed securities [industry], which I saw as one of the driving forces for the [2008] crisis and a mathematical lie. They implied that we had some mathematical, statistical evidence that these mortgage-backed securities were safe investments, when, in fact, we had nothing like that. The statisticians who were building these models were working in a company that was literally selling the ratings that they didn't even believe in themselves. It was the first time I had seen mathematics being weaponized and it opened my eyes to that possibility. The people in charge of these companies, especially Moody's, put pressure on these mathematicians to make them lie, but those mathematicians, at the end of the day, they did that. It was messed up and gross and I didn't want to have anything to do with it. I spent some time in risk, after I left the hedge fund, trying to still kind of naively imagine that with better mathematics we could do a better job with risk. So I worked on the credit-default-swaps risk model. The credit default swaps were one of the big problems [of the 2008 financial crisis] and then once I got a better model, nobody cared. Nobody wanted the better model because nobody actually wants to know what their risk is. I ended up thinking, this is another example of how people are using mathematics, brandishing it as authoritative and trustworthy, but what's actually going on behind the covers is corrupt. GAZETTE: Big data is often touted as a tool that delivers good things — more accuracy, efficiency, objectivity. But you say not so, and that big data has a "dark side." Can you explain? O'NEIL: Big data essentially is a way of separating winners and losers. Big data profiles people. It has all sorts of information about them — consumer behavior, everything available in public records, voting, demography. It profiles people and then it sorts people into winners and losers in various ways. Are you persuadable as a voter or are you not persuadable as a voter? Are you likely to be vulnerable to a payday loan advertisement or are you impervious to that payday loan advertisement? So you have scores in a multitude of ways. The framing of it by the people who own these models is that it's going to benefit the world because more information is better. When, of course, what's really going on and what I wanted people to know about is that it's a rigged system, a system based on surveillance and on asymmetry of information where the people who have the power have much more information about you than you have about them. They use that to score you and then to deny you or offer you opportunities. GAZETTE: How integrated are algorithms in our lives? O'NEIL: It depends. One of the things that I noticed in my research is that poor people, people of color, people who have less time on their hands to be more careful about how their data are collected are particularly vulnerable to the more pernicious algorithms. But all of us are subject to many, many algorithms, many of which we can't even detect. Whenever we go online, whenever we buy insurance, whenever we apply for loans, especially if we look for peer-to-peer lending loans. |
oh thank you :) but studying origin and development is philosophy of mathematics its not meta maths and meta maths if development of algorithm i guess
anonymous
5 years ago
and its basically a computer language @hba btw message me
anonymous
5 years ago
@Hero can u? :)
anonymous
5 years ago
@amistre64 help me
hba
5 years ago
Sure i'll message you mam.Meta maths is not a computer language though.
anonymous
5 years ago
lol its not a computer language its a language used to write mathematical algorithm like i guess you've heard of MATLAB
hba
5 years ago
Oh yeah,I am not so sure about it though.
anonymous
5 years ago
its ok (: m just curious to know
hba
5 years ago
@amistre64 Please pour some knowledge in our empty glasses.
amistre64
5 years ago
meta math was a user defined group that the admins decided to keep when they cleaned up the user created groups
anonymous
5 years ago
lol super complex @amistre64 didn't get u :C
hba
5 years ago
@amistre64 Your feedback did not provide what meta maths is ?
amistre64
5 years ago
the admins tried the idea that users could create their own groups. Users abused the priviledge and the admins took that functionality away. They eventually went thru and deleted most of the user defined groups; but kept a few that had not been problematic
amistre64
5 years ago
mate math is one of the subgroups in mathematics as a result of there being nowhere else to put it, and not wanting to get rid of it
anonymous
5 years ago
what i got to know from wikipedia is as follows m not talking about this group here:-
Metamath is a language for developing strictly formalized mathematical definitions and proofs[1] accompanied by a proof checker for this language and a growing[2] database of thousands of proved theorems covering conventional results in logic, set theory, number theory, group theory, algebra, analysis, and topology, as well as topics in Hilbert spaces and quantum logic.
TuringTest
5 years ago
I have the honor of having named the group, so it means what I wanted it to mean ;)
meta=above
math=math
it's basically for irregular advanced problems that would liley get lost in time if they were in a section as large as the general math group
TuringTest
5 years ago
likely*
TuringTest
5 years ago
by posting questions that require days to solve in a smaller group they are more likely to get regular attention from some of our better users. I didn't look up the meaning on Wikipedia when I made the name, I just wanted to describe that the group was outside of any particular branch of mathematics.
anonymous
5 years ago
@TuringTest then if i have question that deals with the definition of some mathematical term can i put it there for debate?
TuringTest
5 years ago
I guess, but I don't think it's the best place for it, as the group is much more slow-paced. You should probably just post that in regular math imo. |
From fast cars and aeroplanes to computer encryption – mathematics
underpins so much of modern life.
In this episode, Jim Al-Khalili
uncovers how, between the 9th and 14th centuries, mathematicians from
the Islamic world helped mathematicise science and lay the foundations
of algebra.
He looks at the modern mathematics behind flight, and behind
the record-breaking fastest car in the world, tracing the route back
from these achievements to the legacy of the Persian mathematician Al
Khwarizmi.
We also discover the role that the Islamic world played in
giving us the modern numeral system that we take for granted in everyday
life |
MONROE, CT, USA -- In celebration of Pi Day, Masuk High School students, with help from local preschoolers, assembled a pi paper chain made of 75,000 links, using a different color for each of the ten digits of pi -
setting the new world record for the Longest Pi Chain. Photo: In celebration of Pi Day, Masuk High School students, with help from local preschoolers, made The World's Longest Pi Chain. Photo: Autumn Driscoll (enlarge photo)
The previous Guinness world record for the Longest Pi Chain has 65,000 links.
Guinness World Records also recognized the longest paper chain by an individual: it measures 327.1 m (1,073 ft 1.93 in) long and was made by Christopher Potts (UK) in Kimberworth, Rotherham.
When stretched out, the chain is more than a mile long, said Mary Ahlers, head of the high school math department.
"I knew pi was long, but I never knew it was this long," sophomore Austin Vuchla, 15, said.
Pi is an irrational number that represents the ratio of a circle's circumference to its diameter. Its decimal representation never ends or repeats. Mathematicians have thus far computed 5 trillion digits of the decimal expression.
Different shades of colored paper represented different numbers on the paper pi chain. |
2 + 2 = 7 (Because sometimes what you DON'T know matters)
Anyone who knows any math will know that the single most certain truth on earth is that the Cubs will never win the pendant.
Wait, no, I was going to say, the single most certain truth on earth is "2+2=4". It is completely indisputable. Except when it's wrong. I'm just going to come out and say it: 2 + 2 =7.
You might want to stick to your old school math guns and insist that the answer is 4, but I'm telling you I have scientific proof of what I am saying. You see, I have several boxes which I pulled out of the pantry, and then I labeled then numerically based on the order in which I pulled them out. The first box was noodles. That box is 1. The second box was a mix for chocolate cake. The third box was exactly the same as the second, so I labeled them both as "2".
Well, these are big boxes, and each weighs three and a half pounds. (You've probably guessed that I got them at Costco.) When I added up the weight of both boxes, they totaled seven pounds of mix! I'll be eating cake until my NEXT birthday!
Oh, so looking back I guess what I should have said was, The weight of box 2 + the weight of box 2 = 7lbs. It seems I did leave out a bit of context.
What I think we all learned from this is simply that someone with no ability to resist chocolate cake should not shop at Costco. Wait, no, that wasn't…. Well, ok, that too. But what we are supposed to have learned is that the unspoken assumptions or context in any given situation can make a world of difference. In the case of basic math, the assumed context is always that the numbers represent single units, meaning that 2+2=4 means two units plus two units equals four units. We're used to not saying it, because all through grade school that's all it means. This is the normal use of numbers, and frankly, third graders really don't need to do anything else with them. If you're eight years old, you shouldn't be worrying about meters per second squared unless you're a nerd destined to be a super hero. However, when you get into something like Physics in high school, you need to know what the units are. Two WHAT plus two WHAT? Feet per second? Pounds per square inch? These things matter. Getting this wrong causes your three billion dollar space craft to crash into Mars instead of landing there like it was supposed to, and then you and your PhD in engineering wind up living behind a dumpster licking chocolate out of discarded candy wrappers.
Here's some advice: At your next interview, try to pass that incident off as a "Minor Math Error" and not a "Three Billion Dollar Loss."
When you're having an argument with someone, ask yourself, "What is NOT being said?" Try and figure out what YOU have forgotten to say, and what THEY have forgotten to say. In the spaces where you don't say something, other people often fill in the blanks with their own assumptions. Sometimes figuring out what is meant when it hasn't been said can resolve a whole lot of conflict. On the other hand sometimes it can cause a whole lot of conflict. I am here of course thinking about dating. But that's a topic for a different blog.
For a longer discussion about how the need for context effects an understanding of the Bible, see this post here. And thanks for letting be your Rent-A-Friend. |
ART AND THE MAGIC SQUARE, PART V
Friday, April 11, 2014
One of the hottest topics going on in the research of magic squares are water retention magic squares. Developed by Dr. Craig Knecht, these squares are analyzed from the perspective that a larger integer corresponds to a taller height than a smaller integer. In this way, smaller integers can be surrounded by larger ones creating what are called ponds.
According to Dr. Knecht, magic squares in the Luo Shu format produce the maximum number of ponds. The blue cells identify ponds. The drainage path for the cell in green is long, eventually spilling off the square at the yellow spillway cell.
Here is the link for the Wiki page describing water retention magic squares. And here is another link to Harvey Heinz's fine web site on magic squares. On this web page, posted in 2008, the water retention square is referred to as the "topographical" magic square. Harvey is no longer with us but his memories live on through his love for magic squares. |
Links
Imagine a cantankerous amateur mathematician on his deathbed leaving in the margins of a book an assertion of a theorem. Because Pierre de Fermat, a lawyer by formal profession, corresponded, collaborated, goaded and vexed the great mathematicians of his day—no less than Newton, Descartes, and Gauss, and proposed so many original mathematical ideas, his assertion was given credibility. And the 350-year quest for a short proof, any proof, of Fermat's Last Theorem was on.
The assertion is a simple one—reminiscent of Pythagoras' theorem learned by any first semester algebra student. Rather than hypotenuse squared = a squared + b squared. Fermat's Theorem is that X^n + Y^n = Z^n where n cannot equal 0 nor can n>2. His exuberant margin notes indicated there was a simple short proof. In 1999 Andrew Wiles and a team of mathematicians proved Fermat's theorem (and incidentally collected a lot of prize money), but Wiles' proof was not a short proof. It was more than 100 pages long.
Some people who study the history of mathematics have insinuated that Fermat never had a short proof for his theorem. It has been postulated that he was a very competitive and argumentative man who knew he was dying, so he left the margin note to vex his competition in the mathematical world. Wiles and his team had to use mathematics that had not even been invented when Fermat was alive in order to prove Fermat's Last Theorem.
However, Fermat was somewhat of an enigma. He remained throughout his life an "amateur" mathematician, (mostly because he did not always offer up publishable proofs of his work). Peter L. Bernstein, in his book "Against the Gods", described Fermat as "a mathematician of rare power. He was an independent inventor of analytic geometry, he contributed to the early development of calculus, he did research on the weight of the earth, and he worked on light refraction and optics. " Fermat also corresponded with Blaise Pascal and together they developed the theory of probability after a gambler posed to them the conundrum of why if he betted on a five being rolled on a die four times, in the long run he would win, but if two dice were rolled and he was betting on double sixes the chances were much lower. Fermat's crowning achievement was that he is credited with developing the modern theory of numbers.
If you are feeling like you want a challenge, the search is still on for a short proof of Fermat's Last Theorem. A mathematician of my acquaintance believes it will be solved not by a mathematician but by a creative and clever individual. There is a $1 million dollar prize offered by a Texas businessman named D. Andrew Beal to any one who can offer a short proof of the theorem. |
Saturday, April 24, 2010
"The importance of the creation of the zero mark can never be exaggerated. This giving to airy nothing, not merely a local habitation and a name, a picture, a symbol, but helpful power, is the characteristic of the Hindu race from whence it sprang. It is like coining the Nirvana into dynamos. No single mathematical creation has been more potent for the general on-go of intelligence and power." – G.B. Halsted
Zero is a special mathematical concept. As previously discussed signs represent actions or quantities that we can point to and say that is what the sign represents. Zero is a special sign that can't be seen, measured, or even conceptualized. In the number system all positive numbers are signs that represent quantities that we can empirically experience. For example if we look at the number seven, I can see seven apples or seven people. Although the concept of the sign that represents number seven is abstract its quantity can be seen and measured. Negative numbers are also concepts and even though the quantity is absent it can be conceptually quantified. If I can measure seven apples I can conceptualize the physical absence of seven apples. Zero is different; zero is a sign that points to another dimension, a pure concept existing solely in our consciousness. Negative numbers are the absence of the quantity they represent, zero is nothingness.
Monday, April 19, 2010
To describe the unique metaphysical qualities of zero requires painting a number of brushstroke ideas in an effort to reveal the larger picture. Each idea, in itself, might not be extraordinary but in totality they form an image that points to another dimension. The next few posts will represent the rational argument, a philosophical claim developed with empirical reason. And because the physical world is a world of opposites or in common terms 'a duality' a rational claim in itself is an incomplete explanation therefore later posts in this series will be a spiritual assertion.
Signs and Symbols
To maintain clarity it's necessary to define the difference between signs and symbols. Signs give literal meaning whereas symbols have secondary amorphous meanings. Examples of signs are abbreviations; nouns in the English language are also signs. When I write 'put the book on the table' the words book and table are signs that refer to physical objects. Traffic signs such as a stop sign are signs and not symbols. The order to stop is a literal command and can be seen phemenologically in the action of stopping. Symbols, on the other hand, are objects that embody secondary meaning above and beyond their literal meaning. One example of a symbol is the cross as a symbol of Christian faith. Embodied in the cross is the story and the life of Jesus and the entirety of the Christian Faith. To look at the cross and to think of a physical body on a cross of wood is to misunderstand the meaning of the symbol. The meaningful aspect of symbols arise in your consciousness not from your senses. Other potent symbols are the yin and the yang, the Star of David, the Crescent and the Moon, and even the swastika. In my argument I am focusing on the special characteristics of Zero. The first unique characteristic of zero is it's both a sign and symbol. The two are interlocked and cannot be separated.
Wednesday, April 14, 2010
Zero is a concept that exists only in our minds. We can't measure zero, we can't experience zero, and it doesn't have a physical presence. Yet this concept that doesn't 'exist' has been instrumental in creating the world around us and is essential in the meaning of our lives. Over the course of the next couple of posts I will explore the connection between zero and ultimate reality, first in a rational argument and then in spiritual terms.
The sun, the moon, the sun, the moon, day after day we are experience the wonder of our existence. The most meaningful symbol of our lives is the circle, a timeless myth that exists in all cultures. A universal truth. Over and over we experience circles in the shape of the sun and the full moon. The significance of the sun and the moon are more than their physical presence. They signify the passage of our lives, new beginnings, peace, and nature, the creators of life. The circle is powerful symbol filled with deep meaning. When we create a circle we are creating a symbol that represents something else, the sun, a pie, a wheel, the concept of zero. In cultures with a shared history the meaning is conveyed automatically without the need for explanation. I propose that it isn't a coincidence that we use the circle to represent zero but a profound connection between the physical and the non physical dimension. To support my proposal rationally I will begin by defining what symbols are, look at the connection between symbols and their meaning, give some examples of how they are used, and conclude by stating how zero is really a symbol for ultimate reality. |
In mathematics, the Fibonacci numbers or Fibonacci series or Fibonacci sequence are the numbers in the following integer sequence: (sequence A000045 in OEIS) or, alternatively,[1] By definition, the first two numbers in the Fibonacci sequence are 0 and 1[...] |
Before all the instruments and theorems listed in this article were available the most used tool for measurement were an individuals hands. Astronomy was no different. There are 9 hands from the horizon to zenith if held at arms length, palms facing so they can be seen and stacked on top of one another during the count. This makes 36 hands in a circle and when each hand is divided into tens then there are 360 divisions. Thus 360 degrees in a circle, 90 in a right angle and so on and so forth. Michael Edwards (Michaelmwe3wm) — Preceding unsigned comment added by Michaelmwe3wm (talk • contribs) 21:47, 5 October 2012 (UTC)
The history section is using circular logic. It claims that 360 was chosen because of the number of days in a year, and then goes on to claim that there were 360 days in a year because there were 360 degrees in a circle. 63.228.4.70 22:25, 11 April 2006 (UTC)
Basically it is not known why 360 degrees was originally introduced. The use of 360 degrees implies that certain important angles like the right angle and the angle of a equilateral triangle can be expressed as an integer. This may be a reason that 360 degrees has not died out long ago. Entropeter (talk) 10:25, 12 September 2010 (UTC)
Who did come up with the invention? As well as who, what time was the system invented/adopted.
My little tidbit about the calendar may provide some clues. - Nolan Eakins
I doubt there were too many peoples who literally thought the year consisted of 360 days. Rather, 360 was a conveniently round number (especially in terms of a sexagesimal numbering system) which was CLOSE to the number of days in a year...AnonMoos 01:44, 29 October 2005 (UTC)
According to my old maths teacher they calculated the number of days in a year by lighting a candle everyday at dusk and putting it out at sunrise then saving up the bits that were left over a few years and comparing the lengths. Obviously this method wasnt very accurate hence 360 instead of 365(point whatever). Probably an urban myth but maybe someone can confirm/refute ? 213.40.112.230 (talk) 20:16, 23 September 2008 (UTC)
This is so overwrought to be patently absurd... it would have been rather easier to just, you know; count the number of days. —Preceding unsigned comment added by 86.131.44.14 (talk) 23:49, 2 November 2010 (UTC)
They wouldn't be able to count the number of days because they wouldn't know when to stop counting. If this is true, then I'd imagine they use the candles to determine the length of days. Lining up the candles in the order burned, they would be able to see a sinusoidal wave of sorts. One wavelengh would then equal one year and the number of candles in a wavelenth would be the number of days in a year. Of course, this wouldn't be exactly accurate because it would be near impossible to light and put out the candles exactly at dusk and dawn each day. It's really not so absurd as you might think. I have never heard of this theory before so I don't really have anything substantial to back up this hypothesis. It's an interesting idea though. — Preceding unsigned comment added by 50.44.89.231 (talk • contribs) 00:27, 28 September 2011
Per the wiki articles on candles and the history of trig, candles did not appear in Europe or the middle east until a few hundred years after the 360 degree circle was known to be established at the latest. — Preceding unsigned comment added by 71.63.69.193 (talk) 07:54, 15 July 2012 (UTC)
Do calculators seriously support this? i've never seen it. It would be very useful to place an approximate year on that peice of information, such as "Whilst this idea did not gain much momentum, most scientific calculators made after 1934 still support it. (Note: I made up that year)
I certainly don't ever remember seeing a scientific calculator (e.g. a calculator that actually supports trig) that didn't offer the grad as an option for its trig functions. but then i'm only 20. Plugwash 01:45, 19 December 2005 (UTC)
The only big exception I've seen is the ubiquitous TI-8x graphing calculators, which don't offer the grad even though TI's cheaper scientific calculators do. Krimpet 14:19, 23 January 2007 (UTC)
In the calculators TI-83, TI-84, and Voyage-200 from Texas Instruments one can choose between degrees, radians, or grads as default. The successor TI-89 has a function that converts degrees to gradians but only degrees and radians can be choosen as default.Entropeter (talk) 10:40, 12 September 2010 (UTC)
why is each degree divided into 60 parts and each minute into sixty seconds???
This is pure speculation, but these subdivisions may also go back to the Babylonian base-60 system. If so, they defined minutes and seconds as naturally as we would define tenths and hundredths of a degree.
i would also like to know why they chose the terms "minutes" and "seconds" - the history section on this page or the sexigesimal page does not address this... —Preceding unsigned comment added by 65.93.145.61 (talk) 01:12, 24 September 2007 (UTC)
I heard that a minute was the first minute (mahy-noot) division of a degree (or hour) and a second is the 2nd. I haven't looked for a source for this, but someone else can if they like. 75.85.51.140 (talk) 09:42, 14 March 2010 (UTC)
Minute comes from Latin, minure, that means decrease, and minus has the same origin. Second also comes from Latin, secundus, that means 2nd. It would be interesting to know how old the division of degrees into minutes and seconds is.Entropeter (talk) 10:48, 12 September 2010 (UTC)
Surely this is a bit pointless, having two subsections on the two major alternative units? I will edit it into one section with a slight introduction to each and then following through by suggesting the user click the links, because surely they have their own pages? Help plz 15:59, 25 June 2006 (UTC)
Hi, I edited the Babyloninan history additions, which are cool. I can't help but wonder, though, whether some of that belongs under Pi instead of here. It's probably more relevant to just say that the sixty-fold divisions of the degree relate to the Babylonian numeral system, and that the Babylonians were the pioneers in careful astronomical measurements; everyone later built on or generalized their work.
The tablet picture does not correspond to what's in the text. Can we find a picture of the real tablet or at least a reference number?
As an aside, the Egyptians certainly knew that the year had >360 days; they even had a cute myth about the discrepancy. WillowW 10:42, 27 June 2006 (UTC)
I agree; I don't see how the pi information has anything to do with the number of degrees in a circle, except for the fact that Babylonians used a base-60 numbering system. At the least, I think the bit about the "number of days in the year" should lead the history section, with the pi business moved later in the article (if not removed). I started making the change myself, but I have no idea how to segue into the pi stuff, so I'll leave it alone lest I drift into Speculation. I'm guessing that people stuck with 360, rather than 365, because of the Babylonian base-60 numbering system; again, I would rather not add that without any proof, which I do not have.
(Oh, and do tell about that cute myth; sounds like it would be a good addition to the page.) --ScottAlanHill 23:08, 3 September 2006 (UTC)
I agree with the above discussion that the entire Pi related section does not belong in the history of the degree, or to the degree in general. I plan to remove it wholesale unless someone objects and can explain why it should remain here. Crum375 21:22, 21 September 2006 (UTC)
There is also a serious lack of reliable sources for virtually everything in this article and apparently over-abundance of original research till proven otherwise. That really needs to be fixed ASAP. Crum375 21:34, 21 September 2006 (UTC)
I found this reference in an online forum from 1995, which seems to be the source (copyright issues?) for the PI-derivation explanation for the degree that was included in the article. But at this point, I just don't see it more than speculation and I don't see a good quality source, and to me it's really not very convincing. I think the astronomical derivation, from the amount the stars seem to advance in the sky every night in their annual trek around the celestial pole (about 1 degree per evening, with an error of less than 1.4%), and the fact that some old calendars actually counted the year as 360 days is very convincing and logical. We do need better sources, though. And I would accept the PI-derivation method (copyright addressed) if someone could explain better how it derives the degree (the 1995 message doesn't do for me) and provide a better source. Crum375 22:58, 22 September 2006 (UTC)
Presumably the only point here is that the radian is a common measure of angle competing with the degree. π enters into this competition. This is a rare exception to the general rule that unit conversions between conservative and liberal units are by definition rational, for example an inch is an integer number of microns, the velocity of light is an integer number of meters per second, a pound is an integer number of micrograms, Avogadro's number is on the road to becoming an integer, etc., etc. --Vaughan Pratt (talk) 05:36, 5 April 2011 (UTC)
My opinion is that the angle of the Equilateral Triangle tool was a reference angle because however you handle the tool you get the same angle so it makes it a handy tool.
So the reference angle was accorded a number of degrees of 60decimal = 10sexagesimal, as it is today, equal to the base of the Sexagesimal numeration system in use by the ancient Mesopotamian Civilisations and that still today pattern the minute and second arc divisions.
The advantatge of this base is the large number of divisors, facilitating calculations as stated in the wikipedia article Sexagesimal.
How many times an equilateral triangle angle fits in a circle ? Exactly six (since the sum of the triangle angles totals half a circle), which multiplied by the reference angle gives you 360 ! Griba2010 19:34, 4 January 2007 (UTC)
It is published: st-andrews.ac.uk - Babylonian_numerals >>one theory is that an equilateral triangle was considered the fundamental geometrical building block by the Sumerians. Now an angle of an equilateral triangle is 60 degrees so if this were divided into 10, an angle of 6 degrees would become the basic angular unit. Griba2010 11:16, 5 January 2007 (UTC)
I read through your reference (both 'numerals' and 'math'), all I find is speculations (no archeological or historical references for Babylonian geometry), and it doesn't tell us anything (that I can find) about the origin of one degree, which is the subject of this article. Crum375 12:48, 5 January 2007 (UTC)
My personal guess is that the angle of the equilateral triangle was the real standard. It is the easiest angle to reproduce with fidelity, so a good standard.
Any angle was expressed in terms of it following the sexagesimal system. Later in time, the first sexagesimal part became the main standard as many angles were less than the reference one, and a discrete number is always better to express, when talking, than a fractional one. Moreover, a written quantity less than 1 was easily read as integer since there was no sign for the preceding 0 and the meaning had to be taken from the text context. Later on, with finer tools, came the minutes and seconds accuracy. Griba2010 23:18, 6 January 2007 (UTC)
That sounds like a logical guess, but we need better sourcing for it. The fact that the stars in the sky seem to advance by 1 degree every evening around the celestial pole (with ancient calendars using 360 days per year) is also convincing. With the Babylonians not even having a '0', just a 'space', the whole subject of their supposed expertise in sexagesimal math seems shaky to me - since 0 digits would show up periodically, sometimes in groups, while doing measurements. A space is woefully inadequate to represent more than one 0 - when you get to 2 or 3 togther, it's really impractical. My fear is that we may be reading more into it than what was really there. (I do realize that 0 'density' in sexagesimal numbers is 6 times less than decimal ones). Overall we are very short on sources here. Crum375 23:27, 6 January 2007 (UTC)
What is the ancient and modern definitions of the degree? I have seen the following: 1/60 of an angle from an equiangular triangle. 1/90 of a right angle. 1/180 of a straight angle. 1/360 of a circle. 1/360 of a complete rotation. 180/π radians. 360/2π radians.
Although they are all equal, only one can be the definition at a time in order to prove the other are equivalent. Zginder 21:26, 23 April 2007 (UTC)
Right now the content related to the various articles relating to measurement seems to be rather indifferently handled. This is not good, because at least 45 or so are of a great deal of importance to Wikipedia, and are even regarded as Vital articles. On that basis, I am proposing a new project at Wikipedia:WikiProject Council/Proposals#Measurement to work with these articles, and the others that relate to the concepts of measurement. Any and all input in the proposed project, including indications of willingness to contribute to its work, would be greatly appreciated. Thank you for your attention. John Carter 21:01, 2 May 2007 (UTC)
I think I'm going to be alone on this one, but I've always had a problem with people stating that there are 360 degrees in a circle.
Here's why:
People say "there are 180 degrees in a triangle". What they mean is the sum of the internal angles of a triangle is 180 degrees. Similarly, there are 360 degrees in a rectangle.
In general, there are 180 * (number of sides - 2) degrees in a polygon.
Now here is where it starts getting debatable. I see a circle as a polygon of infinite number of sides.
Hence, as the number of sides tends to infinity, the number of degrees also tends to infinity.
So there are not 360 degrees in a circle, otherwise it would look like a square :-P
It's more correct to say: "There are 360 degrees in a revolution.", which obviously doesn't relate to the sum of the internal angles.
I heard the question "How many degrees are there in a circle?" on The Weakest Link and I was upset that they allow questions which have, at the very least, debatable answers. What do you think?
The Strongest Link would have answered instantly at 400 words a minute, "What do you mean? There are two possible answers: 360 degrees and infinitely many degrees, depending on what you mean. So what do you mean? Or are you the Weakest Link?" --Vaughan Pratt (talk) 04:03, 5 April 2011 (UTC)
I recently examined a 1904 pattern uglomer (lining plane) sight in the St Petersberg Museum of Artillery, Engineers and Signals. This circular sight was divided into 600 dividions and there was no secondary or micrometeer sclae to divide these units. Obviously this is a source of the Russian 6000 mils.
Next in Sweden I again examined sights, on a 1930s vintage howitzer the azumuth scale used 6300 mils, however, the elevation scale used a degrees symbol and each 'degree' had four divisions, the three 'dividers being marked '12', '24' and '36', impying a degree divided into 48 subunits.
The 6400 mil circle seems to have been a French invention about 100 years ago. Nfe 01:50, 23 October 2007 (UTC)
A definition of an angle would be that an angle is the union of two rays that have the same endpoint. The sides of the angles are the two rays, while the vertex is their common endpoint. stands for an angle. You can put it in front of three letters which represent points. The first and third letters represent points on each of the rays that form one of the sides. The middle letter represents the vertex. As you can see in the diagram, each point is represented in the written form. The letters can go either way - that is, first and last letter are interchangeable. So, that angle could either be ABC or CBA. Since B is the vertex, it is always in the middle of the two letters. You can also name an angle by just the letter of its vertex. So, for the example in the picture, the angle could also be labeled B. That's only if there are no other angles that share the same vertex. There is a third way to label angles. In the third way, each angle is designated with a number, so the example could be labeled 1 or 2 or whatever you wanted.
Angles are measured in degrees. The number of degrees tell you how wide open the angle is. You can measure angles with a protracter, and you can buy them at just about any store that carries school items. Degrees are marked by a ° symbol. For those of you whose browsers can't interpret that, a degree symbol looks like this: . I tend to just write it out instead of using the symbol because it's quicker on the computer. There are up to 360 degrees in an angle. As you can see in the picture below, the 360 degrees form a circle.
There are a few more basic things you should know about angles. First of all, the space inside an angle of less than 180 degrees, is a convex set, while the space outside of one is a nonconvex set. The opposite is true for an angle of more than 180 degrees (but less than 360 degrees). The side of an angle that is started at would be called the initial side, and the side that an angle ended at would be called the terminal side. The measure of ABC is written mABC.
When measuring angles, you usually go counterclockwise, starting where the 3 would be on a clock. That would be called a zero angle because there is nothing in it - just a single ray going directly to the right. The next important type of angle is called the acute angle. An acute angle is an angle whose measure is inbetween 0 and 90 degrees. An example would be the 45 degree angle in the picture. The next important type of angle is the right angle. This is probably the most important type of angle there is because of all the spifty things that you can do with one. I won't go into all of them here. (I have to save something for later articles!) A right angle is an angle whose measure is exactly 90 degrees. Continuing around the circle, next is the obtuse angle. An obtuse angle is an angle whose measure is inbetween 90 and 180 degrees. The 135 degree angle in the diagram is an example. The last major kind of angle is the straight angle. A straight angle is an angle that measures exactly 180 degrees. Thus the name - the two rays form a straight line. A negative angle is also possible. This just means that you go clockwise instead of counterclockwise.
A lot of geometry teachers don't go beyond that, at least at first. There isn't much else left to explain, but I'll give it a shot. After straight angles, there aren't any more special angles that you need to know about. A 360 degree angle is an angle that does a full circle. It looks just like a zero angle, but instead of having no degrees, it has 360 of them. (Duh. You can't get more basic than that!)
It is possible to have an angle with more than 360 degrees. To find out what it looks like, all you do is subtract 360 from it until you have an angle less than or equal to 360. (What?! You want an example? C'mon, you people...) For example, if you have an angle that is 546 degrees, you subtract 360 from 546 to get 186. Thus, the angle is the equivalent of a 186 degree angle.
There are a few more terms that you should also know. Supplementary angles are two angles whose measures combined equal 180 degrees. Complementary angles are two angles whose measures combined equal 90 degrees. Two non-straight and non-zero angles are adjacent if and only if a common side is in the interior of the angle formed by the non-common sides. A linear pair is a pair of adjacent angles whose non-common sides are opposite rays. Vertical angles are two angles that have a common vertex and whose sides form two lines. is a bisector of DAC if and only if is in the interior of DAC and mDAB = mCAB —Preceding unsigned comment added by 199.126.187.152 (talk) 03:09, 9 October 2008 (UTC)
You wrote "The letters can go either way - that is, first and last letter are interchangeable. So, that angle could either be ABC or CBA." By "interchangeable" you seem to be saying here that the angle ABC and the angle CBA are equal. You also wrote "There are up to 360 degrees in an angle." And you defined an angle to be "the union of two rays that have the same endpoint." Since these three statements taken together are inconsistent, it would appear that you need to explain what you meant by at least one of them. Since you're right about complementary and supplementary angles, one imagines you can find and fix the apparent inconsistency here. --Vaughan Pratt (talk) 04:34, 5 April 2011 (UTC)
The caption under the diagram in the History section is troubling me. It says "A circle with an equilateral Chord (geometry) (red). About one fifty-seventh of this arc is a degree. 2π such chords complete the circle". I always thought that the chord was the straight line (as marked in red on the diagram), that one-sixtieth of that angle was one degree, and that it took exactly 6 of those to complete the circle. I'm not sure what the name for the shape would be but the diagram should highlight an arc of the same length as the radius for there to be 2π of them in a full circle. --ClickRick (talk) 11:22, 27 May 2009 (UTC)
You are absolutely right. Some half-wit had edited the article and messed this up. I've now reverted it back to the text that was there before, which said pretty much exactly what you write above, that the chord is the straight line (as marked in red on the diagram), that one-sixtieth of that angle is one degree, and that it takes exactly 6 of those to complete the circle.
That's for a 12-hour clock, whose hour hand runs at 30 minutes per minute. For the more usual 24-hour clock in this translation the hour hand only goes 15 minutes per minute, though in both cases the minute hand goes at 6 degrees per minute or 6 minutes per second and the second hand goes at 6 degrees per second. --Vaughan Pratt (talk) 05:03, 5 April 2011 (UTC)
Crum785 changed the text of the alternative units section changing the equation ° = π/180 to 1° = π/180, under the description "more accurate". However, I'd argue this misses the whole point of the way the section was previously phrased. The way it was previously written was the somewhat provocative, but arguably correct statement that the degree symbol is itself a mathematical constant, in the same way as π or e. To be honest, this is how I always thought of it, and I know at least one prominent professor at my university who feels the same way. The new statement is not at all elucidating, and really doesn't show that treating the degree sign as a mathematical constant works in all cases.
Can you find a source saying that your way is more correct? I couldn't find any that explicitly say ° = π/180 is incorrect, and for the above cited reasons, I'd prefer if we could use this instead. Obviously, we don't want to be spreading false information, but as long as ° = π/180 is not false, it seems preferable. I agree the notation is not commonplace, but that doesn't make it incorrect. 129.15.127.237 (talk) 06:50, 8 March 2010 (UTC)
What's at stake here? I learnt back in, what, 2nd grade perhaps, that multiplying anything by 1 gave back the same result, for example 1 times 7 equals 7.
Are you questioning whether 1° = ° ? Or are you saying that 1° is not as syntactically correct as ° ? --Vaughan Pratt (talk) 04:45, 5 April 2011 (UTC)
Is there any reason why we mention this, as opposed to any of numerous other circumstances in which angles might be measured? Does it have some special importance here that I'm missing? (If retained it needs rewording because it is not clear what "it" refers to.) 86.181.169.8 (talk) 04:15, 20 November 2011 (UTC)
Besides degrees of sixtywise, and the metric grade and radian, the following divisions are noted.
1. The division of the circle into signs and degrees to match the months and days, is ancient, and separately in Egypt, in the Chaldea, and in China, (in the far east). Such measures are used to record the movements of the sun and the moon and the stars (planets), against the fixed stars. Although 12 signs are usual, 15 or even 36 signs are recorded too.
2. The egyptian zodiac contains 36 decans. The origin of the 24-hour day comes from the count of 24 hours in the 'decimal day', ten hours of sunlight, and hour each of twilights of dawn and dusk, and twelve hours being the rising of twelve decans at night. It was the greeks that made the hours of equal lengths, and divided them against the chaldean fractions.
2. According to Sir Thomas L Heath (A manual of greek mathematics, p384), the Chaldeans used the division of circles into 360 degrees only for astronomy. Circles in general are rendered as a diameter of 60 ells, with pi=3, gives a circle of 180 ells of 24 digits each. It is Hipparchus who advamced the elliptic division for general use.
3. Angles based on the mil (such as the Swedish and NATO scales), suppose pi is somewhere near 3, and the circle is 1000 units. The use of such says that at 1000 yards, two points separated by a mill is separated by a yard. Pi variously equals 3.15 in Sweden, 3.2 in the NATO countries. In the NATO case, a division with pi=3.2, divides the circle to 6400 mills, which makes the compass points come at 200 mills.
4. An angle-system one encounters on astronomy chards is HMS of right ascession. Here, the circle is divided into 24 hours, of 60 minutes of 60 seconds, rather than 360 degrees. It kind of replicates the standard siderial clock (zenith pointer points to siderial time against RA).
5. Common angles include the use of clock-face units, eg '4-oclock', where up, forward, or north is 12-oclock.
A protractor is a simple instrument used to measure angles, typically on a map. A dial sight or aiming circle is an artillery instrument used to lay a gun in azimuth by means of an aiming point other than the target. It is copiously covered in the relevant professional literature, mostly published by Governments who tend to the be users of artillery. Ugloma is the transliteration of the Russian name.Nfe (talk) 08:46, 2 July 2013 (UTC)
First of all, a word of explanation for the benefit of anyone reading this who does not know the history which led to the above comment being posted. It refers to the fact that, in connection with the Russian army's use of a unit of 1⁄6000 of a circle, until a few days ago the article said "this may be seen on a protractor, circa 1900, in the St Petersberg Museum of Artillery", which has been changed to "this may be seen on an 'ugloma' (dial sight or aiming cirle in English) , circa 1900, in the St Petersberg Museum of Artillery".
I have just spent a considerable amount of time searching for confirmation that this word exists in Russian, and has the meaning you attribute to it. I have looked in two Russian dictionaries that I have, and several online dictionaries. I have searched on Google for "углома" (ugloma) and for various forms that would be expected to be case forms of that word, if it exists, such as "угломах" (uglomakh), "угломой" (uglomoy), "угломов" (uglomov), "угломе" (uglome). The conclusion is perfectly clear: "углома" (ugloma) exists only as the genitive of "углом" (uglom), which means "angle" or "corner", and does not have the meaning you attribute to it at all. Apart from dictionaries saying so, I found numerous actual uses of the word (in various case forms) where the context made it abundantly clear that "angle" or "corner" was the meaning, and not a single one where it could possibly be taken as meaning a measuring instrument. Here are a few examples: [1], [2], [3], [4] and there are many more. The most relevant Russian word I can find is "угломер" (uglomer), which usually means "protractor", but which has a broader range of uses than the usual English meanings of "protractor", and can be used to refer to inclinometers, alidades, and goniometers. Instruments developed from the goniometer are used in the sort of application you refer to, and have been variously referred to in English as "goniometric sights", "aiming circle goniometers", "dial sights", "aiming circles" and "panoramic telescope", so it seems very probable that the Russia "угломер" includes the dial sights and aiming circles that you have given as English equivalents in the article, and I have little doubt that the instrument referred to in the article is some form of goniometer or dial sight.
Having said all that, it is really an academic question what the correct Russian word is, because this is the English language Wikipedia, and in the absence of very good reasons for doing otherwise, we should use English. (The article Napoleon refers to the French troops' horses, not to their chevaux; we don't use a French word just because the particular horses happened to be French ones.) Another point is that the present wording of the article is not likely to be very helpful, because most readers will never have heard even of a dial sight or aiming circle, let alone either the mythical "ugloma" or the real "uglomer". I therefore intend to replace the present wording with one which uses the English expressions for the device, and gives a brief explanation of the term. If you disagree with my changes for any reason, please let me know. JamesBWatson (talk) 12:57, 2 July 2013 (UTC)
Incidentally, I have also replaced "aiming cirle" with "aiming circle", which I trust is OK. JamesBWatson (talk) 13:05, 2 July 2013 (UTC) |
1) With Pythagorean's theorem, dozens of proofs exist (including one by a US President). So you have the choice of which proof you favor.
This theorem may have more known proofs than any other (the law of quadratic reciprocity being another contender for that distinction); the book The Pythagorean Proposition contains 370 proofs.[11]
2) In calculus, certain problems can be solved in more than one way (algebra, etc.)
There are other situations where certain proofs aren't fully accepted by all mathematicians. So what do you think?
PhilX
I once heard a phrase: "Math isn't about finding limits, but rather about finding possibilities." I hold that opinion as a personal axiom.
1) If we look at the nature of number as stemming strictly from 1
2) and all number stemming from 1, in itself being composed of 1
3) and all number being composed of 1 manifesting ad infinitum
4) Ad infinition is 1 revolving into itself to produce all possible numbers.
5) All possible number exist through all mathematical functions, as all mathematical functions are strictly extensions of positive (addition) and negative (subtraction) values.
6) As all mathematical functions are extensions of positive and negative values (founded in basic arithmatic), these mathematical functions provide foundations for further mathematical functions (multiplication, division) ad infinitum in correspondence to the ad infinitum nature of number (considering form and function are interjoined).
7)In theory there are infinite mathematics stemming from a core base synonymous to "1", and these infinite numbers/functions are a result of 1 revolving through itselfHow can "i" be any different than a "point" considering all imaginary structures at the micro and macro level are reduced to a point, with the intermediate being composed of "points"?
The observation I am trying to make, is how can what we understand of number be seperated from the spatial nature of reality we observe?
If all number is strictly composed of "1"....well what is "1"?Nothing to forgive, no offense was made on your part, none was intended on mine.Well, if you had watched the videos and made your own research, you would not need to be asking that question! To understand this one must make effort, and study! The videos I have linked to are short interviews/lectures of physics and mathematics professors; I would not say that they were confusing people when they were showing that these mathematical equations are being used to make sense of the real world in physics!
Now, of course the context and the analytic continuation is important, but for the purposes of this thread, it is still an example which shows that in mathematics we have the choice, at least from my perspective. Moreover, the analytic continuation spot on illustrates my point. Isn't it freedom in mathematics the subject of this thread? The thing is that you mentioned the context principle by bringing up the analytic continuation concept, but then why did you not apply the same context principle to the example with respect to the subject of this thread? This does not show consistency on your part. |
Mathematics Colloquia and Seminars
Voting in Agreeable Societies
Student-Run Applied & Math Seminar
Speaker:
Anthony Caine, UC Davis
Location:
2112 MSB
Start time:
Wed, Feb 3 2016, 12:10PM
Consider a one dimensional political specturm like liberal versus conservative. Did you know that in a society where every two people can find a common ground then there is a stance that will be agreed upon by all people? While this is interesting, such a society does not model, say, the presidential election. Indeed, there exists people in America who cannot agree.
However, maybe out of every 10 people, there are 3 that agree. This motivates the following definition and question, let's call a society (k,m)-agreeable if there are at least m people and for every subset of m people at least k of them agree on a candidate. Then what is the maximum number of people we can satisfy in the election? We can further generalize this to multidimensional "candidates", i.e. we have two dimensional political spectrum (control over property)x(control over person). |
Saturday, November 8, 2008
I admire most
I have my favorite numbers too. My favorite number is 47 - the number of letters in the longest word in the English language. It also combines two of my favorite digits 4 and 7. 5 is also one of my favorite digits.
Why do you ask?
These are my favorite numbers to write.
I like the sharp corners and straight lines. 5 isn't all straight, but it does have corners. It's also another reason that I like Z so much. I've been crossing my z's since Algebra II, when I didn't want to confuse the Zs in my equations with 2s. |
abstract
Primordially a
geometry was a science on properties of geometrical objects and their mutual
disposition. Such interpretation of the term "geometry" is qualified
as physical geometry. A use of only Euclidean geometry generated another
interpretation of the term "geometry", which was interpreted as a logical
construction. Such interpretation of the term " geometry" is qualified
as mathematical geometry. Mathematical geometry cannot use for description of
the space-time, generally speaking. Nevertheless the mathematical geometry has
been used for description of the space-time during the twentieth century. This
circumstance lead to problems in general relativity.
There is
text of the paper in English (pdf, ps) and in Russian (ps, pdf) |
Everyone knows that a square has 4 corners, but how many square faces are there
in a tesseract? And how many tesseracts are there in a 9-cube?
As an explorer of higher spaces, I often need to know the answers to arcane questions like this.
Many years ago I found a quick and easy way of finding them: a triangle of numbers I call
"Cartan's Triangle". And as we shall see, this triangle can open the door to many strange adventures.
If you have a copy of Excel, you can build your own Cartan's Triangle in less than 30 seconds, as follows:
Enter 1 in cell B1
Enter =A1+2*B1 in cell B2
Drag a selection rectangle as big as you want, from cell B2 to, say, cell K10
From the Edit menu, choose Fill Down then Fill Right
The result should look something like this:
Each row in this table represents an n-dimensional square. Row 1 is a point, row 2 is an edge,
row 3 is a square, etc. The columns indicate the number of corners, edges, squares, etc. contained therein.
So, starting at row 1, we see that a point contains 1 "corner" and nothing else.
Row 2 tells us that an edge contains 2 corners and 1 edge.
Row 3 tells us that a square has 4 corners, 4 edges, and 1 square. And so on.
The answers to the above questions are now easy to find.
Cell D5 tells us that a tesseract contains 24 squares.
And cell F10 reveals number of tesseracts in a 9-cube: 4032.
Cartan's triangle works every time, and you can make one as big as you want.
It seems magical that the interlocking complexities of endless higher-dimensional objects
could all be revealed by a single, absurdly simple formula, B2 = A1 + 2*B1.
Where did this formula come from?
Deriving the Formula
By 1988 I realized that the starmaze was built along the edges of a nine-dimensional hypercube
and I needed to know how these things were constructed. There are books that will tell you the number of squares
in a tesseract, but I found that in order to really understand hypercubes I needed to reason
these things out for myself.
One of the attractive things about higher dimensions is that
it is quite possible to do this, even for those without a lot of mathematical training.
I took a rather circuitous route, but had a lot of fun along the way.
Here is how I did it.
First I invented a notation to represent the idea of an n-dimensional square:
= a corner, = an edge, = a square, etc.
I then coined the term "radiate" to describe the way multiple edges spring from a single corner or multiple squares
start from the same edge. For me this captured the essence of how n-dimensional squares are constructed:
Each corner in a radiates n edges;
each edge shared by 2 corners
Each edge in a radiates (n-1) squares;
each square shared by 4 edges
Each square in a radiates (n-2) cubes;
each cube shared by 6 squares
When expressed in this way, it's easy to extend this sequence into the mysterious realm of higher dimensions.
We can generalize this pattern into the following assertion:
Each
in a
radiates (n-k+1) 's;
each
shared by 2k 's
To check this out, try using 2 for k and 3 for n. The assertion then becomes:
Each edge in a cube radiates 2 squares; each square shared by 4 edges
This is all very well and good, but it still doesn't tell me how many squares there are in a cube.
What it does do, though, is tell me how squares radiate from edges. So if I knew how many edges there are in a cube,
I could use my assertion to figure how many squares a cube has.
We know that each edge in a cube radiates (n-k+1) = 2 squares. So if a cube has 12 edges, there are a total
of 12*2 = 24 squares radiated. This is still not right because we are counting each square more than once.
To get the right number we have to divide this value by the total number of edges shared by each square.
Fortunately, our assertion gives us that number as well: 2k, in this case 4.
24 / 4 = 6 squares, the correct answer.
So if we know the number of edges in a cube, we can figure out the number of squares it has.
We can represent this mathematically by defining a recursive function.
Let f (k,n) = the total number of 's
in a where n >= k. Then:
f (k,n) = (n-k+1) * f (k-1,n) / 2k
For recursion to work correctly, we need to provide a stopping point. To find the number of squares
in a cube, we need the number of edges. To find the number of edges, we need
the number of corners. Fortunately, the number of corners in a cube (or any other n-square) is easy
to figure out. There are 2 corners in an edge, 4 in a square, 8 in a cube, etc. The number of corners
in an n-square is 2n.
So now we can fully define our function as follows:
For k>0, f (k,n) = (n-k+1) * f (k-1,n) / 2k
For k=0, f (k,n) = 2n
Now we're getting somewhere! Questions like "How many squares are there in a tesseract are now easy to answer:
A tesseract has 16 corners, 32 edges, and 24 squares. Still, recursive functions are a bit of nuisance.
It would be better to have a straightforward formula that would give us the answer in one fell swoop.
To figure that out, I tried leaving n as a variable and then plugging in successive values of k to see
if I could generalize the pattern:
Wait a second. n! / (n-k)!k! is the formula for (n choose k). So now all this gobble-de gook boils down to something surprisingly simple:
f (k,n) = (n choose k) * 2n-k
(n choose k) refers to the number of different ways of choosing k items from a set of n items. If you have four friends,
Andy, Betty, Carl and Desdemona, but you can only invite 2 of them to join you for dinner, there are (4 choose 2) = 4! / 2!2! = 24 / 4 = 6
different ways of doing this:
Andy and Betty
Andy and Carl
Andy and Desdemona
Betty and Carl
Betty and Desdemona
Carl and Desdemona
In hindsight, it's not surprising that (n choose k) should show up in this formula. When you ask questions like "How many
squares are there in a tesseract?" you are soon faced with the question "How many different ways are there to make a square
using four dimensions?" A square is just an extension along any 2 of those 4 dimensions, or (4 choose 2).
So there are 6 basic ways of making a square in four dimensions, but each of these ways occur more than once.
This is easier to see in 3 dimensions. Imagine yourself standing inside a cube-shaped room. There are (3 choose 2) or 3
different types of square:
YZ Squares perpendicular to the x dimension (which form the walls to your left and right)
XZ Squares perpendicular to the y dimension (which form the walls in front and behind you)
XY Squares perpendicular to the z dimension (which form the floor and ceiling)
Squares occur 2 times for each unused dimension.
In a cube each square uses up 2 of the 3 available dimensions, so each type
of square occurs twice. In a tesseract, each square uses up 2 of 4 available
dimensions, so each of the 6 different types of square occurs 2*2 = 4 times.
This is precisely what our formula tells us:
f (2,4) = (4 choose 2) * 24-2 = 6 * 22 = 24.
Triangles Old and New
When I discovered that (n choose k) was at the center of my formula, I remembered another place where this ubiquitous
little function plays a prominent role: Pascal's Triangle. Pascal's triangle is a triangular array of numbers
with many wonderful properties that make it a favorite tool of recreational mathematicians. It was published by
Blaise Pascal in 1654, but had been discovered centuries earlier by both Chinese and Persian mathematicians.
It's easy to construct a Pascal's Triangle. Start with a "1" at the apex of triangle. Then put a "1" below and to
its left and another "1" below and to its right to form the second row. The third row is "1 2 1". Each number is the sum
of the number above and to the left and the number above and to the right (assume the space around the edge of the triangle
is filled with zeros). When you are done, your triangle should look something like this:
Binary coefficients show up all the time when you're dealing with hypercubes. Suppose you divide
each dimension so that 30% of it is red and 70% of it is blue. A square built this way would consist
of one small red square, two purple rectangles, and one large blue square - coefficients "1 2 1".
A cube would consist of a small red cube, three beams, three slabs, and a large
blue cube - coefficients "1 3 3 1". And so on into higher dimensions.
So where does (n choose k) come into this? It turns out that the formula for the kth number (starting from 0)
in the nth row of Pascal's Triangle is, you guessed it, (n choose k). The triangle works because
(n+1 choose k) = (n choose k-1) + (n choose k).
In thinking about this, I wondered if I could use the same trick to derive the numbers of my formula, f (k,n).
I did the math and quickly discovered a very similar property:
f (k, n+1) = f (k-1, n) + 2*f (k, n)
This meant that I could make a triangle of my own. It would work just like Pascal's Triangle,
but instead of deriving each value by adding the two closest numbers above it, you derive
each value by adding the number above-left to 2 times the number above-right.
The result looks like this:
I am certainly not the first person ever to think of this. The formula for f (k, n) is widely known,
and others have noticed that the numbers could be arranged in a triangle. But I had to call it something,
and since no one else I know of bothered to name it, I decided to call my creation "Cartan's Triangle."
I've found it very handy over the years and, as we've seen, you can use Excel to generate one from scratch
in just a few seconds.
But wait. There's more!
The Third Power
The rows of Pascal's Triangle sum to powers of two: 1, 2, 4, 8, etc. So after creating Cartan's
Triangle, one of the first things I did was to sum each row. The values I got surprised me:
1, 3, 9, 27, 81, etc. These, of course, are powers of three.
By this time I had spent a decade wandering the starmaze, and had encountered powers of two at every turn.
I could speak binary in my dreams. But in all that time I had never before encountered powers of three.
Would I now have to learn to speak, uh, trinary?
As I studied the triangle it dawned on me that the answer was YES! If you really want to understand what a hypercube
is made of, it would be handy to speak trinary.
Each row of Cartan's Triangle represents an n-dimensional
square: a point, an edge, a square, a cube, a tesseract, etc. And each value in a given row represents
a component of that n-square. Row 3, for example, tells me that a cube consists of 8 corners, 12 edges,
6 squares, and 1 cube. 8 + 12 + 6 + 1 = 33 = 27.
So if you wanted to index all the components of an n-square, and assign a unique number to each one, the numbers
would always total to a power of three. I had been using binary numbers to index the corners
of n-squares for years. I now saw that I could use the same approach to index all the components of an n-square.
The most elegant way of assigning those numbers would be to give each one an n-digit trinary number.
Not only would this system assign a unique number to every component, it would also provide a meaningful description
of that component. A trinary index number would be like DNA - even in isolation it would reveal exactly what
kind of object it was, both type and subtype, and exactly where it lived in n-dimensional space. I saw how
it would work in a flash and quickly filled a page with a definition:
A Notational System For Uniquely Identifying All Components Of An N-Dimensional Square
Each component is assigned an n-digit base 3 number, with the right-most digit representing the x dimension,
the next digit the y dimension, etc.
The three symbols used are 0, 1, and X.
A 0 indicates that the component is located at the origin of that digit's dimension.
A 1 indicates that the component is displaced from the origin of that digit's dimension.
An X indicates that the component is extended across that digit's dimension.
Regardless of dimension, all n-squares have one corner located at the origin. That corner
is always assigned the value (0,0,0,...,0) where the number of 0's is equal to the value of n.
For example, the four corners and four edges of a square would be assigned the following values:
The number of X's indicates the type of component: no X's indicates a point, one X indicates an edge,
two X's indicates a square, etc.
The pattern of X's indicates the sub-type. For example, a 3-digit index containing two X's is
a square component of a cube. There are three possible sub-types of such a square: a left/right yz-square,
a back/front xz-square, and a bottom/top xy-square. Each sub-type has a corresponding pattern.
A yz-square has X's in the 2nd and 3rd positions, a xz-square in the 1st and 3rd, and an xy-square in the 1st and 2nd.
The points can be thought of as being located in an n-dimensional binary coordinate system with only
two positions allowed along each dimension: located at the origin (0) or displaced from it (1).
The coordinates can be represented by an n-bit binary number. Regardless of dimension, adjacent
coordinates always differ by a single bit.
Each subtype can be assigned a relative location by forming all non-X digits into an (n-k)
bit binary number where k is the dimensional size (number of X's) of the component type. This can be thought of
as collapsing all occurrences of a component to individual points by removing the k dimensions in which
they extend and locating those points in an (n-k) dimensional binary coordinate system.
To find the (k-1) dimensional sub-components which form the boundaries of a particular k-dimensional
component, hold the non-X digits fixed and, for each X, change the X to a 1, then to a 0.
This defines the two sub-components for each of the k X's in the component.
The Components of a Cube
The best way to understand this notational system is to try it out on a familiar object.
The cube, with its 8 corners, 3 kinds of edge (4 each for a total of 12), and 3 kinds of square
(2 each for a total of 6), is rich enough in components to fully illustrate the power
of trinary indexing.
For each of the 27 components, I list its base 10 number, its 3-digit trinary number,
and its type and subtype. There are three sub-types of edge: width edges extend along the x dimension,
length edges extend along the y dimension, and height edges extend along the z dimension.
Each of the three sub-types of square occur twice: there is a left and right yz-square,
a back and front xz-square, and a bottom and top xy-square.
Base 10
z
y
x
Component
0.
0
0
0
Corner (origin)
1.
0
0
1
Corner
2.
0
0
X
Width Edge
3.
0
1
0
Corner
4.
0
1
1
Corner
5.
0
1
X
Width Edge
6.
0
X
0
Length Edge
7.
0
X
1
Length Edge
8.
0
X
X
XY-Square (bottom)
9.
1
0
0
Corner
10.
1
0
1
Corner
11.
1
0
X
Width Edge
12.
1
1
0
Corner
13.
1
1
1
Corner
14.
1
1
X
Width Edge
15.
1
X
0
Length Edge
16.
1
X
1
Length Edge
17.
1
X
X
XY-Square (top)
18.
X
0
0
Height Edge
19.
X
0
1
Height Edge
20.
X
0
X
XZ-Square (back)
21.
X
1
0
Height Edge
22.
X
1
1
Height Edge
23.
X
1
X
XZ-Square (front)
24.
X
X
0
YZ-Square (left)
25.
X
X
1
YZ-Square (right)
26.
X
X
X
Cube
The Nine-Space Hotel
Now that we have mastered the cube, we are finally ready to sail into deeper waters.
Higher dimensional cubes quickly become a dizzying labyrinth of interlocking spaces.
The trinary notation system is like finding a map to the maze.
Let's take a 9-cube for example. If we look at row 9 of Cartan's triangle we see
that a 9-cube is composed of no less than 19683 components:
512 Corners
2304 Edges
4608 Squares
5376 Cubes
4032 Tesseracts
2016 5-Cubes
672 6-Cubes
144 7-Cubes
18 8-Cubes
The 9-cube itself
I normally conceive of the starmaze as a set of 512 rooms (the corners of the 9-cube) connected
by a total of 2304 one-way passages (the edges of the 9-cube). But let's try exploring the 9-cube
in a different way. Let's try to imagine what it would be like to crawl through the cubes
of the 9-cube. We are, after all, three-dimensional beings, and the rooms that we actually live in
are cubes (more or less).
So let's start inside a particular cube, somewhere in 9-dimensional space. We can imagine it as
a pleasant little room with a rug, a cozy couch, and maybe a few paintings on the walls.
On a side table there sits a Tower of Hanoi puzzle with 3 disks on each of the 3 spindles
and a piece if paper with a Tic Tac Toe game scribbled on it.
Each wall in this room includes a door leading to an adjacent cube. For added fun, there is
a trap-door under the rug leading to lower room. And if we pull on the little chain dangling
from the ceiling a wooden staircase folds down allowing us to climb up to the attic.
Everything seems perfectly normal at first. As we move from room to room, we find small differences:
a red couch instead of a blue one, an odd piece of statuary, someone else's hat tossed casually
on a small side-table. But each room has the same monotonous cubical shape with the same six exits.
The sheer size of the mansion we find ourselves in soon becomes oppressive. There seems to be no
beginning or end to these odd little rooms. (There are, after all, over five thousand of them!)
And then the deja vu sets in. We stride confidently into a second room, then straight into a third,
on into a fourth, and then, without ever changing direction, straight into the first room we left from.
This becomes even more disconcerting when we climb into the attic, climb into the attic's attic,
the attic's attic's attic, and then, puffing and panting, climb one more set of stairs, push aside a rug,
and find... the very room we started from.
This is when the panic sets in.
Hollering for help only makes matters worse since we can hear our own yell coming at us from all six directions.
So after a while we calm down and take a closer look at the room we are in.
What's this? A tiny plaque mounted on the wall with the number 16152. How odd. Upon further inspection
we find three more plaques mounted neatly in the center of each wall, each with a different number.
It turns out that even the floor has its own number, and the ceiling too.
We get down on our hands and knees and peer into a corner. Sure enough, there is a very tiny number painted
there, right on the baseboard. And following the edge up where the two walls meet we find yet another number
printed sideways. Every edge has its own number, though the ones along
the ceiling are quite difficult to read.
Our invesigation is complete when we stand in the exact center of the room and examine the little chain hanging down
from the ceiling. On the end of that chain is a plum-sized crystalline orb, which seems to shimmer inside with
shifting squares. And inscribed on the surface of that orb, with exquisite delicacy, we find
the words: "The Nine-Space Hotel, Room 16179".
Now it the time to whip out our handy Cartan's Triangle. Left yours in your other coat pocket?
Not to worry - you can always make another one. All you need is a pencil, a paper, and the ability to add.
Using the insight gained by your Cartan's Triangle, you begin by converting each number from base ten to base three.
16179 becomes X1101X0X0. This tells us that this particular room extends along the second, fourth, and ninth dimensions,
even though the room still seems to our 3-D eyes to occupy the usual width, length, and height.
Now, at least, we know where we are.
We can also find all the components of our current cube. The three X's in the cubes address represent extensions
in space: height (from bottom to top), length (from behind to in front), and width (from left to right).
If we change the first X to a 0 we are essentially collapsing the height of the room and reducing it to the bottom
face of the cube, that is, the floor. Changing it to a 1 instead produces the ceiling. The same can be done
to the other two X's as well, as follows:
Floor
01101X0X0
3057
Ceiling
11101X0X0
9618
Rear Wall
X110100X0
16125
Front Wall
X110110X0
16152
Left Wall
X1101X000
16173
Right Wall
X1101X010
16176
The 12 edge numbers are just as easy to find.
To find the edge marking the intersection of two walls, compare each digit of their two addresses.
If a digit is the same, retain it. If the digit is an X in one address and a 0 or 1 in the other,
choose the non-X digit. The resulting address will have only one X. For example:
Front Wall
X1101
10
X0
16152
Left Wall
X1101
X0
00
16173
Front Left Edge
X1101
10
00
16146
That initial X tells us that this edge extends in only one dimension,
our Z dimension, from floor to ceiling.
If we change that X to a 0, we get the corner at the bottom of that edge;
if we change it to a 1, we get the corner at the top.
The resulting 8 corners give us the exact boundaries of our cube in 9 space:
Bottom Rear Left Corner
(0,1,1,0,1,0,0,0,0)
2997
Top Rear Left Corner
(1,1,1,0,1,0,0,0,0)
9558
Bottom Front Left Corner
(0,1,1,0,1,1,0,0,0)
3024
Top Front Left Corner
(1,1,1,0,1,1,0,0,0)
9585
Bottom Rear Right Corner
(0,1,1,0,1,0,0,1,0)
3000
Top Rear Right Corner
(1,1,1,0,1,0,0,1,0)
9561
Bottom Front Right Corner
(0,1,1,0,1,1,0,1,0)
3027
Top Front Right Corner
(1,1,1,0,1,1,0,1,0)
9588
Thanks to Cartan's Triangle, we now know EXACTLY where we are.
But there is even more information concealed in the address.
The trinary address can also tell us what KIND of cube we are in.
We saw earlier that there were three different kind of squares in an ordinary cube, each occuring twice.
In a 9-cube there are no less than 84 (9 choose 3) different kinds of cubes,
each occurring 64 times. This is where the total number of cubes in a 9-cube comes from:
84 times 64 is 5376. We know that our particular cube is of the 9-4-2 type, since that's where the X's are.
This may explain the changes in decor from room to room. Perhaps all 9-4-2 cubes have red couches.
We can also find our position in relation to the other 63 9-4-2 cubes. If we collapse dimensions
2, 4, and 9, the result will be a 6-dimensional space, with all the 9-4-2 cubes reduced to points
occupying the 64 corners of a 6-cube. When we simulate this collapse by removing the three X's
in our address, we are left with a 6-bit number: 110100 or 52. This means that our cube is on the
53rd corner of the 6-cube (since the numbering starts with 0), coordinates (1,1,0,1,0,0).
So now we know that our cube is the 53rd instance of a type 9-4-2 cube. And we also know the
addresses of all the other 9-4-2 type cubes. We are finally ready to start traveling to other rooms.
It is at just this moment that we see a beautiful little girl wander into the room directly
in front of us. She smiles mischievously at us but instead of walking through the door she reaches up to
touch something next to the door. We have something on our side of the wall as well, a
funny little black knob surrounded by seven colored dots.
We had noticed these before next to all the doors, but had assumed they were merely some odd
kind of decoration. As the little girl touches this thing, it becomes clear that it is a
switch of some kind, currently pointing to the green dot. And the knob on her side of the
wall must be attached to ours, because as she twists her knob from green to blue, our knob
moves as well.
As the knob switches from green to blue a distinct clicking sound is heard. At that
precise moment something astonishing happens. The room we were just looking at changes
in an instant. Suddenly it's a different room altogether as if we've flipped from
one television channel to another.
And the little girl? She has vanished altogether. But she must be somewhere because the
knob is still turning. Click - it points to teal and once again the room we see through the
door changes, yet there is still no sign of the little girl.
The knob is still turning. Click. Click. Click.
The knob jumps to the magenta dot, then to the yellow dot, then to the brown dot. Three more empty
rooms come and go. And then there is a final click to the seventh dot, which is colored bright red.
And this time something even stranger happens. This time our door disappears!
The knob is still there and has finally stopped turning, but the place which a moment earlier had
held a wide archway with rich wooden trim was now a solid plaster wall with no trace
of an outline or anything to suggest a doorway had ever been there. In the distance
we can hear someone giggling and the sound of running feet.
Oh dear. This 9-cube is even spookier than we thought. In order to understand what just happened,
it might be better to retreat for a moment to a simpler world, the relatively cozy world of the
4-cube, or "tesseract".
A Crooked House
The tesseract provides a gentle introduction to the art of moving from cube to cube
within a higher dimensional space. One of the most well-known and entertaining descriptions
of the insides of a tesseract comes from Robert Heinlein's 1940 short story "-And He Built
A Crooked House", available in Clifton Fadiman's excellent anthology
Fantasia Mathematica.
In this story an eccentric architect builds a home in the shape of an unfolded tesseract.
When an earthquake causes the tesseract to fold back into its natural form, the architect's
client and his wife, who are seeing the home for the first time, begin to have some very
odd experiences.
In its unfolded state, this tesseract looks like an upside-down crucifix with sidebars
extending in four directions. The garage and entrance is on the first floor, with stairs
extending up to a central room which opens onto a kitchen, drawing room, lounge, and dining room.
Above this is the master bedroom, and above that a study at the top of the tower.
The floor plan of the house looks something like this:
I have assigned trinary index numbers to each room. If you study these numbers you will
begin to get an idea of how the numbers change when you move from one room to another.
The digits are in wzyx order, so the RIGHTmost position is the x direction (0=left, 1=right),
the next digit is the y direction (0=back, 1=front), then z (0=bottom, 1=top).
We'll save the weird w direction for later.
Every time you move from one room to another, one of the X's in the address changes places
with a 0 or 1. You can think of this is a two-step process. First reduce one of the three X's
to either a 0 or 1 - this is the equivalent of choosing which of the 6 walls you want to move
towards. Then turn the other non-X digit into an X to step into the next room.
When you climb the stairs from the entrance to the center room, the X in the
z position changes to a 0 and a new X appears in the w position. From the center, moving back
or forth involves swapping the X in the y position, and moving left or right involves
swapping the X in the x position. Another shift of the z position X takes us up
to the third floor and another shift at the same position takes us to the top of the tower.
The easiest way to visualize a tesseract is as a small central cube with six pyramid stubs
extending from each face and merging to form the six faces of a larger outer cube. That's
how the central projection looks in 3-space; in 4-space the pyramid stubs are actually cubes and all eight cubes,
even the "inner" and "outer" cubes, are the same size.
Here's where the weirdness begins. In the folded up tesseract, the master bedroom corresponds
to that small inner cube. The center room is the pyramid extending from its bottom face
and the kitchen, lounge, drawing room, and dining room are the north, south, east and west pyramids.
The study is the final pyramid on the bedroom's upper face. And those six pyramids all
converge to form the six faces of the inside-out entrance cube!
So if you go up from the 4th floor, you'll wind up on the 1st floor. In Heinlein's story,
a window in the west wall of the lounge looks into the dining room from the south;
this is because the west lounge wall and the south dining room wall are actually the
same wall in 4-space.
The floor of the kitchen forms the north wall of the entrance,
its ceiling forms the north wall of the bedroom,
and a window in the kitchen's north wall would look into the north wall of the study!
It's worth noting, by the way, that the tesseract is more than just these 8 cubes.
The 8 cubes form the surface of the tesseract, just as 6 squares form the surface of a cube.
But just as there is a whole space inside the surface of a cube, so there is
a whole space inside the 8 cubes of a tesseract, a vaster and more mysterious space in which
none of those 8 cubes partake. But let's get back to our cubes.
Every room address has 3 X's. By changing any one of these X's to a 0 or a 1, you can
move in one of six directions, corresponding to the six faces of the cube. If we follow
this simple logic, we can construct a complete chart or graph of all possible movements, with
rooms as nodes and lines drawn whenever two rooms share a common wall.
The resulting graph reveals the stunning symmetry of the tesseract:
Heinlein's story includes a great chase scene as the architect pursues a mysterious stranger
in a straight line through the drawing room, kitchen, dining room, and lounge without ever
getting any closer to his prey. He then realizes that he has been chasing himself.
This path appears in the above graph as a rectangle, as do other straight line circuits
like the climb from entrance to center to bedroom to study and back to entrance.
Notice that in this diagram, each square represents a cube-shaped room, and each line
represents a wall. Each room node has exactly six lines projecting from it, corresponding
perfectly to the six square faces of a cube, for a total of 24 lines in all
(8 rooms times 6 divided by 2, since otherwise each line would be counted twice).
Each line connects two nodes, just as each wall connects two adjoining rooms.
This is exactly as one would expect and matches exactly with line 4 of Cartan's Triangle.
As predicted, a tesseract has 24 squares (walls) and 8 cubes (rooms). But if you think
that you now understand the way walls behave in higher spaces, you will be in for a shock.
Patterns that seem clear and well-understood in one dimension are often utterly undone
in the next. The journey from three dimensions to four was strange enough, but the journey
to the fifth dimension is stranger still.
The 5-Cube
Let's have another look at Cartan's Triangle. This time, pay special attention to
the diagonals indicating the number of walls and rooms in each higher space:
Do you see what is happening? The number of rooms are increasing at a faster rate
than the number of walls. By the time we get to the ninth dimension, there are more
rooms than there are walls!
How can this be? The answer is that in dimensions higher than four, each wall is shared
by more than two rooms. In the fifth dimension, each wall is shared by three rooms.
A doorway in any wall connects rooms A, B, and C. So when you leave room A, you might
walk into room B. But then again, you might walk into room C.
Actually, this same phenomenon did occur in Heinlein's story. The reason it happened in his
tesseract and not ours is that we are treating our n-cubes as closed systems. In Heinlein's story,
his tesseract also had connections to normal 3-space. In fact, he posited a space that curves
just like a piece of paper might wrap around a cube. Flatlanders living on that piece of paper
might enter the cube in one part of their "space", travel a short distance inside the cube, and
exit on a piece of the paper that seems far away to them. In a similar fashion, one window of
Heinlein's house looks out over Los Angeles while another looked straight down into Manhattan.
Heinlein handled this awkward situation by supposing that the choice about whether you move
from A to B or A to C is controlled by your subconscious mind and appears essentially random.
I find this unruly, so I suggest instead that we install little knobs next to each door
so that we can consciously switch from one option to the other, like the channels of a TV set.
I'd also prefer to do this in a symmetrical way, so that whenever room A is connected to room B,
room B is also connected to room A.
My first attempt to do this was not entirely successful. I thought that since a given
wall in room A could lead to either room B or room C, I would need a knob with two possible channel
positions. The "black" channel would lead to B and the "red" channel would lead to C.
The knobs controlling any given wall would always stay in synch.
I tried drawing a diagram of room
connections just as we did for the tesseract. I knew going in that the diagram would be a
tad more complicated: a 5-cube contains 40 rooms instead of 8, and since each wall now has a
knob that lets you switch between two possible destinations, there are 12 lines projecting
from each room instead of 6, for a total of 240 (40 * 12 / 2) lines. I found a symmetrical
way of assigning two channels to each wall so that half the lines were black and half
were red. The result looked like this (click to see the full diagram):
What a mess! To get a better grasp of the underlying relationships, I next tried drawing
only the black lines. This was much better. If you click this diagram you will see that the
rooms are neatly partitioned into two sets of ten rooms (with 00 or 11 in their addresses)
and one set of twenty rooms (with 01 or 10 in their addresses). In the larger partition, I
turned the lines green whenever they connected a 10 to a 01:
The diagram reveals that there are ten types of room inside the 5-cube, each occurring 4 times
(00, 01, 10, and 11) for a total of 40 rooms. But there was still something that didn't add up.
Recall that I said there were 240 connecting lines in the 5-cube diagram, half red and half black.
If each wall leads from either A to B (black) or A to C (red), it seems like I should have 120 (240 / 2) walls.
But if you look at Cartan's triangle, you will see that there are only 80 walls in a 5-cube.
What did I do wrong?
The Right Colors
The number of channels per cube wall in an N-cube is N-3 (1 in a tesseract,
2 in a 5-cube, 3 in a 6-cube, etc.) And so the total number of cube-to-cube connecting lines
in an N-cube diagram is simply the total number of cubes times the number of walls per cube (6)
times the number of channels per wall (N-3) divided by 2 (so we don't count the same line twice).
According to Cartan's triangle, this number is increasing faster than the actual number of walls
connecting the cubes. If we divide the number of connecting lines by the number of walls,
we find a familiar sequence:
N
Connections
Walls
C / W
4
24
24
1
5
240
80
3
6
1440
240
6
7
6720
672
10
8
26880
1792
15
9
96768
4608
21
This sequence (1, 3, 6, 10, 15, 21, ...) is simply a list of the triangular numbers
(1, 1+2, 1+2+3, etc.). What did this mean? Something painfully obvious, no doubt.
I began using actual trinary addresses of particular walls and adjoining
rooms, just as I did with room number 16179 above, to see exactly how all these things
are connected in different higher dimensional spaces.
I drew diagrams which showed exactly which rooms connected to which through a given wall.
These diagrams all turned out to be what is known as "complete" or "universal" graphs,
that is, graphs in which each node is connected to every other node. And that's
when the light bulb finally went on. The number of edges in a complete graph of
order N is the Nth triangular number. When N=3 you have a triangle of 3 nodes and 3 edges.
When N=4 you have 4 nodes and 6 edges. The reason the number of edges increases faster
than the number of nodes is because the edges represent every possible combination of two nodes
(N choose 2).
Here is the mistake I made. When I added channel-changing knobs to every wall of the 5-cube
I assumed that each wall was doing double duty, connecting room 1 to either room 2 or room 3
(and 2 back to 1 and 3 back to 1) depending on the channel selected. But I forgot that this
same wall ALSO connects 2 to 3.
So when you are standing in front of a doorway in room 1 of a 5-cube, not only does that door
sometimes lead to room 2 and sometimes to room 3, sometimes there may be people that you can't
even see using that same door to walk between room 2 and room 3! In order to make this
work, my channel knobs would need more than just two colors.
In my diagrams, I used the color of an edge connecting two nodes (rooms) to represent
the channel setting on the wall (from either side) when those two rooms are connected.
In choosing colors for the edges, I had to make sure that no channel setting ever led
to more than one room. The minimum number of colors needed to accomplish this is
a value known in graph theory as the chromatic index.
When n is even, the chromatic index of the complete graph on n nodes is n - 1
But when n is odd, the chromatic index of the complete graph on n nodes is n
For any 3-cube in an N-cube, the number of channel positions for each wall (the number of edges per node)
is N-3, and the number of nodes in the complete graph is N-2. The above rule about the chromatic index
tells us that the number of colors to choose from will be N-3 if N is even or N-2 if N is odd.
For the tesseract (4-cube) things were deceptively easy: 1 channel and 1 color, which meant that
channel selectors were not even needed. For the 5-cube, we need 2 channels per wall as expected,
but the number of colors needed is 3.
The knob in room 1 will have two valid destination settings,
say red and green. The knob in room 2 will also have two destination settings, but instead it will use
red and blue. And the knob in room 3 will allow travellers to choose from either blue or green.
That's two channels per wall, but 3 colors (red, blue, and green). And since the three channel knobs
in the three rooms are all controlling the same wall, each knob has to show all 3 colors.
So if someone in room 2 changes the knob from red to blue, the knobs will change to blue in all
three rooms.
This is fine for the people in rooms 2 and 3, who both have blue as one of their
two valid destination settings. But room 1 only has red and green as valid destinations,
so what happens when the knob in room 1 changes to blue? In that case, the door will
disappear altogether, since the wall no longer connects room 1 to anywhere.
This bizarre outcome only happens in odd-numbered N-cubes.
The same rules apply for higher dimensions. For the 6-cube we need 3 channels per wall but can use
the same 3 colors we used in the 5-cube, so every color is a valid destination and there
are no disappearing doors. In the 9-cube, which is where our story started,
each door has a knob with 6 valid destination settings, but since 9 is an odd number, 7 colors are
required and each knob will have a seventh color that makes its door go away.
The only remaining problem is finding the correct arrangement of colors for each value of N.
Fortunately this is not too hard to do. The following chart sums up the colors needed for each
N-cube. In each case, the diagram shows how the rooms sharing each wall are connected:
For each chart I have chosen colors that are symmetrical (they reflect across the diagonal)
and unique (no color appears more than once in any row or column).
This ensures that if the red channel on a given wall connects room 1 to room 2, red will also connect
room 2 to room 1 and there will be no other red channels on that wall.
The colors can be assigned in a mechanical (but hard to describe) way that can be extended
to any number of higher dimensions.
Now we finally have enough information to understand how that little girl in the Nine-Space Hotel
made our door disappear. Room 16179 has the same channel knob as room 5 in the 9-cube
diagram above. When we first encountered her, the little girl was standing in an adjacent
room with a type=6 knob. We could see each other, which meant that wall must have been
set to the green channel.
If you look at row 6 column 5 of the color chart (from the girl's point of view) or
row 5 column 6 (from our point of view) you can see what happened each time she flipped
the channel. When she flipped from green to blue, her door disappeared. But our door
switched from room 6 to an empty room 7 - which is why the girl disappeared from our point of view.
She then flipped to the teal channel, which meant that she now was looking at room 7 and
we were looking at room 1. Switching to magenta caused her to see room 1 and us to see room 2.
She switched to yellow then brown. Now she was seeing 3 and we were seeing 4. Her final
switch was to the red channel, which meant she was seeing room 4. But room 5 does not have
a red destination, so now it was our turn for our door to disappear. The girl was free to run
giggling into room 4 while we stared amazed at the place where our door has just been.
The color table also allows us to finally draw the correct diagram for the 5-cube, using
three colors instead of two. Click the following close-up to see the full diagram in all
its glory, with red, blue and green connecting lines instead of the black and red lines
of my original attempt.
The three rooms shown in this close-up, XX11X, XX1X1, and XXX11 all happen to share the same
wall. The algorithm I used to assign edge colors is to first remove the two Xs in common to
any two adjoining cubes, then, depending whether the remaining X is in the 1st, 2nd, or 3rd position,
I assign it a 1, 2, or 3 and refer to the color chart to determine the edge color.
As you can see in this case, a red line connects XXX11 (room type 1) to XX1X1 (room type 2),
a blue line connects XX1X1 to XX11X (room type 3), and a green line connects XX11X back
to XXX11. This exactly matches the 3-room triangle diagram for the 5-cube above, and works
perfectly for every other set of 3 rooms sharing a common wall as well.
The resulting diagram is just as messy as it was before. But, as before, we can gain a clearer
picture of the 40 cubes depicted by restricting the edges to those of just one color.
I did this for red, green, and blue colors and combined them into the following diagram
(click to see it full-size).
As you can see, red and blue links both partition the rooms into six distinct sub-graphs, while
green keeps everything fully connected. All three cases, though, only include 36 of the 40 cubes.
There are 4 cubes with no red connections at all, 4 (different) cubes with no green connections,
and 4 more with no blue connections.
Using this technique and the trinary index numbers derived from Cartan's triangle, we have
created a complete map of the cubes in a 5-cube, which could be printed out and used to plot
a path from any cube to any other. The same technique could be used to create just such
a map of the Nine-Space Hotel, though with 5,376 rooms and 96,768 connecting lines of 7 colors,
we would need a larger sheet of paper.
Tic Tac Toe
By showing you not just the finished results, but also my own mistakes and contortions
along the way, I hope I have conveyed what it is like to explore higher spaces
and how useful Cartan's triangle can be. Now, as a finale, I'd like to apply trinary
indexing one more time to make a simple game more interesting.
There are a number of possible candidates for which trinary indexing is a natural fit.
The first game that occurred to me was the Tower of Hanoi.
The three spindles in this game can represent a 0, 1, and X. The disks placed on
these spindles are digits which vary according to what spindle they reside on,
with the digits ordered according to the size of the disk.
In this way, every possible position in the Tower of Hanoi puzzle can be assigned an
N-digit trinary number where N is the number of disks. It is then possible to
correlate every position of the puzzle with a particular component of an N-cube.
Each move becomes the change of a single digit, which corresponds to moving a
corner to a different location, or changing that corner to an edge, or changing
that edge to a square.
If the starting spindle is "0" and the ending spindle is "1", the entire puzzle
becomes one of finding a path from the origin (0,0,...,0) to the opposite corner
(1,1,...,1). Whenever you are forced to use the third spindle, the component will
increase in dimension, from a point, to an edge, to a square and so forth - and
removing disks from the third spindle will have the reverse effect.
The solution to the puzzle, then, becomes an animation showing a point
growing and hopping and shrinking and hopping and growing again, until it
finally shrinks back down to a point. Looking at the puzzle in this
way might help you see how to solve it, and adds an extra layer of meaning
to every step along the way.
But on further consideration, I thought of an even simpler, even
more familiar game that I could corrupt with my higher-dimensional hijinks:
the venerable game of Tic Tac Toe.
Tic Tac Toe employs the same three-by-three grid used by the Starmaze.
Each of its nine cells can be in one of three possible states:
X, O, or empty. If we assign X cells the digit X, O cells the digit 0,
and empty cells the digit 1, then each position in a Tic Tac Toe game
can be reduced to a nine-digit trinary number.
Our next decision is what order to place the nine cells in.
We can number the cells any way we want, but as is my custom,
I prefer to use the Lo Shu.
This ancient magic square adds an extra element of charm.
So we'll place the rightmost digit (the x dimension) in the
cell below the center, the next y dimension digit in the top
right corner, etc.
Tic Tac Toe starts with an empty grid. This corresponds to a
corner with coordinates (1,1,1,1,1,1,1,1,1) or 9841 in base 10,
a point as far away from the origin as possible.
Every move that player O makes (let's call him Oscar) will change one of these 1s to a 0.
This will simply move the referenced component one step closer to the origin.
Every move that player X makes (let's call him Xavier) will change one of these 1s to an X.
This will change the corner into an edge, the edge into a square,
and so forth.
So Oscar and Xavier each seek to change the component in a different
way. Oscar always tries to move the component closer to the origin.
Xavier always tries to raise the component to a higher dimension.
The goal of the game is to reach a winning component that corresponds
to three Os in a row for Oscar or three Xs in a row for Xavier.
This component will be either a square (which can only happen if Oscar goes
first and wins before Xavier can place his third X),
a cube, a tesseract, or a 5-cube (which can only happen if Xavier goes
first and wins by filling the last empty cell).
To see how this works, let's follow a typical game, step by step.
We'll let Oscar go first.
First move. Oscar places an O in cell 3. The component has now shifted from
9841 to 9832 or 111111011. This is a corner one step closer to the origin in
the z dimension.
Second move. Xavier responds by placing an X in cell 4. The component number
is now 9859 or 11111X011. This is an edge along the w dimension extending from
(1,1,1,1,1,0,0,1,1) to (1,1,1,1,1,1,0,1,1).
Third move. Oscar places an O in cell 6, resulting in component 9616 or 11101X011.
This effectively moves the edge one step closer to the origin along the 6th dimension.
Fourth move. Xavier extends the edge to a square by placing an X in cell 2.
The component number is now 9619 or 11101X0X1. The square extends across the w
and y dimensions.
Fifth move. Oscar, who is perhaps not the brightest bulb on the tree,
responds by placing an O in cell 1. This subtracts 1 from the component number,
changing it to 9618 or 11101X0X0. The square still extends across the w and y
dimensions, but now occupies a position one step closer to the origin.
Sixth move. Xavier wins the game by placing an X in cell 9, thereby
extending the square along the 9th dimension until it becomes a cube.
The winning cube number is 16179 or X1101X0X0.
Sound familiar? This is the very room we began our journey from in the Nine-Space Hotel.
To anyone who made it all the way to the end of this very long, very strange page,
I salute you. I hope you will join me in the exploration of higher space. And if you
ever lose your way, I hope Cartan's Triangle will help you find your way home. |
2 + 2 = We Must Raise Taxes Because White People Are Bad
Since moonbattery is a totalitarian ideology, nothing escapes from its poisonous lies — not even mathematics. Teachers unsure how to pass off indoctrination in race-based Marxism (a.k.a. "social justice") as a math class can find resources to assist them at Radical Math:
There are at least two related ideas behind "Social Justice Math". The first is that you can use mathematics to teach and learn about issues of social and economic justice. The second is that you can learn math through the study of social justice issues….
A subspecies of Social Justice Math is "Ethnomathematics," defined as:
The study and celebration of mathematical practices from various countries and cultures from both historical and contemporary perspectives, including: symbolic systems, spatial designs, games and puzzles, calculation methods, measurement in time and space, architecture and design, problem solving, etc.
With all this important material to cover, it's no surprise if teachers don't get around to boring stuff like multiplication tablesComparing how money spent on military operations could be used to support other important causes (ex: if a bomb costs $10 million and a it costs $10,000 to provide health care for an entire family for a year, how many families could get health care for the cost of this bomb).
Here's how geometry can be combined with "environmental justice":
Determine the density of toxic waste facilities, factories, dumps, etc, in the neighborhood.
By now it should be obvious why moonbats cannot be left in control of education. This means prying it out of the fist of Big Government and the unions that bankroll Democrat politicians.
Eventually progressive math will dispense with numbers altogether, replacing them with feelings. |
When a puzzling disease devastates her beloved father, mathematics prodigy Mona Gray isolates herself and turns to the world of numbers for comfort. |
If you own an Apple phone, it's quite likely that you've spent hours (maybe not all together) asking Siri challenging questions that 'she' probably won't have an answer to. Or just trying out different answers that Siri would give for the heck of it.
There would also have been times when your mathematically challenged self would have taken the easy way out and transferred that bugging Math problem on to Siri. Well, that's what this guy did, kind of. Except that what followed is simply amazing.
In this video, that has gone viral since it was shared on Facebook on January 21, a guy asks Siri: "Siri, what is one trillion to the 10th power?" As Siri calculates the answer and responds, two girls sitting in the background groove to with response in the most fascinating way. (Hint: What do you think happens when Math and music combine?) |
Mathematical Physics
The Prime Number is Connected to the Quantum-Mechanical Basic Equation.
The prime number is connected to the quantum-mechanical basic equation.
Mathematician Euler discovered a prime number and a connection with π (Japanese yen) for the first time. The left side of a go board of the following equation that I had only with a prime number is equal to π2/6. I transformed the following equation and had the equation of the area of Japanese yen. Then it became the equation that the prime number equation (zeta function) of the oiler assumed a prime number a radius. Here, a prime number and the correlation with what I set were provided on the top of the pulsation wave pattern of the figure of prime number, physics fusion as if I showed it to a figure of of the Lehman expectation proof that I contributed from an association between Schrodinger equation and circular motion of the elementary particle pulsation principle correlation chart in the online posting before last time. The prime number has a quantum-mechanical basic equation, the connection that are close to Schrodinger equation |
Keep calm and use correct scientific methodology.
Prime Time
In the modern world, pure mathematicians unfortunately do not make that much money because they are perceived as 'useless', working purely in the strive for a beautiful proof or an elegant derivation. I feel as if many areas of mathematics have gradually pushed their way into the realm of art, since they exist purely for aesthetics or intellectual curiosity. Of course, mathematicians do maths not for the money, but for the beauty. However, if you're looking for a cheeky way to make a bit of quick cash in mathematics, there is one way…
At the start of the new millennium, the Clay Mathematics Institute in Peterborough set out a series of seven mathematical problems which had challenged mathematicians for generations. They were considered to be the most 'important classic questions that have resisted solution over the years'. All of the questions are very theoretical and abstract in nature, which contributes to their difficulty because it requires an extremely diligent pure mathematician who can 'think outside of the box' to be able to solve them. For each problem, the first person to provide a solution to it would be awarded $1,000,000 as a reward. Now if that's not an attractive incentive, then I don't know what is. But I kinda lied earlier about it being easy. It is most definitely not easy. It is very, very hard.
So far, only one of the seven problems have been conquered. This was the Poincaré Conjecture, which states that "Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere". In more simpler terms, "Any finite 3-dimensional space, which doesn't have any 'holes' in it, can be continuously deformed into a 3-sphere". This conjecture, or hypothesis, was proven by Grigori Perelman in 2003, and after review, was confirmed in 2006, which lead to him being offered a Fields Medal, the pinnacle of mathematical achievement equivalent to a Nobel Prize. Later, in 2010, Perelman was awarded the Millennium Prize for solving the problem and was offered the million dollar prize. On both occasions, he declined the offer. He believed that he did not deserve to be given an award for solving a problem that countless others had contributed towards. The Poincaré Conjecture had been worked on by generations before him, but he was just lucky enough to have been the one to finish. "I'm not interested in money or fame; I don't want to be on display like an animal in a zoo."
Out of the seven Millennium Problems, the Riemann Hypothesis is arguably the most important for number theory. It seeks to understand the most fundamental objects in mathematics – prime numbers. Prime numbers are the very atoms of arithmetic. Any positive integer can be made by multiplying together two or more prime numbers. Despite how fundamental prime numbers are for the basis of number theory and mathematics, we've never managed to find harmony in their weird, disjointed music. Every new prime number we discover seems to appear randomly, as if Nature chose it by flipping a coin. This random and unpredictable nature of prime numbers makes it the prime pursuit (see what I did there?) for mathematicians who have a desire to find order in numbers.
What the Riemann Hypothesis claims to offer is an incredibly accurate approximation for the number of primes under a certain integer, which would allow us to map the distribution of prime number a lot more efficiently. The proof of this Hypothesis is so important because so many other theorems rely on it. For the past 150 years, countless theorems have needed to say "if the Riemann hypothesis is true…", so being able to prove it would immediately validate the consequences in these theorems as true. To go into any depth on the Riemann Hypothesis would prolong this post far too much, so I will leave it until another day.
You may be thinking, "What's so special about these primes anyway?"
The fact that any number can be factorised into a bunch of prime numbers makes primes vitally important to modern communications. Most cryptography used in modern computers works by using the prime factors of large numbers. The large number used to encrypt the data can be publicly known, but in order to decrypt it again, only the prime factors of that large number can be used. Since prime numbers stay ever elusive, we do not have an efficient way to find the prime factors of very large numbers. For a hacker to compute the factors manually, it would take so much time that we say that it is impossible. A modern super-computer could chew on a 256-bit factorisation problem for longer than the current age of the universe, and still not get the answer. It is possible that as we develop new mathematical strategies or advanced hardware technology like quantum computers, we are able to prime factorise large numbers much faster, which would effectively undermine and destroy modern encryption.
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10 thoughts on "Prime Time"
It's rare that you find pure scientists (and I firmly believe that mathematics is a science) who study science for the money. They wake up wanting to learn, wanting to explore, wanting to become more broad in their knowledge of the world, everything within, and everything beyond. We're an odd breed that's often misunderstood, and most of us are fine with that. We'll just go back to our hobbies, like Dungeons and Dragons, Raspberry Pi, guitars, developing biceps…
I completely agree with what you've said. Scientists are always asking questions, always wanting to understand more about this universe we live in, driven by nothing more than pure curiosity. Hope you enjoyed the post! Thanks for the xkcd – it fits here perfectly!
~Harvey
As a practitioner of one of the lesser sciences in Jonny's xkcd I have always suspected that mathematicians see the world quite differently than the rest of us. They see a world of the subtending form(s) of the universe rather than this simple object-and-color-world that most of us live in. It is a world of pure beauty, as you say, with annoying bits, like a jigsaw puzzle of impossible shapes with even more impossible shapes that are supposed to fit in, but no one knows where. Yet.
Okay, I had to Google three terms to make sense of this. (Language Arts major.) If the Riemann Hypothesis is discovered/proved and a method of patterning is revealed for the occurrence of prime numbers, then current encryption methods will be cracked? Did I get that right? So, what would replace the current system…any thoughts?
Sorry, I may not have explained it very clearly. Even though the Riemann Hypothesis has not been as yet proved, it has basically been assumed to be true by all mathematicians for a long time. Being able to actually prove the Riemann Hypothesis can be seen as purely for mathematical perfection and intellectual curiosity. It brings us one step closer to really understanding the mathematical oddities that are the primes. In terms of encryption, this will have no impact. I mentioned in the final paragraph that the growth of quantum computing could spell disaster – but with new technology comes new systems and new solutions. We'll always find a way.
~Harvey
I'm certainly no mathematician by trade, but in my spaghetti-brain way of thinking, could the confusion of prime numbers be simplified using a different numbering system? Something radically different from base-ten?
Over the past few centuries, this is essentially what mathematicians have been trying to do. By attempting different approaches, a new perspective on the primes can be found. One example of this is the use of pure geometry instead of number theory. This interpretation of the Riemann Hypothesis shows the prime numbers as 'zero points' on a graph, and as long as we can prove that all the primes lie on a single line, then the hypothesis will have been proven. I will aim to go into more depth on the topic in future posts, so stay tuned! Thanks for the great question!
~Harvey
Harvey-I am honored by your response. This is fascinating work you are doing, and you obviously have the mind-set to persist! My area of interest centers around how our brains process and create–sometimes having to unlearn entire systems of thought-organization in order to re-assemble our conceptualizations. In my field (education), the challenge existing beyond those potentially newly-assembled concepts is then trying to communicate those concepts–especially difficult if words for those concepts don't currently exist. I am cheering you on and looking forward to your future posts. |
mathematical physicist
Russian translation: Математический физик (матфизик |
Mathematics + Unconditional Love = ?
The Journal for Urban Mathematics Education (JUME) is a peer-reviewed, open-access journal, that recently published an article I wrote that explored the question, "What does teaching mathematics as agape, or unconditional love, look like?" Think about it…if a teacher started with unconditional love for students, as defined by agape, what would it look like to teach mathematics? This article was an attempt at answering that question from a theoretical perspective. Here is a link to the article and and here is how to cite it |
NEWS AND VIEWS
Local News
Five scavenger hunt forms were handed in on November 9th. Overall winner and first place in the category of students who have not taken Calculus was freshman Mehdi Bandali. Second place in this category was Jennifer Findley. The first place math major winner was Asmee Elmkhanter and the second place was taken by John Apodaca. First place winners took home a $30 gift certificate from the Auraria Book Store and the second place winners a $10 gift certificate from Starbucks.
By popular demand, the undergraduate committee together with the Career Services Center is putting together a Career Day tentatively scheduled for early February. We will invite a panel of local employers to discuss what kinds of opportunities they have for math graduates (B.S. in Mathematics) and what they are looking for in successful candidates. Stay tuned for more details.
Math News
"Numbers are Male, Said Pythagoras, and the Idea Persists," by Margaret Wertheim. The New York Times, 3 October 2006.
A recent report by the National Academy of Sciences points to "widespread bias against women in science and engineering," Margaret Wertheim writes in a recent New York Times article. However, she notes, "there is reason to believe that when it comes to the mathematically intensive sciences like physics and astronomy, it is not just bureaucracies that stand in the way." Rather, she suggests, "the problem goes back to the ancient Greeks, particularly to Pythagoras."
After describing some of Pythagoras' great ideas, including "all is number," Wertheim describes how, like many Greek cults, the religion of Pythagoreanism was dualistic: reality consisted of the mind/spirit/transcendent realm versus the body/matter/earth realm. For Pythagoreans among others, the former was associated with maleness---and doing mathematics---and the latter with femaleness. In the 12th century, universities, founded to educate the clergy, by definition excluded women. In addition, "the Pythagorean association of mathematics with transcendence was easily imported into a Christian context... Thus, from the start, women were excluded from this academic field and its associated sciences."
Many women entering the sciences in the 1970s continue to be "stunned at how slow change has been," notes Wertheim. One of these is Gail Hanson, distinguished professor of physics at the University of California, Riverside, and winner of the W.K.H. Panofsky Prize in physics. She is one of the subjects of Out of the Shadows, a recent book about the "lives and work of 40 outstanding female physicists of the past century." Referring to her own as well as other female physicists' experience of bias, Hanson adds, "And when you get prizes, you're often treated even worse. Men can tolerate a woman in physics as long as she is in a subordinate position, but many cannot tolerate a woman above them."
--- Claudia Clark
While the weekly television series NUMB3RS may be increasing people's awareness of mathematics, it is not the only game in town. In this article, Ivars Peterson reports on two other sources of mathematical entertainment.
The first is The Math Factor, a weekly program of "newsy and entertaining math snippets" presented by mathematician Chaim Goodman-Strauss on University of Arkansas public radio station, KUAF. In a recent segment, Goodman-Strauss and program host Kyle Kellams discussed the Poincaré Conjecture with geometer Jeff Weeks. Other program topics have included "cardinality, encryption, paradoxes, puzzles, [and] rates of change," according to Goodman-Strauss. You can listen to podcasts of The Math Factor by searching for "math factor."
For a "different sort of edifying experience", Peterson suggests an Internet radio station that plays "all science and math songs, all the time!" The host is Greg Crowther of the University of Washington in Seattle. Check it out. Peterson notes that listeners must subscribe to the service, or can purchase individual tracks.
--- Claudia Clark
AMS Feature Column Current Column (archive)
These web essays from the American Mathematical Society are designed for those who have already discovered the joys of mathematics as well as for those who may be uncomfortable with mathematics. |
Wednesday, 1 June 2011
Intuitions Regarding Geometry Are Universal, Study Suggests
All human beings may have the ability to understand elementary geometry, independently of their culture or their level of education. This is the conclusion of a study carried out by CNRS, Inserm, CEA, the Collège de France, Harvard University and Paris Descartes, Paris-Sud 11 and Paris 8 universities. It was conducted on Amazonian Indians living in an isolated area, who had not studied geometry at school and whose language contains little geometric vocabulary. Their intuitive understanding of elementary geometric concepts was compared with that of populations who, on the contrary, had been taught geometry at school. The researchers were able to demonstrate that all human beings may have the ability of demonstrating geometric intuition. This ability may however only emerge from the age of 6-7 years. It could be innate or instead acquired at an early age when children become aware of the space that surrounds them. This work is published in the PNAS.
8 comments:
I tend to believe anything this group of authors say (in particular, I'm a huge fan of Dehaene), so if they say so, it must be true! But incidentally, they are also the authors of the famous 'Log or Linear?' paper, which argues that the same people (the Amazonian tribes that Pica studies) do NOT have a liner intuition of numbers, but rather a logarithmic one.
So Kant is partially vindicated: no innate intuition of numbers as linear, but probably an innate intuition of space that looks like 'our' geometry. Numbers would then be more of a 'cultural invention' than geometrical concepts.
Go read the stuff and we can talk about it! :) Here is a summary of the main results:
Abtract of 'Log or Linear?' (Science, 2008, same authors in different order)
The mapping of numbers onto space is fundamental to measurement and to mathematics. Is this mapping a cultural invention or a universal intuition shared by all humans regardless of culture and education? We probed number-space mappings in the Mundurucu, an Amazonian indigene group with a reduced numerical lexicon and little or no formal education. At all ages, the Mundurucu mapped symbolic and nonsymbolic numbers onto a logarithmic scale, whereas Western adults used linear mapping with small or symbolic numbers and logarithmic mapping when numbers were presented nonsymbolically under conditions that discouraged counting. This indicates that the mapping of numbers onto space is a universal intuition and that this initial intuition of number is logarithmic. The concept of a linear number line appears to be a cultural invention that fails to develop in the absence of formal education.
Hi Catarina, I've read Dehaene's book The Number Sense, and some of this sort of work. The problem is that they're making certain claims that don't make much mathematical sense. E.g.,
"The mapping of numbers onto space is fundamental to measurement and to mathematics."
1. What mapping? 2. What do "the" "mapping", "numbers" and "space" mean? 3. How do they define "mapping"? 4. Is there a unique mapping? If they say so, how do they prove this? What assumptions are needed? 5. By "numbers", do they mean the structure (R, <)? Some other structure? A proper initial segment of the positive reals R? Some metric structure on R? A topological structure? 6. Let R* be the positive reals. log_{10} is a function f: R* -> R. What does "logarithmic" mean in this context? 7. What is a "symbolic number"? (seems like a use/mention confusion). 8. By "space" do they mean standard models of Euclidean geometry? Non-standard models? Or physical space? Something else? (They show no knowledge of the standard literature on representation theorems.)
"This indicates that the mapping of numbers onto space is a universal intuition and that this initial intuition of number is logarithmic."
9. What does "the" mean? 10. What does "mapping" mean? 11. What does "onto" mean? (They probably mean functions from initial segments of (R*, <) to a line segment in 1-dimensional models, but they don't say so.)
"The concept of a linear number line appears to be a cultural invention that fails to develop in the absence of formal education."
12. What is a "concept"? What is the mass or velocity of a concept? 13. If a concept is massless and velocityless, how is it "invented"? What does that mean? 14. What does "linear" mean? 15. What does "number line" mean?
Jeff, I know what you mean, I have a similar kind of reaction when I read what the psychologists studying deductive reasoning say about logic and deduction. Still, in this case I think it's slightly different, as Dehaene does have a degree in math (from the Ecole Normale Superieure, no trivial feat), so he knows his math (probably much better than I know mine, with a modest minor).
Psychologists and cognitive scientists often seem to think that we philosophers are being 'pedantic' when we ask questions such as 'what is a "concept"?', but at least some of these questions do have an impact on the research they do. I'd be curious to hear what you think when you read the whole article (if you haven't yet), as it's hard to judge just from the abstract how solid the research is from a philosophical point of view.
Generally, and again wrt my contact with the literature on the psychology of reasoning, my attitude is: even if the conceptual framework is not as clear as it should be, it doesn't mean that the empirical data are all to be discredited. The hard part is to filter the good stuff through!
Hi Catarina, actually, I think the empirical results of these studies are generally very interesting - plus other work by authors in cognitive & developmental psychology, like Elizabeth Spelke, Alison Gopnik, etc., going back to Piaget. It's the philosophy that bothers me - "crazy talk", as Roy says.
The term of art here is "concept" and the underlying subjectivism or psychologism. I don't know what concepts are, and maybe in some sense, concepts are "constructed". In another sense (e.g., Fregean functions), concepts aren't constructed. Are concepts subjective patterns of associations? Can two different minds both "grasp" the same concept? Are concepts applicable to things? Should we say, (1) The classical concept of negation is different from the intuitionistic concept of negation. or (2) The concept of classical negation is different from the concept of intuitionistic negation.
The first (1) encourages the view that there are many *concepts* of negation. (2) encourages the view that there are many *negation operations*, all related somehow (by "concept of classical negation" we mean "classical theory that governs negation"). I don't have any good answers to these questions, but I'd like to know how the mind "grasps" concepts.
"I don't have any good answers to these questions, but I'd like to know how the mind "grasps" concepts."
Well, I guess that's pretty much what we would all like to know, psychologists in particular :) Do we, philosophers, have truly robust theories of what concepts are? I guess we don't. But if this means that, in the absence of such a conceptualization, all psychological research will be 'crazy talk', then that would be a problem too, I think.
I don't mean to say that all uses of the term 'concept' and related terminology by psychologists should be tolerated, but just that these considerations seem to me to introduce a significant risk of foundational skepticism. *That* we humans cognize is beyond doubt, but the question of *how* we cognize is daunting, and we seem to be forced to use the term 'concept' as a blanket term to refer to the objects of cognition.
And just to clarify, what term would you prefer to use to describe the idea (concept?) of the series of the natural numbers as evenly-spaced and linear?
Hi Catarina, my fault - I'm being very very unclear! Two separate things: (i) the subjectivism/psychologism (Roy's "crazy talk"); (ii) the unclarity about the mathematics.
On (i), it's not the talk of the concepts (though what concepts are is very moot). I happily talk of concepts, notions, ideas, etc. The trouble is the subjectivism or psychologism: the psychologistic assumption that certain mathematical entities *are* (mental) concepts, or intuitions, or inventions. For example, they write:
"Is this mapping a cultural invention or a universal intuition?"
Usually, a mapping f from X to Y is a set of ordered pairs (a subset of the Cartesian product X x Y). Can a function be an "invention"? How can a function be an "intuition"?
Cf., would someone say: "Is the millisecond pulsar an invention or an intuition?" Only an idealist about physical entities would say such a thing.
"what term would you prefer to use to describe the idea (concept?) of the series of the natural numbers as evenly-spaced and linear?"
Quite happy with the word "concept" (and similar, like "notion", etc.)! Not happy with the psychologism about mathematical objects. But your question focuses on the interesting claim by Dehaene et al about mathematical cognition, which seems interesting but is difficult to follow.
So, I don't quite get the phrase "the concept of the series of natural numbers as evenly-spaced and linear". Actually, I think they mean "beliefs about the numbers as being ..." or "manner of visualizing" (and not "the concept" - or maybe they simply have a rather holistic concept of concepts).
Here "linear" doesn't mean a property of orderings (an order (X, <) is linear if < is transitive, irreflexive and trichotomous). I think it means "linear" as a property of functions on the real numbers R (i.e., functions f : R -> R of the form f(x) = ax +b.). Also, it's not clear what "evenly spaced" means, for the linear order (N, <) of the natural numbers isn't "evenly spaced" (though it is discrete). There is no metric defined on it. But they must mean some *metric* structure? (The definition of "evenly-spaced" discrete point set I think would be that the points all lie on a line, and the smallest steps are congruent.)
When they say "space" I'm still confused. So, let's assume they mean the ordered real numbers (R, <). But they might mean Euclidean 3-space, or maybe the 1-dimensional real line construed some other way.
It may be that people's minds do, for some reason, attribute extra properties to, say, (N, <). For example, it might be "visualized" as embedded in (R, <) by inclusion, along with the natural metric on (R, <), which induces a metric back onto N. Namely, d(x,y) = |x-y|. Then, under an embedding of (N, <) into (R, <) with this metric, the "distance" between n and n+1 is always the same (d(n, n+1) = 1, for all n). But perhaps people think of (N, <) as being embedded into (R, <) not by inclusion, but by some other function, call it j: (N, <) -> (R, <). Then, if I understand what they're saying: j(n) is approximately a log(n) + b. This seems eminently plausible. |
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Professor uses magic tricks to teach students math
By IBT Staff Reporter On 06/03/09 AT 4:33 PM
A professor from a British university has created a new way to help students to learn math by teaching them magic tricks.
Professor Peter McOwan from Queen Mary's School of Electronic Engineering and Computer Science of the University of London has produced a series of videos entitled 'Maths in Magic' and 'Hustle' in conjunction with More Maths Grads (MMG).
MMG is a three year project that aims to increase the number of students studying mathematics and encourage participation from groups of learners who have not traditionally been well represented in higher education.
"It's fascinating how many great magic tricks and more worryingly con tricks work using hidden mathematical principles", explained Professor McOwan.
"The videos were made to help show how the power of maths can entertain and mystify, and how if we aren't careful can even part us from our hard earned cash |
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This photographic print is digitally printed on archival photographic paper resulting in vivid, pure color and exceptional detail that is suitable for museum or gallery display.
Babylonian cuneiform numerals. Key to the clay-pressed Cuneiform numerals used in the later Babylonian period (2000BC to 75AD). The Babylonian system was a combination of both the decimal and the sexagesimal systems. As some symbols had multiple values (for example, 1 could also stand for 1/60, 60, or 60 squared) the system was often confusing, but it did find application in pure mathematics and astronomy |
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Tag Archives: infinity
One other fascinating discovery about infinity in One Two Three… Infinity that was new to me is that the number of points on two lines of any length is the same. Also, that the number of points on a plane, and even three dimensional space is the same.
First of all, what I mean by "number of points… is the same" is what you would naturally think: They can be put in a 1:1 (one to one) relationship, for example:
Set 1: A, D, A, M, Z
Set 2: J, U, L, I, E
Set 1 has five letters and set 2 has five letters. We are only concerned with how many items there are in each set, and not what the letters are. The fact that the letter 'A' is repeated twice in set 1 and that neither set has any letter in common are unimportant. To see if the two sets are the same size, or if one is bigger than the other, we pair off the items in the two sets in any order we choose:
J - Z
U - M
L - A
I - D
E - A
We find that both sets are the same size because they can be put in a 1:1 relationship.
Simple enough. So, here is the mind blowing visual proof that two lines have the same number of points:
In this diagram the two lines of different length AB and AC are joined at A. The line CB connects the endpoints, and every line parallel to CB, such as DE, connects a unique point on AB with a unique point on AC and vice versa. So, even though there are an infinity of points on both lines, they can be put in a 1:1 relationship.
In case that's a little too informal, let me just add that CB and all its parallels are just graphs of a line function. You don't even need to know exactly what the function looks like, just that the input is a point on one line and the output is a point on the other, and that for every input there's a unique output. To me, it seems entirely counter intuitive, but the logic is inescapable, two lines of unequal length have the same number of points.
End of part 1. Coming in Part 2 – Use this one weird trick to map all the points on a plane to all the points on a line.
Different Types of Infinities (Wherein I will try to convince my friend, Jeff, that there are different types of infinities, and that some are 'bigger' or 'more abundant' than others.)
I'm going to concentrate on showing that integers and real numbers fall into two different categories of infinity. This is an informal proof, meaning that it lacks the rigor that would qualify as a proof for a mathematician, but should (hopefully) be convincing to a layperson. If you are interested in delving deeper, I recommend starting with this video by Vi Hart.
The guy who did the breakthrough work in the mathematics of infinity was Georg Cantor. Before Cantor came along infinity was taken to be a single concept of numbers going on forever, but Cantor showed that the picture was much more complicated and weird. In fact, some of the leading mathematicians of Cantor's day rejected Cantor's work and pilloried him.
The Proof
First, integers can be seen as a special case of real numbers that have all zeros after the decimal. We typically ignore the decimal when writing down integers, but another way to write the number 5 is 5.00000000000000… and 3843974937 is 3843974937.00000000000000… (the ellipses "…" denote that the zeros repeat forever). You can see in this diagram that the integers are a subset of the rationals which are in turn a subset of the reals. So, although both sets are infinite, one is a subset of the other.
Now, let's look at a line segment (i.e. a line of finite length) on the number line, even the smallest line segment we can imagine, will encompass an infinity of real numbers, but only 0 or a finite number of integers.
For example, this line segment from 3.10… to 3.20… contains 0 integers, but an infinity of real numbers, including the irrational number, π (3.1415926…).
This line segment, from 1.0… to 1,000,000.0… contains 1,000,000 integers, and an infinity of real numbers.
Cantor developed several proofs, including the diagonal argument, showing the one to one relationship between integers and rationals, and also how the reals cannot be put into a one to one relationship with the integers. There are plenty of websites and videos that explain this concept. So, I'm going to present my own 'cinematic proof':
Imagine your are in a desert that stretches to the horizon in all directions. Each grain of sand represents a real number, and you are searching for the sand grains that are integers. You are about to pass out from the heat when you see something in the distance. It starts as a dot wavering in the heat waves but eventually you can make out that it's a person on camelback. Eventually, Omar Sharif is standing before you. He presents you with an easy to follow map to the location of all the integers in the desert, and a scoop that can be adjusted in size to scoop as many of the integers as you want. When you scoop the sand and sift out the integers, you wind up with a small pile, and the sand you discard is a dune that piles up to the sky. That is how the integers and real numbers relate to each other. |
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Thursday, June 11, 2009
Polya's Army
Problem Solving is a big deal in any math class I teach, and I, like most math teachers, use Georg Polya's problem solving phases as a framework. Though I used to teach it as a four step process, I now recognize it as four phases, which problem solvers can progress through in many different ways, back tracking and skipping. The ultimate reference on this is Polya's book How to Solve It. My handout (adapted from Dave Coffey's) is here; it focuses on Polya's questions. (Questioning being another important comprehension strategy.) The modern day successor to Polya as a researcher and teacher of problem solving is Alan Schoenfeld. The link leads to his site where he generously shares a lot of his research and work. I recommend at least skimming Learning to Think Mathematically: Problem Solving, Metacognition, and Sense-Making In Mathematics (pdf link), a novella of a paper. Around pp 60-67 there's an amazing section on novice and expert problem solvers and teaching interventions.
One of my calc students was taking his final early, on his way to the month duty for army reservists, and commented on how he remembers Polya's steps by a connection with the troop leadership procedure. He sees: |
We have now moved on from looking at the arts in TOK to looking at how TOK can be applied to the area of knowledge of mathematics. In our first math TOK lesson, we looked at how we could define math in a way that would work in the realm of theory of knowledge, as well as looking at math in general and comparing our TOK approach to math with how we approached the are of knowledge of the arts. For this reflection, we have been provided with several prompts, many of which are based on a TEDx video about intuition and creativity in math.
The first prompt is to distinguish the differences between a conjecture and theorem. A conjecture would be defined as an opinion or conclusion formed on the basis of incomplete information, and therefore could be considered similar to taking an educated guess.A theorem is a statement that has been proved based on previously established statements. An example of a commonly known and used theorem would be the Pythagorean theorem of right angle triangles.
The second prompt for this reflection is to distinguish what the speaker in the ted talk provided means when he states that maths dominates intuition and tames creativity. I believe that by this he means that math relies heavily on one's intuition in the solving of problems in this AOK. "Taming creativity" could mean that creativity is applied in mathematics in a carefully measured sense. Mathematics is a discipline that has many rules that must be followed and therefore one's curiosity cannot roam free when solving mathematical problems and must be tamed in a sense that you are following the rules set. The next prompt related to the TED talk video is related to the claim made in the TED talk that "maths is eternal" and whether this gives maths a privileged position in TOK. I believe that this does not give maths a privileged position in TOK as many of the AOKs such as art and natural sciences are eternal as well, and as long as there is human existence these areas of knowledge will exist.
I this past TOK class we examined map like and story like knowledge, and how these two types of knowledge relate to the areas of knowledge of science and the arts. Map like knowledge uses equations, formulas, or more quantitative knowledge, while story like knowledge is obtained using qualitative knowledge such as tales and parables. The prompt for this reflection regarding these two types of knowledge is "Knowledge in the arts is clearly story-like whereas knowledge in natural science is clearly map-like."
I would agree that scientific knowledge is more map like then knowledge in the arts. Natural Science disciplines such as chemistry, physics, and biology, which we study in school, often require formulas to be applied to come to conclusions/find solutions, which is a characteristic of map like knowledge. Also, knowledge and information in science in most cases is testable, another characteristic of map like knowledge. Testability is a characteristic of map like knowledge as map like knowledge makes reference to maps, which can be tested by going over the same terrain multiple times. False theories in disciplines that follow map like knowledge can also be disproven by a single finding.
Knowledge in the arts would in most cases be story like knowledge, mainly because it is extremely difficult to quantify knowledge in this area of knowledge. It is also very hard to critique art, as everyone has different interpretations of the meaning behind art pieces, and also have different preferences in art, whether it be different preferences in music, visual art or architectural design. This would relate the story like knowledge as claims in story like knowledge are much more difficult to prove wrong, because they deal with unique persons.
Our task for this post-class TOK reflection is to read over three texts, two of which are short essays, summarize the points of these texts and to answer the prompt "How do both of these essays reflect what is presented in chapter reading about truth in art?" In our most recent TOK class, we were looking at truth in art, and how we could define art using more descriptive statements, as opposed to saying something like "art is objective" which is extremely broad. We also continued to compare art with science, particularly in how we define or describe each area of knowledge.
The essay "Art and Truth", which was given to us as one of the prompts for this reflection, covers the thesis "Though not traditionally a major topic within aesthetics, the relationship between truth and works of art is of considerable interest in the context of Theory of Knowledge." This essay analyses the claim that art can convey truth by examining hat society believes are the roles of artists, giving examples of different artworks that have been created, and outlying the motivations behind these works and how these motivations relate to the roles of artists in general. It also talks about common perspectives on different "genres" of art, such as photography.
The second essay we were assigned to read, "The Truth, the Whole Truth, and Nothing but the Truth: The Merits of Art versus Science". This essay lists significant literary texts throughout history and discusses the theme of "truth" in each of these texts. Some of the books that are featured are "Night" by Elie Wiesel, "To Kill a Mockingbird" by Harper Lee and Shakespeares "Macbeth". It then contrasts truth and the meaning behind literature with painted artworks by artists such as Picasso. It also talks about truth in art in relation to truth in different areas of knowledge such as history and science.
Both texts look at truth in an untraditional way, talking about how truth in literature and painted art, for example, is very different from conventional truth, or what we may think of as truth if not putting much thought towards it. One interesting part of the chapter reading that relates to some of the points brought up in both essays is the idea of "human truth". The reading used Shakespeare's"Macbeth as an example of how art can display human truth. It talks about how the words in Macbeth impact the audience and describe this impact as speaking a "deep and vitally human truth". A key takeaway from both of the essays that I had was that art can display literal truth, but often other "forms" of truth feature as well, giving more meaning and depth to how we view truth especially in the area of knowledge of art, which is one that can be viewed as quite abstract and generally difficult to interpret.
In our most recent TOK class we were looking at the area of knowledge of Art for the first time formally, and discussing what we believe is art, and how does one determine is art, more specifically what is good art. For this classes' I folio reflection, we have been tasked with choosing a claim and first outlining, then an argument for and against this specific claim. The claim that I have chosen is "Unlike The Arts, Science tells us something valuable about the world."
Science offers far more to humanity in terms of the knowledge that it gives us when compared to art, as scientific information helps us understand abstract concepts about the world, of which without we would not have power, the internet, or any other modern inventions. Science and scientific knowledge helps us to understand so many different aspects of the world, and science can be applied not only to subjects viewed as natural science such as physics, chemistry and biology, but also to topics such as engineering or even economics in some ways. Although art may be able to tells us things about the world, such as giving unique perspectives about the scientific world, it is much less influential than science when it comes to inventions that help to improve quality of life for people around the world.
Although art may may not have as much influence on inventions or assist our understanding of the natural world as much as science does, it does play a role in developing our understanding of the human world. Art helps one develop their conceptual knowledge, such as understanding our emotions and the effect of our emotions on others, as well as our moral knowledge, or ability to judge what is right and what is wrong. It can also help with furthering ones aesthetic knowledge, and the ability to see patterns and relationships between objects. These among other types of knowledge all would work together in making someone a more open minded and knowledgeable individual, therefore with no doubt enhancing our knowledge of the world and especially of human interaction.
In our most recent TOK class we discussed the ideas of bias and problems with observation in the realm of natural science. For this reflection, we were given two readings related to the development of knowledge and to bias and observation to look over in order to assist with our response to the prompt "science is objective and descriptive, while the arts are creative and interpretive".
This claim is one that is quite difficult to analyze because science and art are such large topics, with many different disciplines fitting under each. I believe that in the case of science, scientists attempt to be both objective and descriptive, however, in reality, there are many factors that could influence the way that scientists observe and make conclusions, including observations of past scientist, the views of the greater scientific community, or the religious/cultural views of the scientist. It is extremely difficult in any situation to remain completely objective while being able to make conclusions that will lead to descriptive and insightful analysis. However in comparison to the area of knowledge of art science is definitely more objective, and in-depth or descriptive. Although there are certainly art pieces that could be seen as objective, or things considered art that is objective, the majority of artworks either embody the views or emotions of the artist or in some cases is trying to bring attention to an event or movement. Also, art in many ways is interpretive and not structured, so I would say that I would agree with what the claim says that are is "creative and interpretive". However science in some aspects could also be creative and interpretive, and art could be objective and descriptive, so I would say that although I agree with both statements, I believe that they don't only apply to each specific area of knowledge.
In our last TOK class we discussed the differences between science and pseudoscience, more specifically what subjects can be and cannot be defined as being scientific, and which we would consider pseudoscientific subjects. For example, we compared subjects such as astronomy and astrology, and what differs these two fields of study, and which we could consider to actually be scientific. The prompt for this reflection is to analyze the claim "it is unsurprising when we hear that experts in Art can't always agree what 'is' and 'is not' Art. We might say that the distinction between what 'is', and what 'is not' art, is not always clear. Similar to the question of what is art, the distinction between science and pseudoscience is also not clear.
I would say that in the case of science versus pseudoscience, the distinction between the two is reliant on a few factors, one of the most important being how you define science, or more specifically the AOK of natural science, as this is open to many different interpretations. The definition of natural science is debatable, as is what can be considered natural or pseudoscience. In my opinion, pseudoscience differs from science as although it may appear to be scientific, is mistaken to have followed the steps of the scientific process, in relation to how information is gathered and discoveries are made in thiese disciplines. I believe that in order to decipher between topics that are scientific and pseudoscientific, it is important to state how you define each term, in order to make it clear of what the criteria for each of these two topics.
In the previous TOK class, we looked at the area of knowledge of Natural Science. The prompt for this reflection is "Reflecting on our discussions in class, and with inspiration from the TED video (that we were given to watch), what distinguishes Natural Science from other AOKs?.
I believe that natural science is distinguished from other AOKs by how controversial this area of knowledge is. So many ideas and findings in this AOK, as well as the actual definition of Natural Science, can be debatable and open to the interpretations of different people around the world. In our class, we did an activity where we debated whether a particular statement was applicable to the realm of natural science. Initially, much of us were divided on each statement mainly because we were all defining natural science in different ways, therefore meaning that to each of us different aspects of the world applied to natural science than others in the class thought. I think that natural science is such a controversial and debatable area of knowledge mainly because we can in most cases never be 100% certain that our discoveries or ideas are correct because it is very difficult to confirm things such as collision theory as the reactions take place between such tiny molecules. Also, in the past, what has been considered as the correct idea or understanding of different phenomena of natural science has been based on the popular opinion of those at that time in history. For example, when Charles Darwin first came up with his theory of evolution he got a lot of criticism because everything he was saying contradicted the popular belief of how we and the rest of the animals on earth became how we are today.
During today's TOK class we looked at the WOKs (Ways of knowing) of faith and intuition, and for the majority of the class discussed the question "should faith be considered a way of knowing". For this reflection, we have been tasked to discuss faith and intuition as WOKs, and provide an outline for the limitations of each as WOKs and also an outline justifying their inclusion as WOKs.
A limitation of faith as a way of knowing would be that everyone has their own individual perspectives, meaning that most people have different interpretations of what faith is exactly, or different beliefs and place their faith in different ideologies. This could cause biases when issues such as conflict of religious interests are being discussed. Many people put their faith in different religions, and they usually believe strongly in the ideologies of the religion that they follow, therefore being biased towards what their religion believes rather than looking at the positives of another idea or trend. A limitation of intuition as a way of knowing would be that often ones intuition can lead to them carrying out a selfish act, as one can argue that intuition can be thought of as the instinct of self-preservation.
This next paragraph will be working to justify faith and intuition as ways of knowing. A justification for faith as a way of knowing is that without it, much of the area of knowledge of religion would be hard to comprehend. Also, we often put our faith into the people that come up with scientific ideas, as we usually have not experienced the phenomenon that they discovered, and therefore put our faith in them as we are believing that their descriptions of these phenomena are how they actually take place. To justify intuition as a way of knowing, I would say that intuition is almost the shorter term representation of faith, and therefore without faith, intuition would also not be able to be viewed as a way of knowing, for the same reasons as described previously in this paragraph.
In our most recent TOK class, we explored the roles of imagination and memory in the development of one's knowledge. In groups, we thought about how each of these ways of knowing can impact someone's development of knowledge. For imagination, my group came up with three points, the first was that it can help us to perceive or relate events around the world to events in the past, and look at the similarities of the causes of historical events and current events, therefore being able to have an idea of how a current event would play out. The second point was it helps us to visualize certain concepts, such as the value of mathematical variables, such as x, and the final point being that, in the eyes of some people, religions were formed based off of the imagination of people and their abilities to create ideologies that are abstract and were not seen before the times of their creation. We also thought up three points about how memory can be used to develop knowledge. The first point is that it helps us to connect events using our memories, the second is that history is based on memory, recounts and documentation and the last point is that much of indigenous knowledge is based on memory.
The second part of this reflection will discuss the prompt "despite the imperfections of imagination and memory as ways of knowing, the Areas of Knowledge have developed in such as way as to overcome them". In relation to the area of knowledge of mathematics, I believe that this prompt is true as I feel that the only limitations to the development of knowledge in mathematics is the lack of basic mathematical knowledge, as complex mathematics even in more abstract cases is a combination of usually much simpler formulas or rules, and imagination and memory can only help in the pursuit of knowledge in mathematics. If one was to look at this prompt through the lens of natural sciences, I again believe that the imperfections of imagination and memory would not impact the development of knowledge and would assist with it, if anything. |
Calculating Number of Days Passed Since the Introduction of Gregorian Calendar
This study is an algorithm of calculating number of days passed since the introduction of Gregorian Calendar for any given date using simplified formula. It consists of nine algebraic expressions, five of which are integer function by substituting the year, month and day. This formula will calculate the n^th days which gives a number from 1 to ∞ (October 15, 1582 being the day one), that determines the exact number of days passed. This algorithm has no condition even during leap-year and 400-year cycle.
Category:General Mathematics
Calculating Days Difference Between Gregorian and Julian Calendar
This study is an algorithm of calculating days difference between Gregorian & Julian calendar using simplified formula. It consists of two integer function by substituting the year. This formula will determine the exact number of days in any given Year as of December 31. This algorithm has no condition even during leap-year and 400-year rule.
Category:General Mathematics
Mathematics as an Interconnected Whole
Different thinkers suggested varied images and descriptions of mathematics. Platonists believe that mathematical objects exist as Platonic Ideas and mathematicians only discover them. Nominalists think that mathematics is the contents of mathematical manuscripts, books, papers and lectures, with the increasingly growing net of theorems, definitions, proofs, constructions, and conjectures. Pragmatists assume that mathematics exists in mentality of people and when mathematicians introduce new objects they invent and then build them. An interesting peculiarity of the situation is that all these opinions and some others are true but … incomplete. The goal of this work is to explain this peculiarity presenting a complete vision of mathematics as an interconnected Whole.
Category:General Mathematics
Counting the Number of Days in Any Year
This study is an algorithm of calculating the number of days in any given Year in Gregorian & Julian calendar using simplified formula. It consists of seven algebraic (3 for Julian) expression, six of it are integer function by substituting the year. This formula will calculate the number of days which gives a number from 365 to 366 that determines the exact number of days in a given Year. This algorithm has no condition even during leap-year and 400-year rule.
Category:General Mathematics
Calculating the Day of the Week – Direct Substitution
Abstract. This study is an algorithm of calculating the day of the week for any given date in
Gregorian & Julian calendar using simplified formula. It consists of eight algebraic 6 for Julian
expression, five of which are integer function by substituting the year, month and day. This
formula will calculate the modulo 7 which gives a number from 0 to 6, i.e., 0Saturday,
1Sunday, and so on, that determines the exact day of the week. This algorithm has no
condition even during leap‐year and 400‐year cycle.
Category:General Mathematics
Counting the Number of Days in a Month-Year
This study is an algorithm of calculating the number of days of the Month-Year for any given Month and Year in Gregorian & Julian calendar using simplified formula. It consists of eleven algebraic (6 for Julian) expression, all of it are integer function by substituting the year and month. This formula will calculate the number of days which gives a number from 28 to 31 that determines the exact number of days in a given Month-Year. This algorithm has no condition even during leap-year and 400-year rule.
Category:General Mathematics
Feasible Mathematics
From algorithmic information theory (and using notions of algorithmic thermodynamics), we introduce *feasible mathematics* as distinct from *universal mathematics*. Feasible mathematics formalizes the intuition that theorems with very long proofs are unprovable within the context of limited computing resources. It is formalized by augmenting the standard construction of Omega with a conjugate-pair that suppresses programs with long runtimes. The domain of the new construction defines feasible mathematics.
Category:General Mathematics |
These are the musings mucking about in my cranium. Proceed with caution. And maybe bourbon.
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Poly-dodeca-icsosa-and so and on and on and on…
A polyhedron is is what you see in Cairo. The Pyramids. It's basically a bunch of angles and corners that all equal the same length.It's a three dimensional shape with flat plains and sharp edges (Four flat plains and…six…? sharp edges….I think) …but always, always…everything mathematically works.
A dodecahedron is a polyhedron writ larger. Twelve flat plains with sharp angles (18?) _that will always equal the same length and width and size…
And then there's the Icosahedron. Twenty flat faces, thirty sharp angles (maybe…?). And if you roll a natural 20 you've vanquished (usually). And if you roll a natural 1?
Too bad. So sad. Roll another character. You're dead.
A squared plus B squared equals C squared.
I understand the Pythagorean theorem, for as ridiculous as that may sound.
It's not…
It isn't even. There's a right angle you have to account for that may go on for days and days and days…but…at some point, if you apply the theorem…you'll get the answer.
And you can eventually work out the square in the angle too…
For whatever good that may do…
Life…Isn't just Mathematics or Fractiles (no matter how much I may love Mandelbrot…because it's just a recursion that goes on and on and on…and…that's not LIFE…)
Geometry…Mathematics…Science, Psychology, Philosophy all try to and fail to take into account (despite the volumes and volumes and volumes written on human "behavior")…
We actually…every one of us…are not just Science.
Can we be taught/driven/counseled/forced/beseeched/forced by what we have known until the point…even though…no matter how hard we try….yes. We "are who we are"…?
Life…may be Mathematics by Nature.
By Nurture?
It will never, ever, be as simple as a Right Angle.
Or a Twenty-sided die on which you stand or fall depending on how well you roll.
And no matter how you want to bag it up and stuff it or toss it…toss…it… |
Participant to some european programs for making short stays and
schools; including Wroclaw university Poland (august 2000), University
Edinburgh UK (april 2000), Mathematisches Forschungsinstitut Oberwolfach
(2002) |
What if Flatland wasn't a plane
I had a couple of requests to bring the squircles to a New Year's Eve party:
I threw a couple of other 3d prints into the bag, too, because I thought the kids there would like more than just squircles. The 4d shapes generated a lot of conversation. Bathsheba Grossman's "Hypercube B" in particular:
Those conversations got me thinking about common ways of understanding higher dimensional shapes if you live in a lower dimensional world. The sphere passing through a plane generating a series of larger and then smaller circles is one common way of explaining how a 2d being could understand what a sphere looks like. So, today we discussed what a sphere passing through two other 2D shapes would look like.
We started by talking about the problem of a sphere passing through the plane and then briefly talked about what we thought a sphere passing through a "v" shaped piece of paper (a bent plane) would look like:
Next we went to Mathematica to see the shapes for ourselves. The first set of shapes we explored was the sphere passing through a "V". I should have published these videos with the computer in much higher quality – sorry that the computer text is basically impossible to read.
For the sphere passing through the "V" we see shapes that look nothing at all like normal slices of spheres:
The next set of shapes we looked at was a sphere passing through a cone. In this video the sphere is centered directly above the point of the cone:
Finally we looked at how the shapes from the previous video change if the sphere is not centered on the point of the cone. Here you get some pretty strange shapes. I think if you encountered these shapes in the wild it would be really hard to imagine that they would assemble into a sphere!
So, a fun project to start the new year. Turns out there are lots and lots of different "slices" of spheres passing through flat, 2d shapes! |
Yes, it's 3/14/14. In a sense, this whole month can be a celebration of Pi, because it's 3/14, but today, 3/14, is the official day of celebration (set aside by Congress in 2009 in one of its least and most irrational actions to date).
Wynberg celebrated this day in three ways:
An interhouse teacher/learner competition where contestants recited as many the numbers following the decimal point as they could,
Icing the pi symbol on a variety of cookies and cupcakes and having them on sale to raise money for the Academic pillar,
Creating the figure pi in Wellington Quad using nothing but Matric learners.
The Pi Recital challenge proved to be an exciting and interesting one. Many of the participants surprised the audience with their talent of reciting an endless amount of decimals and as each participant was doing so, there was a silent tension in the air. Much cheering and support was received by this small event. We are all extremely proud of the participants, but have to make mention of the top teacher, Mrs T Pelser who remembered 98 of the decimal points, and Nicole Wentzel (Grade 11B) who remembered a staggering 202 of the points. We salute their efforts
The cake sale as well as the Pi formation on the Wellington Quad grass were successful. The cupcakes and biscuits were sold out within minutes and the Matrics managed to form a large and perfect Pi on the grass in an orderly fashion.
Overall, it was a successful Pi Day – definitely one to be remembered. |
Math Through the Ages, A Gentle History
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A 4 page book review. William Berlinghoff and Fernando Gouvea offer a fascinating and insightful view of the history of mathematics in their text Math Through the Ages: A Gentle History For Teachers and Others. The authors' intention in this work is to give their readers "a general feel for the lay of the land perhaps to help you become familiar with the significant landmarks" (Berlinghoff and Gouvea 5). In short, this is an overview of math history, a "brief survey" of what is in a gigantic topic (Berlinghoff and Gouvea 5). However, the appeal of this book goes deeper than this, as the authors often offer insight into the topic that helps to explain historical details, helping the reader see how math concepts developed and evolved. No additional sources cited.
Filename: khmtta.rtf
Mathematical Errors in 7th Grade
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This 3 page paper provides an overview of common mistakes made in seventh grade mathematics problems. This paper relates the reasons for these mistakes and their impacts for learning. Bibliography lists 1 source.
Filename: MH7GrMat.rtf
Mathematician Leonhard Euler's Refutation of Pierre de Fermat's Conjecture
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This is a 3 page paper discussing Euler's refutation of Fermat's conjecture. In 1637, French lawyer Pierre de Fermat wrote that he had "discovered a truly marvelous proof which this margin is too narrow to contain" in regards to a mathematical statement which had been unproven for over 1000 years. The basis of Fermat's ("Last") theorem or conjecture began with that of the Pythagoras equation [x.sup.2] + [y.sup.2] = [z.sup.2] which he proved "had an infinite set of whole number solutions" which related to the lengths of the sides of a right-angled triangle. Pythagoras did not know "how many solutions existed if the exponent in his equation were a number greater than 2". Fermat claimed that "for any exponent greater than 2, there were no solutions at all". During his lifetime however, Fermat often did not supply "proofs" of many of his theorems but many mathematicians since his time have been able to prove his claims to be correct except for that in relation to the Pythagoras equation. Swiss mathematician Leonard Euler (1707-1783) did however work further on many of Fermat's theorems and "later proved that there are no solutions when the exponent is 3" and "unfortunately, an infinite number of cases remained and the case-by-case method was doomed to fail". While Fermat's Last Theorem proved to be difficult to prove, Euler managed to disprove and refute other assertions such as "2^(2^n) = p, where p is a prime number" and found that it is only true for the first four cases provided by Fermat.
Bibliography lists 4 sources.
Filename: TJEuler1.rtf
MATHEMATICS AND THE ART OF WAR
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This 8-page paper examines how mathematic applications have been used to help analyze and calcluate wars. Bibliography lists 3 sources.
Filename: MTmat |
Math Circles
19Nov09
The University of Waterloo's CEMC holds Math Circles in which they invite local grade 6-12 kids to come to the university one evening a week for math enrichment activities. In the past years I have participated for 2-3 weeks per year, running sessions on the topics of Game Theory and Conics (ellipses, parabolas, etc). This year my topic is Graphs! Here are six small graphs (you can ignore the numbers on the edges):
The first session, yesterday, was very enjoyable (at least to me, but it also seemed to get positive feedback from the other participants as well).
In my experience, it can be tricky to guess the appropriate level of discourse when preparing the lessons, mostly since I have no prior experience with most of the kids in the session. This caused me to "lose" about a 1/3 of my first session to covering background material: proof by induction. (It's not really a "loss" because I would say that the time spent was still fun for me and beneficial for the students.)
In the course of developing the online notes to accompany the presentation, I wanted to see if there was a good resource online with an appropriate explanation of "proof by induction." There are certainly hundreds of books including the subject, and given the size of the internet, one might conclude that there are thousands of articles online about induction? Well, it doesn't quite seem to be that many. I found that Wikipedia had the best all-purpose description (with an example, and without getting too abstract) and there is another online at Carnegie Mellon University. I was hoping to find something good at the Art of Problem Solving website but their description was a little too terse. It looks like there is still some room on the internet for well-written mathematical enrichment material, I am glad the Math Circles content is available to help fill that hole. |
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