Monthly Archives: November 2014

Math Education: Why we are where we are

   I have from time to time read about all the horrors of our current educational system and the ongoing ways that people have come up with to ‘fix’ our broken system. Yes, our system definitely has its faults, and I’ll touch on those in a moment (I think the largest is a dictatorial set up of learning), but have you ever wondered why we are where we are in terms of our education and math education in particular?

Fredrick the Great of Prussia by Anton Graff, via Wikimedia Commons.

   The most common system of education in the United States and the world is based off the Prussian education system. The Prussian education system, which began in 1716, has an odd and strangely juxtapositional beginning. One half of the juxtaposition was a high minded religious ideal and the other half was all about governmental control. Before universal education was brought about by the Prussian education system, education was expensive and you got what your parents could afford. According to  Brendan McGuigan on Wisegeek “The Pietists,(1) among other religious factions of the time, believed that the deepest understanding of God could come only through a personal reading of the Bible; therefore, literacy was important for all people, not just the wealthy.” What seperated the Pietist movement from other creeds that called for literacy was that the Pietists wanted to create schools for all and not just for literacy, math was also considered to be a fundamental part of education. The Pietist movement was started by Johann Kaspar Sehade (2) in 1686 in Dresden, the capital of Saxony in the Germanies.(3) 

    The second half of the juxtaposition was that King Fredrick the Great of Prussia was looking for a way to consolidate power in his newly formed country. King Fredrick took the ideas of Johann Sehade and modified them to cement control of his newly formed dynasty through blind obedience to those in power.(4) (You will obey the teacher…or else.) The Prussian system was not all bad; it introduced the idea of universal education to the world, and some would argue that the advent of universal education heralded the start of the industrial revolution (5) or if not the start then definitely the continuation and acceleration of the revolution into the world we know today.

The Prussian educational system was brought to the United States by Horace Mann (4) in 1843 and has been used for compulsory universal education ever since. I remember that on my first day of college at the University of Arizona I was shocked when at the end of the freshman orientation a questionnaire was handed out by a representative of the state board of education that asked for our opinions on what should or should not be taught to high school students. I don’t remember all of the subjects, but two of them were the swastika and the Holocaust. The reason I was shocked was they wanted to know my opinion on something and not to just a regurgitation of knowledge. This was the first time my opinion was considered of worth at all by any organization, much less the government. The point of this flashback was to point out that the inherent obedience training that was put into the Prussian system is still there in a subtle way; it says without words that you (the student) know nothing, and your opinion is worth even less. At least it had seemed so to me. The Prussian system is very good for rote memorization, and the idea of universal education is unbelievably important, because it is not possible to predict where or when the next super genius world changer will show up.

What does all this have to do with math education? I believe it is best illustrated with a metaphor. Imagine a society in which all of its technology and understanding of the world was based around beautiful and amazing works of art. Now imagine for a moment that this world taught its children art almost exclusively through paint-by-numbers™. Then we for some bizarre reason expect the children to be able to paint masterpieces while they are busy looking for the lines to paint inside of. This is similar to what the Prussian system does for math education. There have been through the years various attempts at reform, some of the latest have been Khan Academy, Google schools, and the new common core standards. These groups and reforms are all attempting to make math a more accessible subject. An aspect to the difficulty of math education that these reforms can not take into account is the societal belief that math is hard and the problem that most of the ‘math’ teachers in primary education are not well versed in mathematics themselves.

   These problems are linked and unfortunately endemic to our society and even though it can not be dealt with until more people really understand math there are some signs of hope. (Quite the catch 22 though, isn’t it?)  Like the recent sea change towards “nerds” and “geeks” in our society.  In recent years it has become more socially acceptable to be smart and, our media is reflecting this. So here is hoping that this societal change of heart really takes hold and we can become much better at math and be proud of it.

1 http://www.wisegeek.com/what-is-the-prussian-education-system.htm

2 http://www.newadvent.org/cathen/12080c.htm

3 it was known as the Germanies at the time because the unification of the German city-states did not happen until 1871 under Emperor Wilhelm I

4 http://www.school.namaya.com/newamericanacademy/images/the-prussian-industrial-history- of-public-schooling1.pdf

5 Me, its argued by me, there may be others but definitely me.

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Turing, Leibniz and Hilbert’s Entscheidungsproblem

Alan Turing. This image is in the public domain in the US because its copyright has expired.

Alan Turing. This image is in the public domain in the US because its copyright has expired.

What’s the first thing you think of when you hear the name Gottfried Leibniz? Let me guess: calculus.  Now what do you think when you hear of Alan Turing?  You might think of codebreaking during World War II, or the new movie coming out about him (The Imitation Game), or maybe you haven’t heard of him.  So why would I mention these two together? Computers of course! Wait, what do these two have to do with computers? Well let’s take a look and see.

The Entscheidungsproblem origins start with Gottfried Leibniz in the seventeenth century.  Leibniz had successfully created a mechanical calculating machine, one of the first of its kind.  This calculating machine led him to question if a machine could be made that could determine the truth values of mathematical statements.  In his research, he found that one would have to find a formal language to create this machine.  In 1900, David Hilbert, a German mathematician, included the following in his 23 unsolved (at the time) problems designed to further the disciplines in mathematics:

“10. Determination of the solvability of a Diophantine equation.  Given a diophantine equation with any number of unknown quantities and with rational integral numerical coefficients: to devise a process according to which it can be determined by a finite number of operations whether the equation is solvable in rational integers. (Winton Newson’s translation of HIlbert’s original problem, as quoted in D. Joyce)”

By 1928, Hilbert had broadened his question about Diophantine equations to a much more general question about mathematical statements in general: is there an algorithm that is universally valid. This created a new idea; is there an algorithm that can tell us if any algorithm will terminate?  The last of these three ideas was the beginning of the Entscheidungsproblem. In May, 1936 Alan Turing wrote a paper called “On Computable Numbers, with an Application to Entscheidungsproblem.”  In this paper, Turning reformatted Kurt Godels results on the limits of proof and computation.  He made a hypothetical device known as the Turing machine and went on to prove there was no solution to the Entscheidungsproblem.  He did this by using his Turing machine to show that the halting problem is undecidable; that it is impossible to know whether a program will finish running or continue forever.

The Turing machine itself can represent a computing machine.  It can change symbols on a strip of tape based on a set of tools.  A Turing machine has 3 main components, the first being an infinite tape. This infinite tape would be divided into cells in which a symbol can be placed.  In the tape there would be a head.  The head accesses one cell at a time, while moving either right to left or left to right.  The third component would be a member were there would be a fixed finite number of states.  After having these three components you have three actions: 1) write a symbol, 2) move either left or right, and 3) update its current state.  The formal definition of a Turing machine is defined as a 7-tuple.  The seven elements of the tuple would be as follows: a set of states, an input alphabet, the tape alphabet, the start state, a unique accept state, a unique reject state, and a transition function.

A Turing Machine, without infinite tape. Image: Rocky Acosta, via Wikimedia Commons.

A Turing Machine, without infinite tape. Image: Rocky Acosta, via Wikimedia Commons.

Turing’s work on the Entscheidungsproblem and the Turing machine can be thought of as the birth of computer science and digital computers.  During World War II the idea of the Turing machine was used and manipulated into a simpler form, as well as into an actual electronic computer.  This led to machines such as the counter machine, register machine, and random access machine.  All of these machines launched us even further into the computer era.

It is interesting to see that the birth of the modern computer came from the Entscheidungsproblem, an idea that Leibniz had first thought of.  Why would I think this is interesting?  Leibniz had also worked on binary numbers and arithmetic, which is similar to what is used today in modern computing. It seems that Leibniz was ahead of his time.  Alan Turing seems to have just taken his ideas and brought them to our times.  We can see that without Turing we wouldn’t have modern computers the way they are. This means we wouldn’t be able to do any math that requires a computer to help with computations.  Think, how many times have you used your computer to access the Internet to get the answer to a math problem you were unable to solve? Not only that, but studying math wouldn’t have been as easy.  Knowledge that is passed through the Internet wouldn’t be possible without computers, with no YouTube to help show how to solve math problems, with no Khan Academy or Wolframalpha, and no easy access to any knowledge of any past essays that were written.

Sadly, Turing’s end wasn’t a happy one.  Living in England in the early 20th century as a gay man led him to commit suicide.  Leibniz lived almost twice as long as Turing.  It makes you wonder if we could have had even more interesting computing machines or ways of thinking of computational mathematics if he had lived a full life past the age of 41.

Sources

History on Turings life – http://www.math.rutgers.edu/courses/436/Honors02/turing.html

Hilberts Problems – http://aleph0.clarku.edu/~djoyce/hilbert/problems.html http://mathworld.wolfram.com/HilbertsProblems.html

Turings paper “On computable Numbers, with an application to the Entscheidungsproblem – http://plms.oxfordjournals.org/content/s2-42/1/230.full.pdf+html

Gottfried Wilhelm Leibniz on wikipedia – http://en.wikipedia.org/wiki/Gottfried_Wilhelm_Leibniz

Wikipedia article on the Turing Machine – http://en.wikipedia.org/wiki/Turing_machine

Cryptography – Keeping Our Online Secrets Safe Since the 90s

ElectronicMediaPieChart

A breakdown of time Americans spend with electronic media. Image: Courtesy of http://www.statista.com/chart/1971/electronic-media-use, Felix Richter

We live in an era where the internet is king. Between our cellphones, tablets, game consoles, laptops, and other devices, the average American adult (18+) spends 11 hours per day ingesting electronic media in some way, shape, or form.  I’m sure we can all admit that on a weekly basis we access or create data that we don’t necessarily want the public to see. Whether it be our bank account or credit card information, our Facebook interactions, our emails, our tweets, our PayPal activity, or even our browsing history. That being said, I’m sure some of us take our internet privacy for granted; but how exactly does are internet privacy remain… private? The answer is simple: modular arithmetic. More specifically, cryptographic algorithms.

A History of Cryptography

Cryptography dates back to Egyptian scribes in 1900 B.C., and it was first used in their hieroglyphs. The Egyptians presumably wanted hide the content of their hieroglyphs from others, and they used very basic cryptography to do so. As you can imagine, this whole “keeping a message’s content safe” idea would become widely popular as mankind become more and more intelligent. The Romans, specifically Julius Caesar himself, created the first truly math-oriented cryptography. He used it primarily to protect messages of military significance. Caesar’s cryptographical ideas would later be used to build out modern day cryptography.

There are two main types of cryptography widely used across the web today: symmetric-key encryption, and asymmetric-key encryption (we’ll go into details later, I promise!). Both of these types of encryption rely on modular arithmetic. We must give credit where credit is due. Friedrich Gauss (1777-1855), birthed modular arithmetic in 1801. Believe it or not, this famous mathematician made most of his breakthroughs in his twenties! For those that aren’t familiar with modular arithmetic, here’s a timeless example (pun intended, wait for it…).  The length of a linear line can have a start and end point, or it can go on to infinity in either direction. In modular arithmetic, the length of a “circular” number line is called the modulus. To actually do the arithmetic, consider this example: Take a regular clock (see, here’s the pun!), consisting of the numbers 1-12 . Clocks measure time on a 12 hour time table before starting back over at 1. The modulus for a 12-hour clock is 12 because it has 12 different numbers for the number of hours. To actually do the arithmetic, take this for example: It’s 8PM and we want to add 9 hours (8 + 9 mod 12). 8 +9 equals 17, however when using a modulus of 12, our number line wraps back around after counting to 12. For this we would count forward from 8 – ie. 8, 9 ,10, 11, 12, 1, 2, 3, 4, 5. So, (8 + 9 mod 12) = 5 AM in this case.

Caesar Cipher

CaesarCipher

A basic Caesar Cipher using a left shift of 3. Image: Matt_Crypto, via Wikimedia Commons.

As I said above, the Caesar cipher has acted as a building block for some of our modern day cryptography. Caesar’s main encryption step is incorporated in some of the more complex schemes we still rely on today. However, the Caesar cipher can be easily broken, or decrypted (more on this soon!). This particular cipher is concerned with the alphabet. The theory behind it is replacing each letter in the alphabet with a different letter some fixed number of positions down the alphabet (this is reffered to as the shift). For instance, with a shift of 3, A would replace D, and B would replace E.

Original: ABCDEFGHIJKLMNOPQRSTUVWXYZ

Cipher:   XYZABCDEFGHIJKLMNOPQRSTUVW

This can be represented mathematically using modular arithmetic. The encryption of any letter ‘x” by a shift ‘n’ can be described as follows:

Encryption:

E(x) = (x + n) mod 26

Decryption:

D(x) = (x – n) mod 26

Brute-Force Attacking:

This cipher is extremely easy to break. There are only 26 possible shifts (26 different english letters). When taking a brute-force approach, it’s only a matter of varying through the different shifts until the message is decrypted. In fact, this process could be optimized by analyzing the encrypted string, finding frequently used letters and associating them with common vowels. That way, you could brute force using intelligent shifts. However, this approach would have to be modified when switching between languages.

Cryptography Online

As promised, I will explain the two types of internet cryptography. First, we have symmetric-key cryptography. This is based on the concept that both communicating parties share the same key for encryption as well as decryption. This key is mathematically applied to a numerical equivalent of the data each party is encrypting/decrypting. It is imperative that this key is kept secret. If another party finds out what the key is, none of the encrypted data is safe anymore. Symmetric-key cryptography uses either stream ciphers (encrypt the numerical representation of the data one digit at a time.), or block ciphers (taking blocks of digits, and encrypting them as a whole). Symmetric-key algorithms have an advantage over asymmetric in that they require less computational power.

AsymmetricCrypto

Asymmetric-Key encryption. Anyone can encrypt data using the public key, but the data only be decrypted with the private key. Image: Dave Gothenburg, via Wikimedia Commons.

As for Asymmetric-key cryptography (aka public-key cryptography) we use a slightly different approach. This cryptosystem implements both a private and public key. The public key is used to do the encryption (just like symmetric key cryptography), but the private key is used to do the decryption. The word “asymmetric” stems from the different keys performing opposite functions. This type of cryptosystem is more user friendly, and requires less administration. This is why public-key cryptography is widely implemented across the web.

The RSA Cryptosystem

The RSA cryptosystem is one of the most practical applications of modular mathematics we see today. In fact, if you look at your browser’s address bar right now and you see an “https” at the beginning of your URL, you’re more than likely relying on an RSA encryption to keep your data secure. RSA was created was created in 1977 by Ron Rivest, Adi Shamir, and Leonard Adleman at MIT. As far as cryptosystems are concerned, RSA in particular is one of the most straightforward to visualize mathematically. This algorithm consists of three parts: key generation, encryption, and decryption. I will be walking through a widely-used example using 3, and 11. Not only does this process use Gauss’s modular arithmetic, it also uses Euler’s totient function φ(n). (A function that counts the totatives of n – the positive integers less than or equal to n that are relatively prime to n.)

Generating the key is the most confusing part, but here’s a somewhat simplified version (don’t get nauseous!):

  1. Randomly pick 2 prime numbers p and q : p=3 q=11
  2. Calculate the modulus : n = p * q  ->   3 * 11 = 33
  3. Calculate the totient φ(n)  : z = (p – 1) * (q – 1) -> ( 3 – 1) * (11 – 1) = 20
  4. Choose a prime number k, such that k is co-prime to z : k=7
  5. n and k become the public key
  6. Calculate the private key : k * j = 1 mod z | 7 * j = 1 mod 20
  7. In the previous step, we’re only interested in the remainder. Since we’re working with small numbers here, we can say –> 21/20 gives us “some number” with a remainder of 1. Therefore – 7 * j = 21 -> j = 3
  8. j becomes the private key

After the public and private keys are generated, encryption and decryption become easy!  Given P is the data we’d like to encrypt and E is the encrypted message we want to generate:

P^k = E (mod n)  – When P (data we’d like to encrypt) = “14”  We get: 14^7 = E mod 33  So E=20

Given E is the encrypted data we’ve received, and P is the data we want to decrypt:

E^j = P ( mod n) – 20^3 = P mod 33  So P = 14

This proves RSA works!

Does the RSA Cryptosystem Really Keep Me Safe?

Theoretically, a hacker could factor the modulus “n”  in the steps above. Given the ability to recover the prime factors p and q an attacker can compute the “secret exponent” “d” from the public key (n, e). Once the hacker has this “secret exponent”, they can decrypt all data sent with its matching public key. RSA keeps us safe from hackers because there is no known algorithm (The NSA probably has one!) that can factor these large integers in a timely manner. In fact, the largest known number ever factored was 768 bits (232 digits!) long, and this was done with a supercomputer using a state-of-the-art implementation. If that doesn’t make you feel safe enough, RSA keys are typically 1024 to 2048 bits (617 digits!) long, so we don’t need to worry about our data getting hijacked. However it is recommended that we use a value of n that is at least 2048 bits long to ensure the encryption is never cracked.

Sources:

http://en.wikipedia.org/wiki/RSA_(cryptosystem)

http://blogs.ams.org/mathgradblog/2014/03/30/rsa/

http://www.studentpulse.com/articles/41/a-brief-history-of-cryptography

http://www.ti89.com/cryptotut/mod_arithmetic.htm

http://en.wikipedia.org/wiki/Modular_arithmetic

http://cunymathblog.commons.gc.cuny.edu/

How and Why RSA works:

https://www.youtube.com/watch?v=wXB-V_Keiu8

Chaos Theory: the Smallest things can have big consequences

One can tell by the name itself what chaos theory might mean. Chaos means something that is unpredictable, random and unstable. There are many known and predictable phenomena in science such as electricity, gravity or chemical reactions; however, chaos theory examines things that are not possible to control. For example, nature:  weather, earthquakes, clouds, trees, tsunamis, and tornadoes. Other than nature, there are human-related unpredictable things, such as the stock market and our brain states. Chaos theory is a field of mathematics that deals with complex systems whose behavior is highly sensitive to the slightest changes in initial conditions. For example, someone clapping their hands could change the weather, so even the smallest alterations can have big consequences.

Chaos theory emerged around the second half of the 20th century. This is because chaos theory has complex systems and these systems contain many elements that move. For this reason, computers are needed to calculate all the different possibilities.  How did chaos theory come to be? A man named Edward Lorenz, a meteorologist, created a weather model on his computer in 1960. This weather model consisted of an extensive array of complex equations to predict weather conditions. This model always gave different sequence of numbers that represented weather conditions. One day he became curious and ran his own tests to see what the outcome would be. After running a sequence, he started running the same sequence halfway through, re-entering the numbers the first sequence had given him at that point. The results were not what he was expecting; they were entirely different from his first outcomes. The second time, he entered numbers that were rounded to three instead of six digits (for example, .506 versus .506127). Since the difference between these numbers is not much, he expected the results to be only slightly varied. However, that small error gave completely different outcome. Form this he concluded that even the slightest differences in initial conditions makes prediction of past or future outcomes impossible. 

butterfly effect

Image: J.L.Westover.

There are many principles of Chaos. One of them is the butterfly effect, also described by Lorenz. It is said that even a small butterfly flapping its wings in America can create a hurricane in Japan; if the butterfly did not flap its wing at the “right” time in space then the hurricane would not have happened. Even the smallest behavior has a direct effect in the future. Another principle is unpredictability. Since it is not possible to know all the initial conditions of a complex system in adequate detail, we can’t possibly know the outcome of those. As explained above, even the smallest change in numbers can lead to a big errors in prediction; outcomes  can be completely different from what is expected.

We can never know for certain when we might have a storm or tsunami until few days before it’s about to happen. Similarly to the weather, chaos is present in our daily life. For example,  the bus you usually take was late and you decided to take another bus, and randomly you meet a person, and you both start talking, he makes an impression on you, you go on a date with him, fall in love, get married and grow old together. Now imagine that the person had a similar situation: he decided to take this bus rather than his usual bus and met you. What if he never got on that bus at the right time to meet you, and what if you had decided to wait for you usual bus? It is scary to think about how one small decision makes such a big difference in your life.

Work Cited

http://whatis.techtarget.com/definition/chaos-theory
http://fractalfoundation.org/resources/what-is-chaos-theory/
http://www.abarim-publications.com/ChaosTheoryIntroduction.html#.VFUVDfnF9Pc
http://www.crcnetbase.com.ezproxy.lib.utah.edu/doi/pdfplus/10.1201/b11408-31
http://bjps.oxfordjournals.org/content/60/1/195.full.pdf+html

Code Breaking: Bletchley Park and Bill Tutte

While brainstorming ideas for a blog post, I found myself wondering if math has ever directly saved lives. After looking into many options, I ran into a story about a place called Bletchley Park. It was known as the Fortress of Secrets and was said to have saved millions of lives yet didn’t even appear on any map. Nicknamed ‘Station X,’ it was solely designated for breaking codes, specifically ciphers. In World War II, direct communication between leaders and various units around the world was a big problem. These orders/war plans were coded and broadcast via wireless radio, but because they could be so easily intercepted, they became increasingly vulnerable. The solution to this problem was the cipher machine. Adopted in 1926, the Germans’ answer was called the Enigma.

A Lorenz machine. Image: Public domain, via Wikimedia Commons.

A Lorenz machine. Image: Public domain, via Wikimedia Commons.

The Enigma was thought by the Germans to be unbreakable and safe for them to use. Its code was especially hard to crack because each time a key was pressed, its internal wiring was changed. In light of this, the British started to recruit brilliant mathematicians to engage in a battle to learn the enemy’s secrets. The Enigma machine required 6 people to operate, so Hitler ordered even more security. Thus, the Tunny cipher machine was born (also known as the Lorenz Machine). This machine generated code with its 12 wheels and only required two operators to send and receive information. To function, it would first apply two keys, encoding the message twice. The first cipher used 5 wheels with, the second used another 5, and then 2 additional wheels would cause a stutter of random letters that would try to throw off unauthorized decoders. Each wheel had a different number of spokes or choices on it which resulted in 23*26*29*31*37*41*43*47*51*53*59*61 = 1.6 million billion possible combinations!

Here is an example of coding one letter into another: The initial letter is “A,” and the cipher code is “K.” They would be “added up,” and if the corresponding symbol was different, then you would mark down an “x.” Inversely, if it was the same, then you would mark down a *. Here we can see A being coded into the letter N:

A= x x * * *

K= x x x x *

———————–

* * x x * = N

The code’s downfall began with the Germans’ overconfidence in the Tunny machine. A 4000 character message was sent, but the receiver didn’t quite get it, so a re-send was requested. The sender failed to change the wheel settings and re-sent the 4000 character message but it was slightly different. This provided Bletchley with a data set with which he could attempt to crack the code. John Tiltman, a mathematician who led the research department at Bletchley, initially worked on this break but passed it onto Bill Tutte.

Bill began by putting the 4000 word message into columns and made a rectangle out of it. He then looked for repetitions/patterns. Every 23 characters there was a rotation, but he then thought maybe it was 25. So he tried to multiply 23*25 to see if the pattern extends along that but it was inconclusive. But the pattern did extend along 574. He then thought maybe it was 41 as that is the prime number of 574. Resonance occurring after 41 strokes made him deduce that the first wheel had 41 spokes. He then went to the next wheel and so forth. Bill Tutte managed to diagnose how the machine worked without ever actually seeing the machine.

He also worked out a statistical mathematical method of breaking the Tunny code, called the “1+2 – break in method.” To use this decoding method required a massive amount of number-crunching and checking. This is where his co-worker Thomas Flower’s brilliance came into play. He conceptualized Bill Tutte’s method and produced one of the first computers, The Colossus, the world’s first semi-programmable computer completely invented from scratch. With this cracked code—and the computer to help crack it—huge battles were won. It is widely credited with turning the tide of the entire war.

World War II is estimated to have cost 10 million lives per year. Cracking the Tunny code was said to have shortened the war by at least two full years! However, everything involving these machines and ciphers had to be kept secret. The brilliant men involved could not publicly get credit for their achievements for quite some time. Eventually, the secrets were declassified, and Bill Tutte was awarded a membership of the Royal Society. In 1987, he finally signed his name in the Royal Society book where his signature lies alongside those of Isaac Newton, Charles Darwin, Winston Churchill.

Sources:

BBC Documentary: http://vimeo.com/31185786

http://www.bletchleypark.org.uk/content/hist/worldwartwo/enigma.rhtm

http://www.bbc.com/news/uk-england-suffolk-29064159

http://www.english-heritage.org.uk/bletchleypark

Making a Pseudosphere

When I learned about models of the hyperbolic “plane” I was intrigued mostly by the Pseudosphere, and how it represents a surface of constant negative curvature, this grabbed my attention so much that I knew I had to have one. And thus began my epic quest to make a rotation of a tractrix about its asymptote.
One thing to note about the tractrix is that it is a curve that you can create through rather simple methods. All you need is to have a string, paper, a pencil, and an object of small but noticeable mass. The tractrix is created by tying one end of your string to the mass. Place the weighted object on the paper as far away as it can go so that your taut string makes a right angle with the edge of your paper. Holding your end of the string in your hand at the edge of your paper, move your string along the edge of the paper so that your hand follows a straight path. As you move the string, the weight will begin to move. Now mark the location of your mass every centimeter or so. This curve is called the tractrix.
There are many methods to create this shape, one of the more popular modern methods for making these sorts of shapes is on a 3D printer. Since I don’t have a 3D printer, I decided that I would have to make this Pseudosphere by hand which is obviously much cooler. The method I chose was a lathe, which is a tool that turns a chunk of wood at high speeds, allowing the woodworker to remove parts of the wood, yielding radial symmetry about the axis of rotation.
The first thing I had to do was determine how large I wanted my Pseudosphere to be. To do that, I looked up the equation for a Pseudosphere, which is defined parametrically.

Parametric equations for a pseudosphere.

Parametric equations for a pseudosphere.

The constant value “a” can be changed at will, so I decided to make “a” be 3 which would give me a main diameter of 6. At a length of 10 inches, five on each side, I would have sufficient width at the ends to maintain structural integrity. The units I decided to use were inches, due to the fact that I am already familiar with them, and my measuring tools are all in inches.

One-inch blocks of wood glued together before lathing.

One-inch blocks of wood glued together before lathing.

For my choice of wood, I had two options. The first was take a big log and try to cut it down to the desired size. Due to the fact that it would take a while to cut a six inch diameter log that is ten inches long down to the proper size, and since large logs often have splits in them, I chose another option. That was taking a series of one inch thick squares of wood and gluing them together in a pattern roughly resembling the shape of the Pseudosphere.
Using a trace of the tractrix, my father and I determined the placement of the blocks and began the process of gluing pieces together in the right configuration. The gluing process took two days to complete, because I didn’t want my project flying to pieces for a lack of patience on my part.  Essentially, the glue had to set up to full strength to withstand the turning.

An intermediate pseudosphere.

An intermediate pseudosphere.

Once the pieces were all glued together, I had the momentous task of taking squares, and rounding them out to a closer approximation of a Pseudosphere. The process wasn’t easy, throughout the whole project I had a sore hand from having to hold onto the tool through all the jerking motions made by the heavy wood slamming into the metal cutting tool.
After having removed all of the corners, I took calipers, and used them to determine the diameter that the Pseudosphere should have at every inch. Cutting down to those points, I began slowly, and carefully working inwards, trying to get the right curvature. Because the natural tendency of all of these tools is to cut a straight line across a surface, I had to pay special attention to the tool’s path, so that I could get a constantly changing curvature throughout the whole length of the Pseudosphere. There were some tense moments when the knife didn’t behave in the way I was intending, and I cut too deeply, but the wonders of foresight helped, because I had, at the beginning given myself a 1/8 inch cushion in my measurements, so that I could sand, and perform the finishing touches properly.

The finished pseudosphere.

The finished pseudosphere.

The sanding was a fun process, the high friction caused a heat buildup in the sandpaper, making it necessary to use a glove. Once we had it all sanded we applied a finishing oil called “Tung Oil” to give the wood a darker, richer color. It also smells pretty good.
In the end I was left with a wonderful mathematical object. There were some questions from people, such as, why I didn’t make the Pseudosphere infinitely long. My response is twofold. One I would have needed infinite wood. Despite the fact that a Pseudosphere has finite volume, the cylinder I would have needed to start with would have needed to be of infinite volume. Also structural stability had a role to play there, I can only make something so thin and rotate it at high speeds without it suffering a physical breakdown.
I do however hope you like my project, and remember, our lives are only as interesting as we decide to make them.

-Travis

Sources
http://mathworld.wolfram.com/Tractricoid.html

Image credits
(I made them myself, don’t tell anyone)

Zu Chongzhi and his mathematics

https://upload.wikimedia.org/wikipedia/commons/thumb/0/07/%E6%98%86%E5%B1%B1%E4%BA%AD%E6%9E%97%E5%85%AC%E5%9B%AD%E7%A5%96%E5%86%B2%E4%B9%8B%E5%83%8F.jpg/1600px-%E6%98%86%E5%B1%B1%E4%BA%AD%E6%9E%97%E5%85%AC%E5%9B%AD%E7%A5%96%E5%86%B2%E4%B9%8B%E5%83%8F.jpg

Image:Zu Chongzhi. Author: Gisling, via Wikimedia Commons.

Zu Chongzhi was a Chinese mathematician and astronomer during the Liu Song and Southern Qi Dynasties. He did a lot of famous mathematics during his life. His three most important contributions were studying The Nine Chapters on the Mathematical Art, calculating pi, and calculating the volume of sphere.

As we know,  The Nine Chapters on the Mathematical Art is the most famous book in the history of Chinese mathematics. In ancient China, most people could not understand “The Nine Chapters on the Mathematical Art”. Zu Chongzi read the book and then he used his comprehension to explain the formulas of the book. Zu Chongzhi, and his father wrote the “Zhui Shu”(缀术) together. The book made The Nine Chapters on the Mathematical Art easier to read. And the book also added some important formula by Zu. For example, the calculation of pi and the calculation of sphere volume. “Zhui Shu” also become math textbook at the Tang Dynasty Imperial Academy. Unfortunately, the book was lost in the Northern Song Dynasty.

Zu’s ratio, also called milü is named after Zu Chongzhi. Zu’s ratio was an early accurate approximation of pi. It was recorded in the “Book Of Sui” and “Zhui Shu”. (Book Of Sui is the official history of the Sui dynasty). According to the “Book Of Sui”, Zu Chongzhi discovered that pi is between 3.14159276 and 3.14159277. Today, we know the actual number is in accord with Zu’s ratio. But “Book Of Sui” did not record the method used to get the number. Most historians and mathematicians think Zu Chongzhi used Liu Hui’s π algorithm to get the number. Liu Hui’s algorithm means approximating circle with a 24,576 sided polygon. Japanese mathematician Yoshio Mikami pointed out, “22/7 was nothing more than the π value obtained several hundred years earlier by the Greek mathematician Archimedes, however milü π = 355/113 could not be found in any Greek, Indian or Arabian manuscripts, not until 1585 Dutch mathematician Adriaan Anthoniszoom obtained this fraction; the Chinese possessed this most extraordinary fraction over a whole millennium earlier than Europe”. Hence Mikami strongly urged that the fraction 355/113 be named after Zu Chongzhi as Zu’s fraction.( Yoshio Mikami)

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Image:Zu Chongzhi’s method (similar to Cavalieri’s principle) for calculating a sphere’s volume includes calculating the volume of a bicylinder. Author: Chen Bai, via WIkimedia Commons.

Zu Chongzhi’s other important contribution was calculation volume of the sphere. Together with his son Zu Geng, Zu Chongzhi used an ingenious method to determine the volume of the sphere.(Arthur Mazer). In The Nine Chapters on the Mathematical Art, the author used Steinmetz solid to get the volume of the sphere. The solid common to two (or three) right circular cylinders of equal radii intersecting at right angles is called the Steinmetz solid.

https://i0.wp.com/upload.wikimedia.org/wikipedia/commons/2/20/Bicylinder_Steinmetz_solid.gif

Image:Steinmetz solid. Author: Van helsing, via Wikimedia Commons.

But the book did not give the formula of how to get the volume of the sphere. Zu Chongzhi used “Zu Geng principle” (another name: Cavalieri’s principle) to show the volume of the sphere formula is (π*d³)/6. In order to commemorate the fact that Zu Chongzhi found the significant contribution of the principle with his son, people called the principle “Zu Geng principle”. “Zu Geng principle” is the same as “Cavalieri’s principle”, but “Zu Geng principle” is earlier than “Cavalieri’s principle”. “Cavalieri’s principle” means two solids of equal altitude, the sections made by planes parallel to and at the same distance from their respective bases are always equal, then the volumes of the two solids are equal.(Kern and Bland 1948, p. 26).

Work cited:

Yoshio Mikami , (1947). Development of Mathematics in China and Japan. 2nd ed. : Chelsea Pub Co;.

Arthur Mazer , (2010). The Ellipse: A Historical and Mathematical Journey. 1st ed. : Wiley;

Kern, W. F. and Bland, J. R. “Cavalieri’s Theorem” and “Proof of Cavalieri’s Theorem.” §11 and 49 in Solid Mensuration with Proofs, 2nd ed. New York: Wiley, pp. 25-27 and 145-146, 1948.

http://en.wikipedia.org/wiki/Cavalieri%27s_principle

http://en.wikipedia.org/wiki/Zu_Chongzhi