Projects

The projects on the "top 10 moments in mathematics" were fascinating. Some have narrowed down the topics to particular topics like "calculus" or even "Archimedes" or "Architecture". Here are the topics in 2011. Compare the topics in 2010. Both years, Pythagoras was the clear winner.

An example of a project: Great Math moments by Alexandra Vinegar.
A project by Karen Sokolow [PDF]

Twelfth lecture: Computing

We had started the lecture with a demonstration, what experimental mathematics is in the context of Benfords law, where lots of questions are still to be settled.

Experimental math journal

Eleventh lecture: Dynamical systems

Here is Paul Abbott' version of my demonstration of nondistributivity:

T[x_] := 4 x (1 - x); A = NestList[T, 0.3`50, 100];
S[x_] := 4 x - 4 x^2; B = NestList[S, 0.3`50, 100];
Transpose[{A, B}]

When looking for "chaotic buttons" on calculators, Matt found that "Tan" using "Rad".

Tenth lecture: Cryptology

Jennifer pointed out that Bernardo Provenzano was caught in 2006 because of Cesar Cypher use.
A current Crypto mystery from the FBI. Thanks to Anne.
A recent article (in german) about cryptography in a Swiss Newspaper. Mentions
1. 500 BC Skytale of Sparta
2. 100 BC Cesar
3. 1910 Cyphermachines.
4. One pad systems.
5. Public key systems
6. Quantum computing

Ninth lecture: Analysis

Here is an improved animation of the Pythagorean tree.

Snowdodecahedrons (Thanks to Christine)

Seventh lecture: Probability

The Roulette doubling paradox, which is at the start of Martingale theory was quite fun. We played in class Roulette with this game and played the doubling strategy a few times. Bet on Black, doubling each time until we win. The Martingale betting system is interesting, because our experience clashes with the mathematics. The system works with such high probability, that we become convinced that it will always do so. If we play with 1 dollar and come to the Casino every night with 40 grands (so that we can bet) and the table stake limit is 50'000, we can go to the Casino for 178 years until we expect to lose. We only get busted 16 times black fails to show up since 2¹⁶=65'536. Comparing this with other risks in live this is a pretty sure thing. If there were no table stakes and we had unlimited funds, the strategy would always work. It does not contradict the fact that the expected win is always zero, because having won larger and larger sums, we also increase the possibility to lose larger and larger sums. There is no loss if we agree to play until we winning event happens and we are allowed to borrow arbitrary much money.

The question is related to risk considerations in economics or environmental questions, where we ask for "reasonable risks", estimate risks which occur with very low probability like a nuclear disaster, an earth quake, an oil spill, a financial melt down, a data disaster in a "cloud computing company", or meteor crash. The risks are often very low, but as longer the time spans, the more probable they become. The question is similar for the Petersburg paradox, where no sane human would pay an entrance fee of 100 dollars even so we expect to win a lot with this entrance fee in the long term. Its just that we would not expect to win it in our life time.

After class, Alexandra showed a nice card trick: sort a deck of 52 cards with alternating colors. Split the so sorted deck into two decks (making sure that the top colors are different) then shuffle once. Now pull off pairs of cards. They always have different colors. This is an amazing trick because everybody thinks that something with the shuffling is fishy. However, the trick always works and there is no cheating. Assume that we have only 14 cards. We have already split the deck. Shuffling corresponds to find a path through all the cards hitting every card exactly once going left, right or down. Here are the two decks:

1         0
0         1
1         0
0         1
1         0
0         1
1         0
0         1
1         0
0         1
1         0
0         1
1         0

Now walk through the two decks, moving from the left or right forth and back making sure no card is left out. Successive pairs of cards still have different colors. After having been shown the trick, I was sure something with the shuffling was done wrong. But its simple combinatorics and our intuition fails to grasp this when we see it in action.

After talking about Monty Hall, Anne mentioned a which is shown on 'deal or no deal' game TV here and in many other countries. Here is the US version and on YouTube.

Alexandra also pointed out a remotely related to probability but it shows that despite some randomness in a problem there can be order: it is the wine bottle problem: You stack identical wine bottles in a rectangular wine rack. At the bottom of the wine rack, there is room for more than three wine bottles, but not four. For the first layer, you place two bottles in the corners and one in between. For the second layer, you place two wine bottles each of which settles and rests upon two of the wine bottles in the first layer. Of course, the two wine bottles in the second layer won't likely be at the same height because that only happens if the middle bottle in the first layer is exactly centered. Three more wine bottles are placed inside to make a third layer with two of these resting on the sides of the rack. A fourth layer is added consisting of two more wine bottles. Like the second layer, each of these wine bottles rests upon two wine bottles in the third layer. So far, only the first of these layers is guaranteed to be perfectly horizontal. The fourth layer could be tilted quite a bit, in fact! Prove that when you add a fifth layer of three wine bottles, it will be exactly horizontal!

Added on August 9, 2011: An article featuring Bayes theorem in the New York Times. Bayes theorem is extremely popular among mathematical educated scientists. Here is my Math 19b handout about it. It is an amazing theorem because the ratio "results it applies"/"effort to prove it" is so big. Mathematicians call the theorem "trivial". Otto Bretscher once told me that Maurice Auslander has once said (Otto had been at Brandeis when Auslander was there), that "Mathematical theories should be built in such a way that theorems do not have proofs". This is the case with the Bayes story. The definitions of a probability space due to Kolmogorov are so crystal clear, the definition of conditional probability P[A|B] = P[A intersect B]/P[B] simple and the consequence that P[A|B] P[B] = P[B|A] P[A] needs no proof because it follows directly from the definition. This is Auslander's vision of mathematics in action. When the above identity is solved for P[B|A] and the trivial substitution P[A] = P[A cap B] + P[A cap B^c] is done, then we get Bayes theorem. The implications of Bayes theorem can be amazing however as Monty Hall Riddles going back to Gardner and variants show. Both mathematitians and nonmathematicians are mesmerized by the theorem because it is often counter intuitive. I only this spring added an other example to the Boy/Girl problem to my Probability text after being taught by one of my students of this class. The example is due to Gary Foshee: "Dave has two children, one of whom is a boy born at night. What is the probability that Dave has two boys". It is amazing that the information of "being born at night" makes the probability bigger than 1/3. See page 30/31 in my text.

Sixth lecture: Set theory

Cardinality: We talked about the cardinality of numbers.

N < Z < Q < A < R < C

where N= natural numbers, Z=integers, Q=rationals, A=algebraic numbers and R = Real numberes, C= Complex numbers. One observation which came up is that the term "natural number" is not uniformly used. Some literature uses "whole number" for 0,1,2,3,4.... and "counting number" for 1,2,3,4,5... Whether the set "natural numbers" contains zero is not always clear. We briefly looked at the task to enumerate the polynomials with integer coefficients. Is there simple way to write down a function f(n) which gives the n'th polynomial counting all. There is one as we know since Cantor, but is there a nice one which one can teach in a few minutes? Paradoxa: We also looked at the Berry paradox which asks for the The smallest integer not definable in less than 11 words. To this, Ann sent this clip. The Berry paradox is puzzling since we can enumerate all the words of length 11. There are maybe 10⁶)¹¹ words of this length. Some of these sentences define integers. We can make a list of all these integers and take now the smallest number which is not on the list.

Source: About proofs Thanks Karen.

Fifth lecture: Algebra

The German journal "Spiegel" reports about an exhibit at the FU berlin, where visitors can play with symmetries. Here is the article.

About the 15 puzzle. For every square n² there is a n²-1 puzzle of course. Here is a 8 puzzle. For a 3 puzzle the structure of permutations is much easier to see

Forth lecture: Number Theory

Paul Abbott sent me the following note:

Here is your code for Gilbreath's conjecture:

s0 = Table[Prime[n], {n, 1, 10}];
diff[s_]:=Append[Table[Abs[s[[n+1]]-s[[n]]],{n,Length[s]-1}],0]
NestList[diff, s0, Length[s0]]

Here is a simpler alternative using Range, Differences, and PadRight:

With[{n=10},NestList[v\[Function]PadRight[Abs[Differences[v],n],Prime[Range[n]],n]]

Gilbreath's conjecture is usually stated in terms of 
the absolute value of the differences. Here is a simple explorer:

Manipulate[NestList[v\[Function]PadRight[Abs[Differences[v]],n],Prime[Range[n]],n],{{n,10},Range[50]}]

ArrayPlot[With[{n=300},Drop[NestList[v\[Function]\[LeftBracketingBar]
Differences[v]\[RightBracketingBar],Prime[Range[n]],n],20]]]

Visualizing the sequence using ArrayPlot one sees the affect of different paddings p.: 
The Rule 90 cellular automaton controls the behavior of rows that contain only the values 0 and 2:

Manipulate[ArrayPlot[Drop[NestList[v\[Function]PadRight[\[LeftBracketingBar]
Differences[v]\[RightBracketingBar],m,p],Prime[Range[m]],n],30],ImageSize->600],{{n,600},200Range[5]},
{{m,300},100Range[5]},{{p,2},Range[0,4]}]

When discussing the Ulam number spiral, we looked briefly for other ways to arrange the number line. One idea was to put the Ulam spiral onto a surface. Alexandra had the idea of adapting the curvature so that patterns would be visible. For the next picture, I placed the Ulam spiral onto a cone. Cubes were placed in space and colored according to whether they belong to a prime or not.

Third lecture: Geometry

After the presentation, we worked on the Morley Miracle. This remarkable result was obtained in 1899 by Frank Morley who taught at Haverford in Pennsylvania and later at John Hopkins. It took years to come up with a geometric proof. You find the animation here. The computer projector in the classroom unfortunately did not render the red colors. Here is the animation of the Pythagoras proof. To work on the Morley miracle, here are the files: [PS], [PDF], [PNG].

After trying ourselves at the proof, we looked in class at the constructive proof of Conway, where 7 triangles are built and then fit together to an original triangle. Morley's miracle is an excellent laboratory for experimentation. While preparing for this lecture, I had asked myself for example, whether the (1/3,1/3,1/3) angle trisection could be changed to (a,1-2a,a) and whether there would still be an equilateral triangle formed. The movie to the right shows an experiment. While it is close, it is clear that the miracle has disappeared.

Frank Morley (1860-1937). Image source.

Panorama pictures taken while working on the Morley miracle. There is geometry involved in unwrapping the picture. Here is what the camera saw:

Second lecture: Arithmetic

After a rather lengthy powerpoint show, we have written some Clay tablets. Here is an exhibit. It was illuminating to actually do it. I did not know before for example what was the reason for the wedge shaped signs. It is very easy to tilt a rectangular stick in a way to write fast and still distinguish the different directions. Here is a Youtube Video explaining it.

We also worked on showing that numbers are irrational with some presentations on the blackboard.

First lecture

In the first lecture, we have looked at a theorem about towns. The theorem tells that if H is the number of houses and W the number of windows in a town, then the total curvature adds up to 6(H-W). It is an incarnation of a theorem called Gauss-Bonnet theorem.

Here is an illustration.

Some observations about the lecture:

The class had been faster than anticipated to compute the curvatures. It might be that even out in the field, more complicated examples could be used. Best would probably be to have students draw their own houses and towns and experiment.

After having done the experiments, we looked at the case where two bricks touch each other which produces a counter example to the theorem. The class could save the theorem by assigning to the intersection point the value 2 and consider the situation as an example with 2 houses. So, by adding an additional rule that one assigns a value 2 to this points, the theorem stays true. An other possibility, which works and was discovered by the class is to assign the value -4 to such points and to consider the two bricks belong to the same house.

The class was most interested, how the numbers popped up. The rule is to take the circle s(x) of radius 1 centered at x and look at its length |s(x)|. Define K(x) = 6-|s(x)| for points in the interior of the house and K(x) = 3-|s(x)| for points at the boundary. One student explored the case of a "plane", where every square is filled with a brick. In that case, there are only interior points and the curvature is 2 or -2. In average the curvature is zero. The plane is is a "flat" object.

One can explore towns also in space. It could be an interesting project for students to explore what numbers one has to attach to the nodes so that the theorem works. The theorem is still true. The circles of radius 1 play an important role. In the picture to the left for example, the total curvature would be zero. There is no window in the sense we have seen before but there is a global "hole" in the doughnut. Assume we take a single connected surface and tile it with squares and add up the curvature, then the total curvature is still 6-6g, where g is the genus, the number of "holes". In the doughnut case displayed to the left there is one hole and the total curvature adds up to 0.

The theorem worked well to explain an important theorem in mathematics on a level which is accessible in a modest time. While the actual Gauss-Bonnet theorem involves quite a bit of mathematics to define curvature, genus, Euler characteristic etc. the graph theory version is much easier to deal with. One can experiment with a piece of squared paper, draw a town and count the number of houses and the number of windows. Mathematicians have fancier names: the number of houses is called b₀, the zero'th Betti number and is the dimension of the cohomology group H⁰. The number of windows is the first Betti number and is the dimension of the cohomology group H¹. The topic touches concepts in algebraic topology, where one tries to understand geometric objects with algebraic tools.

The announcement before the first lecture: Welcome to Math E-320 2011. Here is the syllabus. See also the website from Last semesters course. This class meets Mondays in Science Center, in 113 from 5:30 PM to 7:30 PM. We start on January 24th. John Stillwell's book "Mathematics and its History" is a good read besides what we do in class. In the first class on January 24th, we talk about mathematics in general and see a short presentation. We will work then on a "mystery problem" together. That will occupy us for an hour at least. We finish with a short quiz about the presentation and the worksheet. Come with an open mind. There is no need to prepare. If you want, have a look at this handout.