Math E-320: Fall 2018
Teaching Math with a Historical Perspective
Mathematics E-320: Fall 2017
Instructor: Oliver Knill
Office: SciCtr 432

## Lecture 13: Computer Science

The lecture on computer science had 4 parts: 1) Experimental Mathematics, 2) The history of computing, 3) The future of computing 3) The limits of computing and 4) Artificial intelligence. Let us add a few links to each
• About Experimental mathematics: we started with the Benford law. I actually learned that as an undergraduate from a book of Vladimir Arnold. who gave it as an exercise related to ergodic theory. Also as an undergraduate, I had been fascinated by gradus suavitatis. I learned it from a book of Wille. I wrote this Mathematik.com page in 2000. It was updated last in the year 2012. The piano animation there was rendered in Povray on April 12, 2000. The animation is in Javascript. It actually plays meaningful accords as you can see when looking at the source. And here are some notes on exeriments in number theory. It mentions the almost periodic matrices shown here:
• About the history of computing: here is a page about the TI 59 Hardware hack. It is actually quite amazing that that calculator survived all this brain surgery. But the interface was also very primitive. The joy stick would tap to some keys of the calculator and the external flag would tap into the display. We had been very fond of these calculators. Here is a photo of our Highschool class during a geology trip in south Germany. [I'm in the middle with the bright shirt and to my left is Christian with a white collar shirt who also was a TI 59 owner. Christian and I would race for quite a few mathematical quests (which he usually won) like the Rubik cube or finding experimentally a formula for the quadratic equation before it was covered in class (we derived it independently just by data fitting using lots and lots of examples). We also did the Rubik cube independently. It would have been considered a huge cheat to look up a solution strategy but it takes weeks and weeks to come up with a strategy. ] Here is a great computer museum featuring many old computers. Until 2010, I kept track of my own computers here. In those years, every new computer had been a mile stone. I will soon be up to an upgrade again.
• I wrote once in 2005 a handout on Turing machines from a dynamical systems point of view. A Javascript illustration of Turing machines was implemented in 2000 in Mathematik.com. This website is a bit in a limbo since decades. At one point, I will have to relaunch it but there are other things to do now.
• About AI: In 2003, with support of the Provost's fund, a few students and I worked on some AI which can do mathematics. Here is a writeup. We used pre-existing Chat bots first and spent a lot of time teaching them math which essentially means collecting knowledge. Of course, today, Google or Wolfram Alpha or Siri or Alexa etc are all much smarter. But what was really innovative in that experiment was that we would implement "context" using cookies (this was done by Andrew Chi and Johnny Carlson), also innovative was that the ability to learn (Mark Lezama) as well that the bots could talk to each other. The implementation was very primitive:
• 1. Take the sentence from the partner. Using simple Perl routines, simplify the sentence. For example "find", "compute", "determine" etc would all go to compute.
• 2. The simplified sentence would directly be translated into a directory tree on the Unix machine. For example "What is the line integral of the vector field F = along the path r(t) = from 0 to 2Pi would be translated to "compute/lineIntegral/run({x,y,sin(z)},{cos(t),sin(t),t})" where run is Mathematica code which can solve the problem.
• 3. The machine would then give the result packed into a meaningful sentence like. "The line integral is .....
```[knill:] grep "Simplicial Complex" */*.txt */*/*.txt */*/*.txt|wc
184    1058   15642
```
Fortunately, OS X is Unix based, which makes it convenient to work with. Here is how I count how many PDF's are in my math library:
```cd ~/ebooks; tree|grep pdf|wc
7028   26796  327121
```
Now, I could pack that into Sofia. It can be hardwired as "HowManyBooksMathLibrary/run.sh" where run.sh contains the shell command
```echo "There are `cd ~/ebooks/Math; tree|grep pdf|wc|awk '{print \$1}'` Math books in your library, Oliver"
```
giving the answer of Sofia. It turns out however that one is much more efficient by just typing the command directly instead of asking sofia. Interfaces like Siri, Alexa etc are consumer end products. A good knowledge of a dozen computer programming languages and a decent amount of Mathematics background allows you to data mine much more efficiently than these machines. The scary thing is that the machines learn fast. And they also learn how to learn and how to program. Yes, programs can program too. The probably simplest example is the self reproducing program. I have often programs spit out programs like for example to write javascript experiments. But I did not type that Javascript. I programmed Mathematica to write Javascript. Sometimes, I even had sometimes to write Mathematica write Mathematica, like if a subprocess needed to run in a sandbox using libraries which would destroy the current computation.

## Lecture 12: Dynamical systems

1. Game of life in Javascript (2015)
2. Here is the Mathematica code for the amusical sequence of Conway. It is not known what happens with the orbit of 8. Conway suggests that it grows to infinity as "probabilistically obvious" or "Probvious". Why is it most likely true? The chance that it will fall back becomes smaller and smaller as larger as the numbers become. It is amusing.
```T[x_] := If[EvenQ[x], 3 x/2,
If[Mod[x, 4] == 1, (3 x + 1)/4, (3 x - 1)/4]];
ListPlot[Log[NestList[T, 8, 100000]]]
```
In a blog of 2015 of this course we gave the 100'000'th iterate. Here is the one-million'th. The chance to fall back is much less than 1060000. Experimentally, we see that sequence grows exponentially with a growth rate of 10n/17. Just for amusement, here is the 1 Millionth number. It has already 25'654 digits:
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47568165995506822886924555785139184
```

## Lecture 11: Cryptology

The crypto lecture has three or four parts, depending on how fast we progress:
1. Early Cryptography until polyalphabetic cyphers.
2. The Enigma: (mostly the history of this fascinating story.
3. Public Key: Diffie-Hellman and Rivest-Shamir-Adleman. How do these protocols work?
4. If time permits, we look also at the problem of error correction or what quantum cryptography is (more informal).
Cryptology is a hot area of math. It has lots of overlap with engineering, with computer science and politics! But it is now mostly hard core mathematics. It is a place, where number theory plays a major role.
• BBC from April 2017: Two graduate students stood silently beside a lectern, listening as their professor presented their work to a conference. Usually, the students would want the glory. And they had, just a couple of days previously. But their families talked them out of it. A few weeks earlier, the Stanford researchers had received an unsettling letter from a shadowy US government agency. If they publicly discussed their findings, the letter said, it would be deemed legally equivalent to exporting nuclear arms to a hostile foreign power. Stanford's lawyer said he thought they could defend any case by citing the First Amendment's protection of free speech. But the university could cover legal costs only for professors. So the students were persuaded to keep schtum. What was this information that US spooks considered so dangerous? Were the students proposing to read out the genetic code of smallpox or lift the lid on some shocking presidential conspiracy? No: they were planning to give the International Symposium on Information Theory an update on their work on public key cryptography.
• Trapdoored primes. From that article However, due to a new phenomenon known as trapdoored primes, described in the paper "A Kilobit Hidden SNFS Discrete Logarithm Computation," successful attacks on 1024-bit keys are no longer theoretical. Trapdoored primes allow an attacker to efficiently break certain 1024-bit keys to decrypt communications and cryptographically impersonate key owners to sign data, all unbeknownst to the victim. The security of many encryption systems is based on mathematical problems involving prime numbers so large that the problems are prohibitively hard for attackers to solve -- a discrete logarithm problem. Unlike prime numbers in RSA keys, which are always supposed to be unique, the primes used by Diffie-Hellman and DSA are frequently standardized, and used by a large number of applications. There is the possibility that some of these primes have been trapdoored. These are specially crafted prime numbers, where the special number field sieve, a special-purpose integer factorization algorithm, can be used to solve the discrete logarithm problem that underpins the key's security. It makes breaking a trapdoored 1024-bit prime at least 10,000 times easier.
• Adleman in the news (Nov 1, 2017): how the word "computer virus" became viral.
• History of Alice and Bob. In our class, we use not Alice and Bob and Eve but Ana, Bob and Eve for the simple reason that now, all three are palindromes. From that article: Alice and Bob are the hypothetical communicants in every cryptographic example or explainer, two people trying to talk with one another without being thwarted or overheard by Eve, Mallory and their legion of nefarious friends. Alice and Bob's first known appearance was in Rivest, Shamir, and Adleman's 1978 Communications in the ACM paper, "A Method for Obtaining Digital Signatures and Public-key Cryptosystems." Since then, they have enjoyed many adventures.
• Shamir wins 2017 Japan Prize for his work in the field of cryptography.
• Update: December 15, 2017: Mathematical Backdoors in Cryptology: cite: "Research on mathematical backdoors is much more difficult (mathematical stuff) - and does not attract researchers that need to publish quickly and regularly on fashionable topics," Filiol added. "This is the reason why this kind of research is essentially done in R&D lab of intelligence agencies (GCHQ, NSA...) and [is designed] more for designing backdoors that detecting them."

## Lecture 10: Analysis

• Here is a fractal of Teresa, Rebekah and Jorge and Doni and Oscar:
• Nancy shared this movie:
from the MIT K12 Viceos.
• Here is the Mandelbulb zoom out by Russ McClay shown at the end of the class:
It has been animated for some time now. this is a classic from 7 years ago. Folks are really creative here: Here is a more artistic use of the algorithm. It must be a modification of the Mandelbulb algorithm. One reason, why no mathematics is available for the Mandelbulb is that there is no complex analysis available in 3D. Here is the detailed proof of the connectivity of the Mandelbrot set. I learned the Douady-Hubbard proof in a one of the great Moser Seminars during graduate school. The paper of Douady and Hubbard is from 1982, two hears after the discovery of Mandelbrot and 4 years after the discovery of Brooks-Matelsky. The Moser seminar was in 1985 or 1986. [ It is funny how things change. At that time, 1985 (the arrogance of youth), I had found the Mandelbrot set already "a dusty object" I have seen hordes of computer scientists program and animate the Mandelbrot set already in 1982, as a freshmen. I was in my first year essentially living in the computer labs.] The dynamical systems seminar presentation then had been done by Jochen Denzler who wrote both a senior thesis (on Aubry-Mather theory) as well as his thesis (integrable PDE's) with Moser. I myself actually presented a stability result on higher dimensional complex dynamics in that undergraduate student seminar. This is an area of complex dynamics which is still very much in development now. Moser had great taste in choosing presentation topics. But what the Mandelbulb deals with is the "real world". It is not an iteration in a complex manifold but a real manifold. This reduces the availability of tools. Maybe, one still has to build the necessary foundations, similarly as Cauchy did, when building complex analysis as we know it now.
• Here is an amazing Menger Sponge animation in 511 Bytes by Mathieu 'p01' Henri. The camera travels in the fractal along a cycloid. The rendering is done in Canvas and ImageData changes the Alpha channel. This guy is a fantastic programmer. He manged to program a dragon curve in 121 Bytes: Here is the page.
• Online Mandelbrot viewers:
1. 1,
2. 2,
3. 3.
4. A Terrain generator in Javascript from This page Here is an other one, which allows custimization:

## Lecture 9: Topology

• Here is a diagram in the book of Davis and Hersch about the discovery paradigm of Lacatos:
Here is the book title:
• The topology lecture will have 4 parts: 1. Rubber geometry. We look at topological equivalence of various things, letters, objects etc, especially look at connectivity. 2. Polyhedra. This is the core of the lectures. We will see the Euler Gem Formula V-E+F =2 for spherical polyhedra, look at semi-regular and higher dimensional versions. We want to understand well why there are exactly 5 Platonic solids. 3. Euler characteristic. We will compute this quantity in some examples and see it for discrete networks or triangulations of surfaces. The last section "4. Strange worlds" will show how weird topology can be. There are spheres for which the outside is very complicated. One can turn a sphere inside out without ripping it apart etc.

## Lecture 8: Probability theory

• Reading the submissions for the homework (compute the probability that a hand of cards in the game "set" gives a "set) was very interesting. There were various approaches. Brute force, looking up, a small argument, or a complicated new argument which usually failed. I must say that when I worked on the problem first without looking up the solution, I got several wrong arguments my self. It is a bit tricky to see things from the right angle. This is typical for probabilistic arguments. Lets first look at the small game with 27 cards. Here is my brute force computation of the set of sets. The program just lists all the game, then picks all hands, and the selects all "sets". The answer is 1/25. (I was doing this brute force listing also to make sure that the result 1/25 is correct). The shortest explanation is that if we pick two cards, then, in order to have a "set", the third card is determined. Only one in 25 cases gives a set.
```(* Mathematica lists of "sets" in the small game of sets, Oliver Knill, October 30, 2017 *)
Game=Partition[Flatten[Table[{i,j,k},{i,3},{j,3},{k,3}]],3];
SetQ[{x_,y_,z_}]:=
(x[[1]]==y[[1]]==z[[1]] || (x[[1]]!=y[[1]] && x[[1]] !=z[[1]] && y[[1]] !=z[[1]]) ) &&
(x[[2]]==y[[2]]==z[[2]] || (x[[2]]!=y[[2]] && x[[2]] !=z[[2]] && y[[2]] !=z[[2]]) ) &&
(x[[3]]==y[[3]]==z[[3]] || (x[[3]]!=y[[3]] && x[[3]] !=z[[3]] && y[[3]] !=z[[3]]) );
Hands=Subsets[Game,{3,3}];
Sets={}; Do[If[SetQ[Hands[[k]]],Sets=Append[Sets,Hands[[k]]]],{k,Length[Hands]}];
Length[Sets]/Length[Hands]
```
And here is a similar brute force determination of sets in the large game with 81 cards. Also here, given two of the 81 cards, the third is determined. Only one of the 79 remaining cards gives a set.
```(* Mathematica lists of "sets" in the large game of sets, Oliver Knill, October 30, 2017 *)
Game=Partition[Flatten[Table[{i,j,k,l},{i,3},{j,3},{k,3},{l,3}]],4];
SetQ[{x_,y_,z_}]:=
(x[[1]]==y[[1]]==z[[1]] || (x[[1]]!=y[[1]] && x[[1]] !=z[[1]] && y[[1]] !=z[[1]]) ) &&
(x[[2]]==y[[2]]==z[[2]] || (x[[2]]!=y[[2]] && x[[2]] !=z[[2]] && y[[2]] !=z[[2]]) ) &&
(x[[3]]==y[[3]]==z[[3]] || (x[[3]]!=y[[3]] && x[[3]] !=z[[3]] && y[[3]] !=z[[3]]) ) &&
(x[[4]]==y[[4]]==z[[4]] || (x[[4]]!=y[[4]] && x[[4]] !=z[[4]] && y[[4]] !=z[[4]]) );
Hands=Subsets[Game,{3,3}];
Sets={}; Do[If[SetQ[Hands[[k]]],Sets=Append[Sets,Hands[[k]]]],{k,Length[Hands]}];
Length[Sets]/Length[Hands]
```
• During the lecture, the question came up, how many time the letter "x" appears in the "hound of Baskerville". Actually, it appears 366 times. The first word, with x is in the word "examination" in Chapter I of the book. One can get a text version on the Gutenberg library.
• An other question which came up is how many letters a typewriter had at the time when Arthur Conan Doyle wrote his novel. He lived from 1859 to 1930. As you see, the typewrites from around 1900 had about 30 keys. So, in the monkey typing Shakespeare problem, we really should have about 30^326694 possible novels of the length of the hound of Baskerville.
• I have used the word "odds" as a synonym with "probability" but indeed as pointed out during class, there is a difference. It is explained here: on a page.
• After class, Teri pointed out that the Price is right format is similar but more complicated than the Monty Hall problem.
• In the probability theory lecture, we discussed first the foundations of probability theory as formulated by Kolmogorov: there is a laboratory X, containing points x which are experiments. Subsets are called "events". There is a probability function P which assigns to every event a number in the interval [0,1] such that the probability of the entire laboratory is 1. An example is when we throw two dice. The laboratory contains all 36 pairs (a,b) with a,b numbers in {1,2,3,4,5,6}. An event can be the situation that the sum of the numbers is even. We can get the probability of this event by counting the number of experiments there and divide by the number of points in the laboratory. In the second part of the lecture, we will look at 4 basic constructs in probability theory: permutations and combinations. In a third part, we look at paradoxa, in the fourth part we look at stochastic processes like random walks.

## Lecture 7: Set theory

• Some have pointed out that other Greek thinkers have come up with paradoxa before the liars paradox. Epicurus (problem of Evils) and Aristotle (wheel paradox) came up with paradoxa. We had not talked about them.
• We have seen a short clip about Cantor during class. Danny mentioned that the "story of math" is available also on Netflix. On an other note, the Venn diagram appeared in the Late show of Colbert on October 18, 2016 at around minute 5:30. That had been before the elections when very few would have predicted the outcome of the elections right. We will talk about probability theory in the next lecture. Probability theory comes in situations where one has little knowledge about the model and assumes the unknown parameters to be random.
• In the set theory lecture, we looked at the foundations of mathematics, explore what infinity means and look at paradoxa and theorems which have shaken the foundations of mathematics. This is not completely esoteric or philosophical. The theory has very concrete applications. One of the current problems with Wireless spots (WPA2) is a vulnerability which is based on simple Boolean operations. We will start the lecture with explaining what a set is (using a concrete example) and then learn how to calculus in the Boolean algebra. At the moment, most wireless WPA2 spots are in danger. A rather embarrassing situation. It has been discovered by Researchers at KU Leuven. Heise article.

## Lecture 6: Calculus

• Before doing the quiz, maybe look at the Handout PDF and Worksheet PDF.
• I had tried to pinpoint the time when functions first appeared closer to Fermat. It could have been much earlier: Donna pointed out Nicole Oresme who invented pre-bar charts with beginning concepts of variables and teachings toward calculus in the 1300s! He is also credited with the first proof of divergence of the harmonic series.
• There was a question about the work of Hypatia. It is mentioned on the Wikipedia page but it is mostly commentary and editing work. There is a text "The astronomical canon however". There are some similarities with Maria of Agnesi who wrote also various commentaries, like for some work of Hopital. Agnesi was the second female professor at a university. My statement that she wrote a textbook, might however be an exaggeration.
• There was also a question about the funny hat of Euler which looks like a turban. There is some discussion on this on Reddit: descriptions are given by Florence Fasanelli in the chapter "Images of Euler" in Leonhard Euler: Life, Work and Legacy: "In this picture Euler is portrayed wearing a bag wig tied with a black ribbon. This style was customary for fashionable men from the 1660s to the later 18th century for those who could afford this expensive item of personal grooming. Wigs required considerable upkeep as well: a barber to shave the head and to powder the wig, which was replaced or restyled every year."
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## Lecture 5: Algebra

• My presentation on symmetry was definitely influenced by the book "Symmetry" by Herman Weyl. Here is the PDF of that masterpiece.
• A joke shared by Donna: "What's purple and commutes? An Abelian grape".
• Records of Rubik cube in 2017: Impressive also the solution in 18 seconds blind or the one-handed or solution with feet in 20 seconds. Amazing also the guy who could do 41 cubes blind (starts at about 11:30 minutes)
• We have seen the arithmetic lecture earlier but the zero element plays an important role in an additive group. The question who invented zero was addressed in This NYT article by Manil Suri. It seems that the Bakhshali manuscript in which the first written occurrence of the number zero is recorded is now placed to 200-400 A.D. Here is an article from September 14. In the following video by the university of Oxford, Marcus de Sautoy talks about it. He stresses that 0 is there the first time that zero appeared as a mathematical object.

## Lecture 4: Number Theory

• One interesting question which came up during class was whether the search for mathematical theorems or structure was motivated by "practicality" or "engineering needs". Some mathematics was definitely discovered as such. One could argue for example that the Pythagorean triples were of use as one can build with them exact rectangular structures which can be useful when measuring land or when building a house. In number theory it is a bit harder to justify. Divisibility properties certainly had some influence in the choice of number systems like the sexagesimal system. The Babylonian search for Pythagorean triples is also more number theory than geometry as they were only looking for solutions with integer side length. [This is by the way also one of the reasons why it is false to state that the Babylonians discovered Pythagoras theorem. They explored a very thin set of examples and there is no evidence whatsoever known indicating that they were aware of the general case; not to speak then also about the proof. Being aware of something is the first step, conjecturing the second, proving it something else. In the case of the Pythagoras theorem, there are not even indications that one had been aware of the theorem before Pythagoras. ] We have seen Wilson's theorem and Fermat's little theorem. Where do these theorems have application? It is safe to say that originally, the discoverers had no application in mind. While Wilson's theorem is an if and only statement about primes it is a wonderful example of a "beautiful theorem" without much application. The little theorem of Fermat seems much less important as it is not an "if and only" statement due to the Carmichael cases. Still, this is the theorem which now used everywhere, in any modern electronic device, primes play a huge role in encryption. To generate primes, one can use Fermat's theorem. One can make the probability of hitting a Carmichael so small that it can be used for applications. This Fermat test can be refined, like the Miller-Rabin test. So, historically, the motivation to find such theorems was mostly innocent and pure. Today, one is aware that such results often have applications later in domains not imagined. We will look at "cryptology", the science of codes in an other lecture.
• Due to exams in 21a, only the first of four batches of slides were spoken: they are here and deal with primes.
• A batch of slides not covered this year was about conjectures in number theory. There is a handout PDF which got updated a bit. We have seen Fermat's little theorem (and you prove it). Fermat's last theorem about the unsolvability of xp + yp = zp for p larger than 2 if x,y,z are positive integers has been proven. The Beal conjecture from 1993 generalizes this: it claims that xp + yq = zr has no solution if p,q,r are larger than 2, x,y,z are positive with no common prime factor. The banker Andrew Beal has offered a prize of one million dollars for the solution of the problem. Mathematicians often call the problem the "generalized Fermat equations" or Tijdeman-Zagier conjecture like here. The Beal conjecture has become widely known with this article of Dan Mauldin. The conjecture has started to appear also in books of mathematicians like the book of Ash and Gross ("Fearless symmetry), where the result of Darmon and Merel is mentioned showing that for r=2, p=q larger than 4 or r=3 and p=q, the Beal conjecture holds. More cases are mentioned on the Wikipedia entry. Ash and Gross in their fantastic gem "Fearless symmetry" avoid to mention the banker Beal (there had been some initial disputes whether the conjecture of Beal new or not. It appears however as if Andrew Beal has found a statement which has not been made before. In a Western number theory problem list of December 1992, Granville asked to find "examples of solutions" which is not a conjecture. The Fermat-Catalan conjecture of Andrew Granville from 1995 comes close. In the book "The book of numbers" by "Tianxin Cai" (also a great book), the Beal conjecture is mentioned on page 287. The Fermat-Catalan conjecture claims that the equation has only finitely many solutions. There is a difference however between having only finitely many solutions and no solutions like there is a difference between the Faltings theorem and the Fermat theorem. There is also a difference of asking for solutions or conjecturing no solution but one has to see that Granville came extremely close to Beal a year earlier. But it was only Beal (as it seems now) who was the first asking explicitly for a conjecture.
 Two books mentioning the Beal-Granville conjecture
Here are some notes of Frits Beukers also avoiding the name Beal. But Beal does not have to complain about lack of attention: Here is a search of Peter Norwig at google and here is an article from the Busindess insider:
• An other set of slides about primes are here. It mentions a Gaussian Goldbach conjecture about Gaussian primes: every even positive Gaussian integer is a sum of two positive Gaussian primes. Even means that the prime is divisible by 1+i. An integer a+ib is positive if a,b are both positive. This conjecture would imply one of the most striking open problems in number theory: the Landau problem:

 Are there infinitely many primes of the form n2 + 1.

Maybe a bit more amusing is the story of particles and primes. It turns out that primes in the complex plane behave like Leptons and primes in the quaternions behave like Hadrons. The Leptons feature neutral neutrini and charged electron positron pairs. The Hadrons feature either Baryons (3 quarks) or Mesons (2 quarks). This is all just combinatorics. The analogy is a bit of a stretch but it shows that the seemingly ugly structure in the Standard model in particle physics is quite natural. It kind of explains how the three gauge groups U(1),SU(2) and SU(3) enter naturally: there are only two associative complete division algebras. The complex plane and the quaternions. The unit sphere in the former is U(1), the unit sphere in the later is SU(2). The SU(3) symmetry appears naturally in quaternions coming from exchanging the space coordinates i,j,k. The later symmetry produces "strong equivalence classes" among primes, the former "weak equivalence classes". If one looks at the weak equivalence classes among strong, we get the Hadrons. The combinatorics "explains" why there are only electromagnetic, weak and strong forces in the standard model. It "explains" why no larger gauge groups like SU(5) appear. It "explains" why the particles come in three flavors and "explains" why Neutrini are the only class of particles with flavor oscillations. It is important to note however that these are only mathematical associations and analogies. In general, a theory in physics is only of any value if it can predict or verify an experimental phenomenon quantitatively. The particle-prime allegory does not do that. But as a mathematician, one can also just ignore the physics and study the math. The math of primes in Gaussian Integers and Quaternions will certainly be studied more in the rest of this millenium. Maybe we will see a proof of Landau's result in a few hundred years. Maybe we have to wait two thousand. It could also be happening tomorrow.

## Lecture 3: Geometry

• Mathematica Notebook of the Miracles Proof
• From a recent book announcement: A Mathematical Gallery from Lisl Gaal. Here is a page about Pythagoras:
• One can find on the web claims that the Babylonian mathematics contained Pythagoras theorem. It is a common misconception when proving a theorem that finding a few cases is equivalent to proving the theorem. In reality, knowing a few cases can help to form a conjecture but it is a long way to an actual theorem. What can one find on the Babylonian Clay tablets? 1. There are examples of Pythagorean triples. 2. There is an example of a numerical computation of the ratio of the diagonal to the side length of a square. The Theorem of Pythagoras is much more than that: it is theorem which holds for any triangle with a right angle. One can speculate that some Babylonian mathematician knew or suspected that. But there is no evidence, no source which suggests that. In the next lecture we look at unsolved problems in number theory. There is evidence today for example that every even integer larger than 2 is a sum of two primes. We don't know that this holds yet. It is called the Goldbach conjecture. In some sense we are in the stage of the Babylonians. We might be "certain" (based on our experiments) that the result is true, but it is not a theorem! Lets assume the theorem will be proven in the year 2400. If in two thousand years a future historian would claim that we had in the 20th century a pretty good idea about the Goldbach theorem and so "knew the theorem", then this last claim would be a mistake. We don't know at the current state whether the theorem is true! In the case of the Babylonians, we don't even have evidence that the Babylonians conjectured the Pythagorean statement! There is no Clay tablet which shows this, nor any historian who has found evidence in some texts. They might not even have asked the question whether a^2+b^2=c^2 for any right angle triangle. Asking good questions is an art which still had to be learned 4 thousand years ago. One can say that the Greeks have started to master this art. They started to ask questions like whether odd perfect numbers exist.
• We have wondered during class about visits of the Greek geometers in Egypt. In 535 BC Pythagoras went to Egypt, while the city of Samos was occupied by Polycrates. Donna mentioned that Euclid studied in Alexandria during the reign of Ptolemy I. The connection with Alexandria is even cemented into the name: Euclid of Alexandria. Donna mentioned also that Apollonius of Perga, who is known for his 8 volume work "Conics" seems have lifted the first four volumes directly from Euclid. Apollonius appears so to be one of the earliest known plagiarists!
• One of the questions during the lecture was "How did Mathematicians like Thales or Pythagoras" make a living? The fact that he could afford to sacrifice 100 oxen after finding his theorem suggests, that he was wealthy man. Donna also writes: "We may see a hint in the comment that Pythagorus was an almost religious figure to his pupils. This might imply that mathematics school - like Euclid's - were designed economically like temples. There was no unified religion in these days. The temples acted independently. If you wanted to be a server in the temples, you basically volunteered. If you wanted to be a major priest, you donated quite a lot. It was often the equivalent of a dowry - which was enough to start a home and maintain it for enough years to replace the goods - a significant capital investment. If for example, 10 individuals wanted to become professors in Euclid's school, they would each make a capital investment. the 11th person - the head of the school - could easily life off the proceeds since in addition there would be a regular revenue stream of donations for solving allocations of wills, design problems from architects, etc.etc. " Sounds like the pupils were what one today would call a "partner" in a business ...
• One of the questions during lecture was about the connection between Pythagoras and Egypt. We read in the Mac Tutor History. Here is a thought of Donna: As far as how he was funded, although I can't find specifics on his school, we may see a hint in the comment that Pythagoras was an almost religious figure to his pupils. This might imply that the mathematics school - like Euclid's - were designed economically like temples. Remember that there was no unified religion in these days. The temples acted independently. If you wanted to be a server in the temples, you basically volunteered. If you wanted to be a major priest, you donated quite a lot. It was often the equivalent of a dowry - which was enough to start a home and maintain it for enough years to replace the goods - a significant capital investment. If for example, 10 individuals wanted to become professors in Euclid's school, they would each make a capital investment. The 11th person - the head of the school - could easily life off the proceeds since in addition there would be a regular revenue stream of donations for solving allocations of wills, design problems from architects, etc.etc. In about 535 BC Pythagoras went to Egypt. This happened a few years after the tyrant Polycrates seized control of the city of Samos. There is some evidence to suggest that Pythagoras and Polycrates were friendly at first and it is claimed that Pythagoras went to Egypt with a letter of introduction written by Polycrates. In fact Polycrates had an alliance with Egypt and there were therefore strong links between Samos and Egypt at this time. The accounts of Pythagoras's time in Egypt suggest that he visited many of the temples and took part in many discussions with the priests. According to Porphyry Pythagoras was refused admission to all the temples except the one at Diospolis where he was accepted into the priesthood after completing the rites necessary for admission.
• There is a nice time line about Greek mathematicians here:

## Lecture 2: Arithmetic

The "Story of One" (BBC) as well as "Cracking the Maya Code" (NOVA) are on youtube:
You have to see the movie "Cracking of the Maya Code". As all Nova movies, it is excellentlyl done. It is a fascinating story also featuring the discovery of Tatiana Proskouriakoff done at the Harvard Peabody museum. The film is based on the book of Michael D. Coe "Braking the Maya Code".
• During lecture, the question was raised why the sexagesimal system prevailed. The large number of prime factors of 60 could be a reason. This is also the explanation in the Wikipedia article. The reason could be similar than why the 10 months were expanded to 12 months or why 24 hours were used in a day or 360 degrees are a full circle. Also the Chinese Zodiac is based on a 12 year cycle. What also could have played a role is that 60 not only has many factors but is also a multiple of 10, the number of fingers of humans and a multiple of 12 the number of months.
• During the lecture, there was a question about the nature of the Mayan hieroglyphic writing. It appears that the glyphs were not only carved in stone but also painted on ceramics, written on bark-paper, molded in stucco, carved and molded glyphs were painted. See a picture of a Stucco. The pictures which appeared in the movie clip, were part of the Dresden Codex, a Mayan book. You find a picture of the manuscript here. The Dresden Codex dates from the thirteenth century. It is called Dresden codex because it was hidden in a book of the Dresden library. Maya writings were used until the 16th centuries. The base 20 system is called the "vigesimal system". There were calendar entries and astronomy. The Dresden Codex allowed to date the stone carvings.
• We looked at a movie clip The clan of the bear cave, a movie from 1986. It illustrates the beginnings of counting. Then came tally sticks, which date back 40'000 years ago. There is an interesting tally counting, the "log tally", which is a graph theoretical method. The number is the sum of the vertex and edge count of a graph. It historically first appeared in an article of August 1916 by Calland [PDF]. The log tally is mentioned in this or this blog as well as the wikipedia page on Tally marks.
• There was a lot of press about a recent Article about sexagesimal trigonometry. Here is an example: Babylonians did trig better 3700 years ago
• The book of Bob Kaplan on "0" and the book of Paul Nahin on "i" show that a single number already contains an immense richness, both historically as well as mathematically:
• There are lots of aspects to the irrationality proof of sqrt(2). We have seen a geometric proof and the simplest proof available which uses the fundamental theorem of arithmetic. A more thorough analysis of Barry Mazur.