Mathematics E-320 : Teaching Math with a Historical Perspective,

Encryption

We have looked at various encryption methods in our Cryptography lecture. I had mentioned that it even has happened that Adobe products had Rot13 encryption built in. We have learned how to crack Cesar cyphers using statistics, a method developed by Al Kindi already. Here is a German Spiegel article which mentions Harvard:

The article is only readable after full payment. But people figured out quickly that the paywall is easy to crack. It is just a Cesar cypher! There are already Firefox plugins which automatically decrypt such pages. Here is the encrypted ASCII string together. In Mathematica, it can be decrypted with a "half liner"

t1=StringJoin[FromCharacterCode[ToCharacterCode[t]-1]]

Here is the decrypted article.

Computing

About the dawn of electronic computing features a longer talk of Konrad Zuse

During the lecture, there was a question how to classify Babbage. Was he a mathematician, or an engineer? It is difficult to answer. We consider him today a "Polymath", a person who is an expert in many topics. About Charles Babbage: When looking a AI challenges, we looked at some geometric problems and Nick mentioned this movie which actually appears in many versions:

Dynamical systems

[December 14 added]: We have looked at the Sitnikov problem, a restricted 3 body problem, where the calendar is chaotic. The idea has been taken off in the novel The three body problem by the Chinese author Liu Cixin.
About the butterfly effect:

Cryptology

We saw in detail how the Diffie-Hellman key exchange works. Once understanding this process, we can also understand more modern crypto implementations like the "Signal protocol" which is used for example by chat application "WhatsApp". Here is a recent article in Heise (German). The principles of public key cryptology can also be ported to other groups (in the mathematical sense, we have seen groups in the algebra lecture). Today, elliptic curves are commonly used. Elliptic curves are a place where various mathematical fields come together (number theory, geometry, analysis). An example is described here. The curve 25519 is
```
y² = x³ + a x² + x 
```
where a=486662 and the arithmetic is done in the field Z^p with prime
```
p=2²⁵⁵-19 = 57896044618658097711785492504343953926634992332820282019728792003956564819949.
```
It turns out that one can do arithmetic on such curves in the same way as on integers.
Why cryptology? Wouldn't it be nice to live in a world without secrets and borders? This idea appears in Sneakers (1992), from which we have seen some scenes in the lecture. Cosmo (Played by Ben Kingsley) sitting on a Cray super computer telling about "changing the world": But it does not need the brains of a "Martin Brice" (Played by Robert Redford in that movie) to see what the consequences would be and where it is important:
- Modern cars with electronic locks can be hacked easily Example
- Election results could be modified by external powers. Some computer scientists actually urge to double check the elections which just happened CNN. Indeed, the security of the voting machines is terrible Example, Example 2, Example 3
- Also in recent times, it became clear (Wikileaks, email hackings) that diplomacy or politics, even private conversations must now be considered public. It changed politics it can change history. We have seen that this has happened in the past and that cryptological aspects were vital (Execution of Mary of Scots, Zimmerman telegram WW I, Enigma machine in WW II).
- Imagine your DNA data are stored on a medical computer and this becomes hacked. Assume the data show you have a 30 percent chance to develop a rare genetic disease before the age of 30. Would any potential employer knowing this hire you? Probably not. It is important that medical records stay confined.
- Imagine your bank account being tempered with and random folks taking money from you. Bank accounts need to be secured by strong cryptology.
- Imagine you have a great idea for a company. You need some time to develop that without others copying your idea and implementing it first. It is vital that you can keep the idea off prying eyes for a while. It is important that you can keep this locked up, at least for some time.

At the beginning of the lecture we started with an overview about the landscape of crypto: cryptoanalysis, cryptography, compression, error correction, hash algorithms. A variant of the later (SHA-1) has recently got into the news: Register. The problem with weak hash functions is that it makes it easier to produce fake objects. We have maybe seen (maybe a bit too much) about the enigma during our lecture. Still, if you have not got enough of it: here is two other presentations, by Numberphile:

Analysis

Analysis is a rather large field of mathematics, ranging from Fourier theory, calculus in infinite dimensions to partial differential equations or measure theory. We will focus mainly on giving a bit of an overview. Mathematically we focus on the concept of dimension.
We have seen some movies of the Mandelbulb, the "Youtube star" of fractals. Here is are some other animations not shown in the lecture:

As mentioned in class, almost nothing mathematical seems to be known about the Mandelbulb. A first question is whether it is connected.

Back to probability and statistics

Direct Media Links: Webm, Ogg Ipod, Quicktime. About the movie. Best Quote from the movie: "People are overlooked for a variety of reasons and perceived flaws: age, appearance, personality. Mathematics cuts right through that".

November 9, 2016: the elections from yesterday illustrate how powerful statistical models can be. Allan Lichtman has a system which predicted a Trump win. The system does not analyze personalities nor policies nor opinions. It simply looks at a high-dimensional value function obtained statistically from previous elections, then uses this to predict the future. It turned out that in these presidential elections, the polls were misleading and that Lichtman's system was more accurate than what polls or pundits predicted. Its a bit like in the movie "Moneyball", were experts were shown to be fooled by things like "how a player looks or whether he throws funny". What counted in baseball in the long term is statistically how high a player scores in a parameter space. In this case, the model does not look at the actual politicians but at the party structure. The data come from previous US elections. The players are the parties. It is therefore not the statistical data of the players were compared but the statistical data of the political parties, completely detached from any person or policy. Lichtman's system does not take into account the personality of the candidates, nor the political agendas, not how likable or successful they are. Its just the raw analysis about the situation of the two parties. A parameter for example is how strong a third party candidate is. And this analysis is based solely from previous election cases. This is the power of statistics: Make a high dimensional model from previous cases, then use the model to predict the outcome of an event. Its a bit like datafitting but of course much more subtle as Lichtman, in order to build the system probably had to look at hundreds of parameters and then weed out the most relevant ones. Its like building a value system for chess game situations. Having a good value system is the key to predict the win or loss of a party.

Picture of Allan Lichtman from the American University. Source

Topology

Topology is "rubber geometry". We will see examples showing that there are some strange geometric objects appearing there. It is an interesting field of mathematics which is under heavy development. Here is an article about the clip we have seen, where Thors Hans Hansson explained the concept of "genus".

And here is the Twitter announcement.

These press statements are excellent examples how to communicate mathematics to a general audience. We have also mentioned this example in our Rhetorik page (German) as many experts often fail to communicate effectively. By the way, I'm editing that Rhetorik page since 18 years (spending since 18 years a couple of hours every week on that family project. The blog contains currently 3220 articles related to communication).

Probability lecture

Kolmogorov: "The theory of probability as a mathematical discipline can and should be developed from axioms in exactly the same way as geometry and algebra."

The axiomatic foundations of probability theory was established in the monograph Foundations of the theory of probability which appeared first in 1933 and which got translated into the English language in 1956. Still today, the confusion about probability is wide spread, even so there are maybe half a hundred good books on the subject now (Here is mine). One reason is that a random variable is still considered by some as something mysterious as some kind of device producing random numbers. Kolmogorov made it very clear that a random variable is just a function on a probability space (Ω,P). Probability theory has become a special case of measure theory. It even becomes part of calculus, if one pushes the situation onto the real axes by talking about probability density functions and distribution functions. Now probability is part of calculus. Kolmogorov was also the first to make this crystal clear. First define F(x) = P[ X ≤ x], the cumulative distribution function. Then define the probability density function as f=F', if one can differentiate. In general it is just a probability measure on the real line. This function f contains information about the random variable like the Expectation E[X] which is just the integral of X over the probability space but also m= int x f(x) dx. The variance is E[(X-E[X])²] which is the integral int (x-m)² f(x) dx. So, the probability density functions allow to compute things by integrating over real numbers rather than integrating over a general probability space, which the laboratory. That is very practical, even in discrete settings. For example, if we throw a dice 3 times, the probability space has 6*6*6 = 216 different elements. But the probability distribution is much smaller: it is given by assigning the probability P[X=k] and these values are nonzero for k=3 up to k=18. For example P[X=3]=1/216. In only one of 216 cases, the three dice are 1. So, one can encode the statistic of the laboratory throwing 3 dice with 18 numbers. This is the power of statistics.
While the axiomatic setup of probability theory makes everything crystal clear, the fact that it appeared late in the development of mathematics (for example, only after quantum mechanics was developed), reflects today in the classroom. Students today need a couple of weeks to get comfortable with the frame work and get away from murky feelings and conceptions.
Proof for widespread confusion is the Monty Hall situation we discussed in detail, the riddle "Dave has two kids, one being born at night, what is the probability that the other is a boy" where even PhD mathematicians can get the wrong answer. Conditional probability is sometimes not intuitive. Also independence is sometimes not appreciated or felt: in a Casino frame work, many think that if 5 times red has appeared in Roulette that magically the chance that an other times red appears has diminished. A third source for confusion are coincidences. We have looked at the Birthday paradox. With 23 kids in your classroom, the probability that two have the same birthday is already more than 1/2. This is counter intuitive and in daily life we often label coincidences as "special" even so statistics tells us that they are not. Even intelligent and educated people fell for coincidences where conditional probability and coincidences come together. In that case the confusion is even more basic, as the probability space is not well defined. If you look for something exceptional and do not specify before hand what it is, then it will appear special and unlikely. Feynman joked once: "Imagine, I saw a car with license plate ARW 357. Of all the millions of license plates in the state, what was the chance that I would see that particular one tonight? Amazing!" Yes, it is a joke, but it is a trap into many scientists have fallen into. We have seen in the Bertrand paradox also how one can fall easily if the probability space is not fixed before hand. An other reason for confusion is the appearance of large numbers. We have seen that in the Petersburg Casino, we always lose in experiments even if put a modest entry fee like 10 dollars. The mathematical result is that we will eventually win. But one has to play for a very long time In physics, the paradox appears in the case of recurrence. All basic laws of physics (quantum mechanics, classical mechanics, general relativity) are reversible but where we see an arrow of time in the form of entropy increase. As in the Petersburg paradox, the confusion is the time scale. We would have to wait a multiple time of the age of the universe to win a 1000 dollars in the Petersburg paradox. Obviously we can not wait so long. These are all cases, where mathematical facts and experience do not play well together. But this is only a problem if one is not aware of it.

To summarize, probability theory can be a bit mysterious at first for the following seven reasons:

There is a clash of intuition and fact in conditional probability (Bayes theorem, Monty Hall stories)
The use of words like "random variable" for functions (measure theory, calculus) distorts.
The perception of randomness is distorted by patterns (building a model after doing the experiment: an example is Astrology).
Products of probabilities can be much different than what we would estimate them to be (Birthday paradox)
There are probabilities and probabilities. (Bertrand paradox, we need to fix not only the probability space but also the measure P)
Our inability to access large numbers leads to clashes between mathematical results and experience (Petersburg paradox, second law or thermodynamics)
The need for measure theory (forced by examples like Banach Tarski). It is impossible to assign a probability measure to all subsets of a laboratory, if this set is uncountable.

When discussing combinatorics, we have played in class with the basic notions of permutations and combinations. These problems come in flavors, where order matters or not or where repetition is possible or not. The situations are best seen in examples:
1. How many words can one form with k=5 different letters. The answer is P(k) = k! =5! = 120.
2. How many words of length k=5 can one type on a typewriter with n=30 letters. The answer is n^k=30⁵.
3. How many ways can one chose k=5 letters from n=30 if order does not matter? The answer is B(n,k)=30!/(5! 25!).
4. How many words of length k=5 with different letters from n=30 can one build? The answer is B(n,k) k! = 30!/25!

What is interesting here is that school mathematics has invented its own notation. One writes for example C(n,k) or Cⁿ_k and sometimes, the indices are put before the letter C, sometimes after.

Set theory lecture

When talking about the results of Kurt Goedel, we are led to philosophical questions. Indeed, Goedel was part of the so called Vienna Circle. Other members of this group were Karl Menger (the one from the Menger Sponge), Hans Hahn (the one from Hahn-Banach), Ludwig Wittgenstein (the one who wrote the infamous Tractatus) and Willard van Orman Quine, a Harvard logician who built up his own logical system called "New foundations" (NF), which was an alternative for the ZFC set theory we have looked at. This leads then to questions in the philosophy of mathematics which are important but which are considered for many mathematicians a bit of a "muddy ground". Still, the foundation work of mathematics goes on and is strong especially at Harvard (Putnam and Woodin in more mathematical contexts). We have mentioned that the Continuum hypothesis is undecidable within the Zermelo Frenkel framework (using an argument which essentially uses the Liars paradox by constructing a sentence telling "I can not be proven"). It does not mean that the continuum hypothesis is undecidable in an other, larger framework which can be considered natural. Indeed, some logicians, including Woodin, work on frame works in which the continuum hypothesis is naturally true. In this context the question whether "mathematics is discovered" (Platonism) or "mathematics is invented" (Empiricism) often comes up. My personal take is that like with many fundamental questions, the discovered or invented question is simply silly. It is not well posed because it is not part of what we can access. We can not disprove for example that we are not part of an elaborate computer game like the "matrix". In such a situation, the mathematics, the rules of the game are of course all invented by an "architect". We can also not disprove the contrary. Therefore, according to Karl Popper (one of the best philosophers of science and who was by the way also a member of the Vienna Circle), this is not part of science. Poppers postulate of empirical falsification is still the "gold standard" on what science is. I myself was exposed at ETH Zuerich to a more anarchistic point of view (by the lectures of Paul Feyerabend), and Feyerabend's arguments makes a lot of sense when looking at the development of science. Much important work in mathematics (as well as in other sciences) was at first only a protoscience and justified or put on a firm foundation later. But I also believe that one must distinguish between a "development phase" of a theory, where things are not yet solid and maybe conjectures or models which have no empirical verification or falsification yet. In the end however, one has to pass the Popper falsification status. Otherwise, the theory needs to go to the garbage bin. The Popper definition of what science is should maybe be made stronger that a theory (in mathematics or in physics) merit should be measured by how much quantitative results the theory can produce. General relativity in physics is an example which passed the test many, many times, most recently with the LIGO experiments. In mathematics, all of the basic fields (covered in this course) have proven their merit and power.

Nick mentioned Euler diagrams when we looked at Venn diagrams and sent the following explaining the difference between Venn and Euler diagrams. I'm actually not convinced why one should distinguish cases, where the intersection is empty or not with a different notion. One can perfectly well just work with Venn diagrams alone. Maybe it could be useful when building data structures.

The questions which appear in set theory are also of rather philosophical nature: how can we be sure to have a solid foundation of mathematics. How do we map all truth? Does infinity really exist? Is our universe finite? Do we really need to model nature with structures needing infinity.

Apropos Cosmos, there was just a new book published by Ian Stewart. Calculating the Cosmos. The New York times has an announcement.

Calculus lecture

I have tried to reproduce the Pecha Kucha talk from 2013 in class with the 20 slides turning in 20 seconds. The actual talk had been rehearsed a bit better. Was just uploaded now: Some more remarks and resources are on this page. Here is famous the movie of Ed Burger Last summer, I uploaded a 15 minutes refresher for the summer school class:

Algebra lecture

[Update November 11, 2016] Rubic cube solved under 1 second:

[Update of October 13, 2016: we also talked about groups like the Rubik cube group or the floppy group (192 elements only) in the algebra lecture: its not only important how fast you can solve it, also how big or small can one make them? And here is a 28x28x28 cube: On a computer one can go further. We have seen the 4D cube. Here is a 1000 x 1000 x 1000 cube: [Update of October 10, 2016: basic algebra is in the news but its a problem in linear algebra, the simplest possible case. The German tabloid "Bild" reports about a knife attack triggered by an algebra problem: the problem is

What is my number if half of my number is half of 400?

The controversy started because a teacher dismissed a pupil's solution 400. The fight however happened, after a 19 year old (Robert) posted the problem on his facebook account. A discussion appeared leading to an aggressive encounter with Alexander he does not know. Alexander stalks him later and attacked him with a knife. Why the fight? Most people answer correctly with 400 as if x is the number, the problem asks that (x/2) = (400/2) meaning x=400. An other interpretation (which is the teachers) is to set x=400/2/2 and then evaluate (400/2)/2 which is indeed 100.

This is close to the expression 400/(2/2) is 400. The later non-associative ambiguity of division is illustrated in Pemdas disputes but here it appears pretty clear how to translate the problem to algebra. A question like "What is half of half of 400" would lead to 100. Still, at the heart of the problem is the ambiguity of order of division (a/b/c) which gives different results when evaluated (a/b)/c or a/(b/c). It is a major algebra mistake to mix up the two for example by not specifying which one the problem has in mind. Anyway, "Bild" calls it the "most dangerous math problem in the world"! Well if that does not make algebra exciting. end of addition of October 10].

We derived in detail the formula for the roots of the quadratic and cubic polynomials and only sketched the quartic. Then we briefly told the story about the discovery of the non-solvability of the quintic. There are two amazing tales which come to this, the story of Abel and the story of Galois. Before summarizing these two dramas, also a bit of caution that these two tales are a major reason for the commonly assumed non-sense that "mathematics is a game for the young". This prejudice has been fueled especially by "A Mathematicians apology" of Godfrey Hardy (1877-1947), which was written in 1940, while Hardy suffered from depression triggered by a heart attack (1939). The stories of Abel and Galois are nice and interesting nevertheless; certainly because they are so atypical and extraordinary. The popularity of the two dramas of course fuels arguments for the validity of rules which based only on anecdotal evidence. Other "prejudice rules" are that "mathematicians are introvert" [("Question: Can you tell the difference between an introverted and extroverted mathematician? Answer: The extrovert looks at the other person's shoes") ] or that "math talent is an innate ability" which is the third example of total non-sense. My own take on this is that there might be physiological differences (like the speed with which structure is built up and which has been analyzed well by Jean Piaget), but that there is (through structure and luck) a relatively fast feedback loop present, killing or fueling interest in mathematics and that different paths would happen for the same person if triggered either way. More importantly, it is the ability to build up "good operating system structure" which then allows to build up knowledge and creativity. An awareness of these structures can be helpful too: (I myself read (or tried to understand), the "Society of Mind" by Marvin Minsky in college and read (and could understand), "How to solve" by Polya, (fortunately early on in college, so that it changed my performance in essentially one stroke). I also made already in high school experiments with intelligence tests and could see experimentally, how well and easily one can manipulate and beat them, simply by preparation, essentially doing lots and lots of them. Since then, I consider intelligence tests very questionable. It is this awareness of meta structures (tools to learn, tools to solve problems, tools to find new things, being aware that it is important to be aware), which can be important. And I'm convinced that both Neils Abel and Évariste Galois have gone through many of such evolutionary stages (allowing them to acquire exponentially faster learning and discovery abilities than others). It is not that Abel and Galois "were born a genius". But of course its nice to relate to them like superheros in the fairy tales of Disney or Marvel. This might be a matter of taste but as a teenager, I was not so much impressed by "early genius" tales (I actually found them terribly depressing, especially when growing into an age, where Gauss already made great discoveries) but by success stories like Marie Curie, who succeeded with hard work, determination and grit, something which is achievable. So, maybe first a picture of Curie, who is a more approachable "genius" [I mean this also here in the sense of Zwicky who wrote a book "everybody a genius" which paints the idea that in principle everybody can do extraordinary things. In the case of Curie, the merit is definitely deserved as she worked hard for science and did not just score a lucky punch.] (The picture is from a book by Katherine Krieg (Marie Curie, Physics and Chemistry Pioneer, I unfortunately can no more find the biography (and the author of the book about Curie) which I had read as a teenager which I had then got from the library in Schaffhausen): Marie Curie, picture appearing in the book of Katherine Krieg about Marie Curie

Marie Curie, picture appearing in the book of Katherine Krieg about Marie Curie

But finally, lets get to the super natural heroes Abel and Galois: first Abel (1802-1829): when Neils Henrik Abel was 13, he entered the Cathedral school in Oslo. An inspiring math teacher, Holmboe, turned out to be the trigger for doing mathematics. When Neils father died, shortly after having his political career distroyed, Holmboe supported Neils. When Abel entered the university, he already worked on solving the quintic. He actually believed of having a solution to that 250 year old problem. Abel himself found a mistake later, continued to work on it after graduation and in 1823 finally realized the impossibility of solving the quintic. Interesting that Gauss, whom the paper was sent, discarded the paper without a glance as he believed it to be the worthless work of a crank. His paper "Beweis der Unmöglichkeit, algebraische Gleichungen von höheren Graden als dem vierten allgemein aufzulösen." was published here. in 1826. He met his future fiancee Christine Kemp in 1924. After extensive travels in Germany, Italy, Austria and Switzerland and France, he returned in 1927 to Norway. Having contracted tuberculosis in Paris, he became seriously ill during a trip to see his fiancee and died shortly after in 1929. Tragically, just two days after his death, a letter from Crelle arrived telling that he obtained a professorship in Berlin.

Neils Abel

Christine Kemp

Now to Evariste Galois (1811-1832): in 1823, Galois entered high school Louis le Grand and shortly after, in 1825 became seriously interested in mathematics. The trigger might have been a geometry book of Legendre. At fifteen, we was reading original papers of Lagrange already. [ By the way, we see here already how things have changed. Today, it is even hard for a graduate student to absorb substantial papers of leading research mathematicians. Its just that the amount of material which has to be mastered has grown so much since Abel's time: after calculus and linear algebra, there is much more algebra (especially abstract and commutative algebra), geometry (especially differential geometry and algebraic geometry), topology (especially point set topology and algebraic topology), analysis (especially complex and functional analysis), differential equations (both ODEs and PDEs) and probability theory (including measure theory and stochastic calculus) are the absolute minimum. And then there are fields, which have to be absorbed "en passant" like logic and set theory, category theory, numerical analysis or combinatorics. And having been exposed on a level to brag about in a party is not enough, they have to be mastered on a level to solve serious exercises. This is just mentioned to make clear that the probability of having a 15 year old today doing substantial work on its own is very small. It is still possible although when good teacher places the pupil at the right spot in an emerging new field. ] In 1828 and 1829, Galois twice tried to enter the école polytechnique, but failed in both cases. The first discovery of Galois in the field of polynomial equations was in 1829. He tried several times to publish it, but eventually the memoir was lost. He still published papers, introducing in particular the concept of a finite field. In 1830, during the second French revolution, Galois, after criticizing the director of the école normale, was expelled and soon after joined the Republican artillery unit of the National guard. The following Bastille day in 1831, Galois headed a protest, wearing the uniform of his former unit and was sentenced to half a year in prison. He was released on April 29, 1832. Shortly after, possibly due to a romantic complication with Stephanie Poterin Du Motel, a daughter of a physician. The opponent of the duel could have been du Motel's fiance or then one of Galois' republican friends (if the reason for the dual had been an other one). The letter to August Chevalier is here. It was written on Mai 29, 1832, the night before the duel.

Galois: Image Source: galois.ihp.fr

Letter of Galois, Mai 29, 1932 Image Source: here.

General first lecture update

A scan of the Bosman book [PDF] featuring the Pythagorean tree is now complete. By the way, I first scanned the book using a smart phone scanning app. The quality of the scans was not bad, but the app had problems with the lighting in the case of pages where pictures and text mix. Since I could not cut up the book (as I do when digitizing my own books) cutting the spine was not an option, it was copied on a copy machine and then fed to the scanner which also did the OCR (probably not that well as the book is in Dutch). [Update October 9, 2016: ran the document through the Adobe OCR using Dutch settings. The above link is an updated cleaned out document.] Here are the trees drawn by Bosman (two pages from the book, color inverted, click on the picture to see it larger).

Number Theory lecture

[Update October 13:] we talked about primes and will do so again in the cryptology lecture. One of the problems one wants to solve given a prime is to find for a given b a number x so that 2^x - b is divisible by p. This is the discrete log problem. Nobody knows how difficult this problem is but one believes that it is hard in the following sense: for a given 300 digit number (which are used today for encryption), there no efficient way to find x = log₂(b). It is believed that with current top notch algorithms, an agency like NSA would need hardware worth of several hundred million of dollars to crack such a 1024 bit discrete log problem. An Heise article [in german] reports today about a paper posted October 5, which assures that for some primes can lead to easier problems as the solution can be found using the number field sieve. In a cryptographic setting, the players are usually called by palindromes: "Ana" sends a message to "Bob" and the adversary "Eve" listens to their conversation. In the scenario of the paper, there is a fourth player, called "Heidi" (like a third party producing primes). I actually think, it would have been nice to call also this fourth player by a palindrome, like Idi, the last three letters of Heidi. Now, all the four players Ana,Bob,Eve,Idi would be three letter polindromes. We will come back to these stories in the cryptology lecture.

Because number theory deals with integers, one could have the impression that it is easy. It turns out to be one of the most difficult and technical area of mathematics. In this lecture we look at a few mathematical results as well as some open problems. We also talked about Goldbach, today probably the most iconic open problems in mathematics since it has entered in numerous ways into popular culture (novels and feature movies as we have seen). Here is some recent news about Goldbach and an application. It is the Helfgott proof of the weak Goldbach conjecture: Grothendieck

Scientific American and Slashdot. The Helfgott proof is a landmark in mathematics as it settles one of the important problems in number theory. But if one looks at the article, it also becomes apparent that the proof is not easy. Actually, analytic number theory is probably one of the most technical areas within number theory. The ternary conjecture looks close to the actual Goldbach conjecture but some of the best number theorists believe that the current methods might just not be strong enough to crack the actual binary Goldbach conjecture and that additional ideas are needed. Still, there is hardly any doubt that an eventual proof of Goldbach will be difficult. Why do we know? We don't, but history tells us that any miracle approach is very unlikely! If we look at major open problems in mathematics, then either they were settled with a counter example or then with hard work and heavy machinery or then with an elaborate construction of a theory. A general picture explaining this has been painted in 1985 by Alexander Grothendieck (who without doubt was one of the most creative mathematicians in recent times).

It is the Hammer and Chisle versus Rising Sea approach. It has been mentioned in the McLarty article (which is an iconic document by itself) from which I take this. The allegory compare the theorem with a nut which needs to be opened (one even says in colloquial language: cracking a nut).

Hammer and Chisle principle	Rising Sea principle
The Hammer and Chisle principle is to put the cutting edge of the chisel against the shell and strike hard. If needed, begin again at many different points until the shell cracks-and you are satisfied. Grothendieck puts this poetically by comparing the theorem to a nut to be opened. The mathematician uses the Hammer and Chisle to reach "the nourishing flesh protected by the shell".	The Rising Sea principle is to submerge the problem by vast theory, going well beyond the results originally to be established. Grothendieck imagines to immerse the nut in some softening liquid until the shell becomes more flexible. After weeks and months, by mere hand pressure, the shell opens like a perfectly ripened avocado!

But you see, in both cases, it is hard work and not magic. We see this principle very well also when teaching mathematics. There are problems which can be solved by Hammer and Chisle and then there are problems which can be solved by built up theory. The generalized Pythagorean theorem (Gua's theorem) mentioned in the geometry lecture (and which was used as an exam problem in our exam from last Wednesday), is a problem which can be solved by "Rising Sea principle". The theory is vector calculus. It has become so easy that the class could solve it (most students solved it perfectly well and "cold" in the sense that none had the slightest idea that the problem was coming and nobody seems have seen the theorem before (it is definitely a lesser known theorem)). But our students had the softening potion (here vector calculus) at hand to soften the problem. They were given the Avocado, not the nut. On a pedagogical note, we gave them a "ripe avocado" in that we put numbers instead of variables a,b,c. See the exam [PDF] and the Solution [PDF]. We can not repeat the experiment, but according to experience, if we would have given the problem in a general form rather than with numbers, I would predict that more than half of the class would have been unable to solve it. They are the smartest kids in the world but abstraction can have a paralyzing effect! Its not that the problem in general is more difficult, it actually is easier with variables as one does not have to do the actual multiplication of numbers, but abstraction can be a psychological lock-up. How come? We have to look at history. It took more than thousand years after Euclid until variables became part of mathematics (we have mentioned Viete) and it really only was harvested when Desquartes linked it with geometry.
The algebraic proof of the geometric problem is magic. Not in the Harry Potter sense but by having built a well oiled machinery. [ The Marounder map in the Rowling story is magic but our cellphones can do that too (track your friends in real time if they have enabled their cellphones to give a way their location) but there is an immense technology behind the real thing, one ingredient is GPS, which required us to send satellites to space, building technological components, then there are millions of lines of code, developements of CPUs etc. ] When solving a problem, one is often not aware of the elaborate theory which is behind. Lets look at the Gua theorem from the beginning. I list the steps here just for illustration but these steps (when written out) establish a full proof (from the ground up) of the Gua theorem:

Define the dot product algebraically as (v1,v2,v3).(w1,w2,w3) = v1 w1+v2 w2+v3 w3.
Prove the Cauchy-Schwarz inequality. This is a two line proof: A twitter proof
Define the angle x through cos(x) = v.w/(|v| |w|). This can be done because of Cauchy-Schwarz. Now we know the identity v.w = |v| |w| cos(x).
Define the cross product algebrically as (v1,v2,v3) x (w1,w2,w3) = (v2 w3-v3 w2,v3 w1 - v1 w3, v1 w2-v2 w1).
Prove the Cauchy-Binet identity |v x w|² + (v.w)² = |v|² |w|². Also this can be tweeted if we use a computer algebra system. A brute force proof needs a couple of lines of computations.
Use the Cauchy-Binet identity and Pythagoras cos²(x) + sin²(x) = 1 to establish |v x w| = |v| |w| sin(x), where x is the angle between v and w.
Using the definition of sine and school knowledge (area = base time height) that |v x w| is the area of a parallelogram spanned by v and w.
Now, the nut has become an avocado, we squeeze and know that the area of a triangle as |v x w|/2.
Use Descartes idea of coordinates to translate the geometric area problem into an algebraic problem by taking points A=(a,0,0), B=(0,b,0), C=(0,0,c) to solve the problem.

This illustrates the power of theory. I asked around also outside the classroom whether somebody knows a direct geometric proof of the Gua theorem which the Greeks could have found (they did not have any algebra or trigonometry, not even the Heron formula which could be used to establish the theorem too). By the way, if you should know an elementary Greek geometry argument for establishing the Gua theorem, let me know. It looks not so easy, as the squares of areas have only geometric interpretations as volumes in a four dimensional space, a concept which was definitely also not available at the time of Euclid. One could probably cut up a four dimensional cube into three four dimensional parts and show geometrically that pieces matches the union of three four dimensional cube representing the squares of the area of each side.

Back to number theory. Ideas going back to ancient times, like the Sieving methods (we mentioned Erasthostenes in detail during the lecture) have been refined more and more but they are hardly recognizable as such for a non-specialist as they are very technical. One of the generalizations is the Selberg sieve and you see that even the Wikipedia article waves the hands just to explain the main idea. It is difficult to imagine that a future mathematician can solve Goldbach without having absorbed all these ideas already. It is especially very unlikely that an amateur "Good Will" genius can do it without having learned the ropes. There are exceptions like Ramanujan. But also there, one has to see that Ramanujan worked extremely hard (essentially non-stop) and had access to a library (and later in his life contact to some of the best mathematicians in the world). By definition he was a professional research mathematician, a person who spends the majority of each day with mathematical research.

Geometry lecture

An update from October 28, 2016: in our multivariable course Math 21a, one of the exam problems featured the 3D Pythagoras generalization. One of the 9 problems in the exam [PDF] was to prove the theorem:

You find the solution on the solution page: the area of the triangle with points A=(a,0,0), B=(0,b,0) C=(0,0,c) is half the area of a parallelogram. In vector geometry, one learns that this area is the length of the cross product between the vectors AB, AC spanning it. This is | (bc,ac,ab) |²/2 = (bc)²/4 + (ac)²/4 + (ab)²/4 by definition of the length (here is where the classical Pythagoras enters) as the definition of length is motivated by Pythagoras. Now the three terms can be interpreted as the squares of areas of the sides.
Some 3D printed objects related to geometry:

A blackboard picture done during the proof of Thales theorem:

After proving Thales and Pythagoras theorem, we looked at a less well known theorem related to Phythagoras. It tells that if we cut off a corner from a cube then the squares of the three areas of the triangles adjacent to the corner add up to the square of the area of the face which was cut.

Arithmetic lecture

In this lecture, we look at the development of numbers. How were the numbers introduced, what is the basic structure of the number system? Clay tablets are fun to decipher. They often carried computations. One of the tablets is YBC 7290.

To the left we see 2'20 which is 2*60+20 = 140. Lets call this side length a. On the top this is repeated. This is the height h of the trapezoid and again 140. To the right we see 2, thats a bit ambiguous. The third mark is actually a side of the trapezoid. Remember that there is a place value system but the 0 was not written. It actually is 2'00 = 2*60 = 120. How do we know that? Because in the center we see 5'3'20 which is 5*60² + 3*60 + 20 = 18200. Now if this is the area, then things work out. The trapezoid formula (a+b) h/2 just matches that: indeed, (120+140)/2 = 130 and indeed, (a+b)/2)* h = 130*140=18200. You notice that it is not so easy to decipher. First of all, the scribe (probably a pupil) makes the 10 marks not always horizontal but just slanted. But once you see which ones are the 10 and which ones are the ones, the content of the tablet becomes crystal clear. In this case, there is no doubt. We see here well the progress of mathematics. One was able to compute areas well already. The computation of volume came only later. The first important entry is the Moscow papyrus, where a formula for the volume of a frustrum has been computed. The Babylonians seem not have been able to do that. But remember the time scale. That tablet seen above is maybe 4000 years old.

Here is our solution to a problem in the textbook which we worked on during the class:

"Tablets in a tablet" Tablets in Tablet

Intro lecture

In the intro lecture, we looked at Pythagorean trees [PDF]
We were able to get the first questions pretty easily and even got an induction proof of the area. We also looked at the geometric series of the maximal distance and so established that the tree is always in a finite region. Here is a resource on Albert Bosman (1891-1961), an electrical engineering and math teacher. The book "A.E. Bosman, Het wondere onderzoekingsveld der vlakke meetkunde, N.V. Uitgeversmaatschappij Parcival, Breda, 1957" is in the Harvard libraries. I will bring it in, once it has arrived. [ P.S. The book had arrived and I will post some scaned pages here]

While doing the lesson planning, I wanted first to cover the "prime graph", a topic I was working on in the summer. You find some information here I showed the animation:

But while preparing it became clear that it is not a topic which can be covered in the 20 minute activity allocated for this part on a "lesser known topic" in mathematics. The Pythagoras tree on the other hand worked well.