Math E-320: Spring 2013
Teaching Math with a Historical Perspective
Mathematics E-320:
Instructor: Oliver Knill
Office: SciCtr 432

Dynamics lecture

Cryptology lecture

Cryptology becomes more and more important today. At the moment, secure cloud computing encryption as well as cell phone encryption are the holy grail. While there is secure encryption (we will learn about one today), in these two cases, the encryption has to be fast enough to become both practical and still remain secure. Here is a recent discussion about a cloud computing which became wider known because of a DMCA takedown request. The story illustrates again that security through obscurity does not work. The wisdom of the crowd (like stackexchange) will hunt you down. The story also shows that with DMCA takedown notices, companies shoot themselves into the foot because the Steisand effect will make the story even wider known and lead to bad publicity.

Analysis lecture

Here is the movie clip in which Spok learns about the formula dim = -log(n)/log(r) for dimension. Jeff sent the Snowflakes link. Actually, snowflakes look different in reality: news of a week ago.

Li Zhou (a visitor of the math in movie page) wrote about the movie Silk, in which the Menger sponge appears.

Topology lecture

Here is an example of a surface which is deformed and rotated in space: Youtube. It is a Calabi-Yau manifold and important in string theory.

Probability lecture

  • Jeff has the following link which mentions the Monty Hall problem.
  • I mentioned the book "The pleasures of Probability". Here are the first few pages.
  • To the right you see the cover of a book dedicated to the Monty hall problem.
Added April 17, 2013: Your life in Pi, an article in slate discusses for the public the question, whether pi is normal. Of course, the article is a pun on the movie "Pi". The question is an old one and has to do with randomness. Are the digits of pi random? This looks like an ill defined question but one can perfectly well make sense of it. If one looks at pi as well as all its translates 10n mod 1 we get a sequence of integer strings. One can now look at the "hull" H which is the set of accumulation points of this sequence. The map T(x) = 10 x mod 1 acts on this space H as a transformation. The space H becomes a probability space with a measure P. If we have a subset A of H, define P[A] = limn to infinity (1/n) times the number of Tk(x) which are in H. If we look at the random variables Xk which give for a point x in H the k'th digit, then the conjecture is that these random variables are independent. Furthermore, the space H should be the entire interval [0,1] and P should be the Lebesgue measure, the function on sets for which P[ [a,b] ] = b-a.
This construction might need a bit time to be understood because it is ergodic theoretic, but it is true that any real number defines so a probability space and a sequence of random variables. It is known from ergodic theory that almost all real numbers in the interval [0,1] have the property that they generate independent random variables like that. Of course, not all numbers do. If we take the number 5/7 for example then the hull consists of a probability space with 7 elements on which the probability measure P is the normalized counting measure |A|/7. Now the random variables Xk are not independent because Xk+7 = Xk. We can not use the digits of 5/7 = 0.714285714285714285714285714285714285.... as a number generator.

Set theory lecture

Calculus lecture

April 21: from Tucker, a 1a student:

"An infinite number of mathematicians walk into a bar. The first one orders one beer. The second one orders half of a beer. The third, a quarter of a beer. The bartender looks at them all and says 'You really need to know your limits,' and pours two beers."

Algebra lecture

Here are the Rubik cube records. We only shortly mentioned the "God number" for the cube. Here is a website about it. It is known since 2010 that the diameter of the cube is 20. Here is a Javascript game, I built 14 years ago. And here the "lights out game". Jeff mentioned the song "The Ballad of Gallois short piece on MP3 and the youtube song featuring some group theory.

Number Theory lecture

More about Jennys number (thanks John). This article links to the webcomic, where Jennys number is approximated by
Pi^2(7^(E/1-1/E)-9)  = 867.5309...
  • We also built the Catalan system on the spot with the map
    T[x_] := If[x == 1,1,DivisorSigma[1,x]-x];
    Here is the orbit of a number like 123456
    ListPlot[NestList[T, 123456, 100],Joined->True]

    Geometry lecture

    Much of the class was already acquainted with the results we discussed. We could prove Pythagoras, Hyppocrates, Thales in detail, saw computer verifications of some results in planimetry. The demonstration files shown in class can be found on the website of last year.
    An other theorem which does not appear in textbooks and which I had learned in high school is on the Worksheet. In some way, it provides an other possibility to compute the product of two numbers.
    We were debating whether Hypocrates can be proven without Pythagoras. Here is an entirely visual proof in a picture book edited by Jackson:
    There are of course also illustrations about Pythagoras.
    A book entirely about Pythagoras is by Maor:

    Arithmetic lecture

    Having had no clay handy, we wrote onto Chewing gum. And it seemed have worked pretty well.

    We discussed the dilemma how the Babylonian mathematicians would distinguish between numbers like 603 and 602 because there was no zero available. Mirjanda mentioned place holders which in some sense also introduce zero. If this is counted as zero, then also the Babylonians have dealt with zero. Here is an article with a picture of places, where zero was invented. And this picture shows the placeholder for zero.

    Here is an article about the history of "0". Also this Scientific american article only talks about the zero found in India. I believe the realization that the Mayans had introduced "0" independently and earlier came only later.

    Added March 11, 2013: An interesting article in Slate about an arithmetic question.

    Introduction lecture

    Working with the parabola was fun. Here are a few comments to the lecture and links and sources.
  • Please send questions and comments to
    Math E320| Oliver Knill | Spring 2013 | Extension School | Harvard University