The term project part of the course consists of working with a group
of two other students on a paper about a mathematical topic
which you will present to the class. I've decided to scrap the final exam in the course
and make the final project count more heavily.
of your presentation and the paper you write with other
members of your project team will now count for 35% of your course
presentations should be about 25 minutes long. There will
be four project groups, and the four presentations will be on Friday, December 19 from 12 noon
to 2 p.m.. This is the time that the registrar had reserved for the final exam in the course.
Please have a look at the list of projects below and e-mail me an ordered list of your first three choices. I will then put people in touch with other class members interested in the same project. We will also set up a time so that everyone can meet with me to discuss their project. If you think of a mathematical project not on this list which you'd like to pursue, that would be fine. Please e-mail me about what you have in mind and we will meet to discuss this.
This project has to do with understanding mathematics relating to music, harmony and rhythym. In class we discussed how continued fractions produce the best rational approximations to real numbers. This pertains to pleasing an unpleasing musical intervals, and to how to choose the number of tones in a musical scale. See this web page as a starting point for this topic. There are many other reference on the web about good and bad sounds. This Ted Talk about the worlds ugliest music. This is related to constructing the best sonar pings for use in detecting submarines.
The greeks asked whether the ratio of the circumference of a circle to its diameter was a rationality. This was first proved by Johann H. Lambert in 1768. A very nice Proof due to Ivan Niven requires only some effort and minimal amount of calculus.
This project is about a the mathematics behind a particular print by M. C. Escher. A few years ago, H. W. Lenstra Jr. discovered a pattern behind this print which connects it to so-called elliptic curves. Roughly speaking, the print comes from taking a picture on a doughnut with one hole and covering the plane by infinitely many copies of this picture using the exponential function. The following Escher website gives a quick introduction to this topic. Be sure to check out the animations of what happens when one dives into the center of the picture. Here is a more detailed article about the mathematics involved.
Here are some notes on an analogous form of music. The main feature of this kind of music is that there is a constant A > 1 such the set S of times t at which is a note is played is taken into itself by multiplcation by A. This leads to the fact that if a second player plays the same score at speed 1/A times that of the first player, every time the second player plays a note, the first one will as well. When A = 2, the Pachibel canon illustrates how several instruments can be playing notes at speeds related to one another by powers of 2. When A is the Golden ratio, one has what called Fibonacci music
In class we will discuss briefly Leonardo da Vinci's discovery of the 17 basic wallpaper patterns. This project consists of explaining in more depth the group theory underlying this classification, as well as the discovery of non-periodic "Penrose tilings" of the plane. The study of Penrose tilings has been of great interest to crystallographers, who have found some substances which have similar properties in three dimensions. These tilings also produce some very pretty pictures!.
Fibonacci numbers and the golden ratio arise in various models of how plants grow, e.g. in the spiral patterns of sunflowers. This project would involve explaining some of these models, e.g. this one, and the natural processes which lead to the appearance of Fibonacci numbers and the golden ratio. Here is a set of notes with more pictures.
Beyond the ordinal numbers there is a vastly bigger class of numbers called the surreal numbers by John Conway, their inventor. These include, for example, omega - 1 as well as 1/omega. The main idea in defining surreal numbers is that these should measure the relative strength of the position of one player or another in various kinds of games. If you are interested in game theory and number theory, this would be an attractive project. If this project interests you, I can loan you a copy of the book "On numbers and games" by John H. Conway, which is the main reference in the subject.
This project is about one model for how to make a choice from among a sequence of alternatives being considered. Imagine that you can interview a sequence of candidates for a job of some kind. After each interview you either have to decide to choose the candidate just interviewed or move on the next, and if you move on then you can't return to choose an earlier candidate. What is the best strategy to use if you want to maximize the chances of ending up with the best candidate? What if you want to maximize the expected rank of the candidate you choose? Here is a Survey on the secretary problem. Here is a More advanced article.