Math 110-02, Fall 2008
Final Project Topics

This course has a final project that counts for 12% of the total grade for the course. Some possible topics for the project are listed below. There are two types of project: The first type is a five-to-seven page research topic on some mathematical topic. The second type is some kind of "hands-on" project such as building something that illustrates a mathematical idea, along with a short one or two page paper discussing what you have done. Almost all the projects of the second type are based on sections of Symmetry, Shape, and Space that we will not cover in class or are extensions of sections that we will cover.

You are not limited to the topics in this list. You are welcome to design some other project, in consultation with me. Your project must be related to mathematics, but does not have to have a lot of specifically mathematical content. It could, for example, be primarily historical, sociological, or artistic in nature.

The project will be due at the end of the term, but you will have to choose your project well before that. More information on the schedule will be available later in the term.

Possible Topics for Research Papers

• Math Attitudes. Do a survey of attitudes about math among your friends, and analyze the result in the light of Keith Devlin's ideas in The Math Instinct.
• Innumeracy. There has been a lot of concern expressed over "innumeracy," which is something like illiteracy only with numbers instead of words. You might report on this concern or analyze it in the light of The Math Instinct.
• Gemetria. Write a paper about gemetria, the form of number mysticism that is based on numerical values of letters in words. Gemetria is mostly associated with the Jewish Kaballah, using the Hebrew alphabet, but it is also possible using the Greek alphabet.
• Fractals. Learn more about fractals and their role in nature and art.
• Escher's Art. The artist M.C. Escher was noted for using the mathematics of symmetry in his artwork. We will be looking at this later in the course. You might want to look at this in more depth.
• Flatland. Flatland is a famous nineteenth century book about a two-dimensional world. We will watch an animated film based rather loosely on this book in class. You could read the original book and report on it, possibly comparing it to the film.
• Math and Gambling. Write a paper about probability theory and its application to gambling. You could also look at how probability theory arose historically out of an interest in gambling odds.
• How to Lie with Statistics. I have a copy of a book called How to Lie With Statistics that would make a good basis for a report.
• The Pythagoreans. Investigate the philosophy and history of the Pythagoreans in more depth.
• Music and Numbers. Look into the mathematical nature of music. How are numbers related to harmony and musical scales?
• Mayan numbers and calendar. These are discussed briefly in ZERO, and would certainly be worth a close look.
• Calculating π. Look at the quest to calculate more and more digits of the number π.
• The Search for Large Primes. Look at the search for very large prime numbers. The largest known prime number as of 2006 was 2^32582657-1, which has 9,808,358 digits. See http://primes.utm.edu/largest.html.
• Newton vs. Leibnitz. Investigate the historical controversy over who really invented calculus.
• Famous solved problems. There are a number of famous mathematical problems that went unsolved for many years before they were finally solved. You might want to write a paper that looks at the history of one of these problems and its solution. For example:
  • The Four-Color Problem: Is it always possible to color a map using just four colors?
  • Fermat's Last Theorem: Are there positive integers x, y, z, and n, with n >2, such that xⁿ + yⁿ = zⁿ?
  • Hilbert's Tenth Problem: Is there an algorithm for solving Diophantine equations (for which the only allowable solutions are integers)?
• Famous unsolved problems. There are also many famous mathematical problems that remain unsolved. You might want to look into one of these. For example:
  • Goldbach's Conjecture. Can every even integer greater than 2 be written as a sum of two prime numbers?
  • Twin Prime Conjecture. Is there an infinite number of twin primes? (Two integers N and N + 2 that are both prime are said to be twin primes.)
• Millennium Prizes. You could write a paper on the phenomenon of the "millennium prizes." These are $1,000,000 prizes that have been offered for the solution of certain mathematical problems. One of the problems, the "Poincaré Conjecture," has been solved just within the past few years. See http://www.claymath.org/millennium/.

Possible Topics for Hands-On Projects

• We will have several encounters with the art of M. C. Escher in the course. You will create some simple examples of Escher-style art as homework, but you might try producing some more advanced and interesting mathematical art using Escher's techniques.
• Do some mathematical origami, such as folding a regular pentagon from a square piece of paper, following the exercises in Section 3.3 of Symmetry, Shape, and Space.
• Build some mechanical linkages, following the exercises in Section 3.5 of Symmetry, Shape, and Space.
• Build several kaleidoscopes, using mirrors with several different angles. See pages 135--136 in Symmetry, Shape, and Space.
• Build and fly a tetrahedral kite, as discussed on pages 248--250 of Symmetry, Shape, and Space.
• Do some experiments with three-dimensional mirror symmetry, following the exercises in Section 8.2 of Symmetry, Shape, and Space.
• Draw some optical illusions, or build some 3D models that illustrate optical illusions, as discussed in Section 10.2 of Symmetry, Shape, and Space.
• Build some wire frames for making soap bubbles, as discussed in Section 11.5 of Symmetry, Shape, and Space (and give a demonstration in class!).
• Make some mazes, using the methods discussed in Section 12.3 of Symmetry, Shape, and Space.