You may work alone or with one other person on your project.  Each project will consist of a paper and a presentation, and should have both mathematical and historical content.   In particular, each project should have some historical background of the problem and solution being presented, and should contain the proof of at least one theorem.  You should choose a topic and have a planning meeting with me by Thursday, October 28.  The paper should be about 5 pages long and is due at the beginning of the last class, on Tuesday, December 14.  The presentation should be about 10-15 minutes long, and will be scheduled during the last few weeks of class.

Possible project topics:

Platonic Solids
    This project would study the five regular solids.  It should include some history, and a proof that there are exactly five platonic solids.  If you do this project, you should also build models of the five solids.  You may wish to talk about the symmetries of the Platonic solids, other solids which are not Platonic but also have many symmetries, variations in higher dimensions, or further applications of the formula in the proof.

Fermat's Last Theorem, n=4 case  (history of the theorem, proof of n=4 case)
    This project should discuss the history of Fermat's Last Theorem, from Fermat's original note in the margin to the final proof by Andrew Wiles in 1995.  It should also include a proof of the theorem in the case where n=4.

Quaternions
    The quaternions are a variation of the complex numbers, but are not commutative.  This project should discuss Hamilton's invention of the quaternions, and some of their properties.  It may also include some discussion of the  Cayley numbers or octonians, a further extension which are not even associative!  You may wish to discuss applications to number theory or to physics.

Euler's Theorem and Fermat's Little Theorem
    These are two theorems involving congruences.  This project should include a discussion of Euler's "phi" function, proofs of the two theorems, and some history of each.  You may also wish to discuss Wilson's theorem.
 
The Chinese Remainder Theorem
    The Chinese Remainder Theorem discusses when a system of linear congruences has a solution.  This project should include some history and a proof of the theorem.

Quadratic residues
    This project will discuss which integers are squares modulo another integer.  This project should include biographical information about Legendre and a discussion of Legendre symbols and quadratic reciprocity.

Gaussian integers
    Gaussian integers are numbers of the form a + bi, where a and b are integers and i² = -1.  This project should include a description of the Gaussian integers, and a comparison of their properties to the properties of the integers, including a proof of unique factorization.  It should also discuss when and why Gauss invented them.

Complex numbers and plane geometry
    Functions of one complex variable correspond to transformations of the complex plane.  This project should discuss the history of the complex numbers, the polar form of a complex number, and the plane geometry of some transformations.

Trisection of angles using origami
    We will prove in class that one cannot trisect an angle under the usual rules of ruler and straightedge construction.  However, if one is allowed to fold the paper, it is possible to trisect an angle.  This project should demonstrate how it is done and include a proof that the procedure described does in fact produce an angle which is one third of the original angle.  The project should also include some discussion of the history behind ruler and compass constructions.

Permutation groups
    A permutation group is the group of permutations of a set.  This project should include a description of permutation groups, several examples, a proof that every permutation is a product of 2-cycles, and some discussion of even and odd permutations.  You may also discuss how to represent other finite groups as permutation groups, or representations of permutation groups as groups of matrices.

Rubik's cube and groups
    The Rubik's cube and related puzzles give some interesting examples of groups.  This project should include the history of the Rubik's cube and related puzzles.  It should also include a discussion of the underlying groups, possible and impossible states of the cube, and methods of solving the puzzle.

Irrationality of pi
     This project would focus on a proof that pi is irrational using continued fractions.  It should also include some historical discussion of continued fractions related to pi.  It could also include a discussion of transcendental numbers and why pi and e are transcendental.

Diophantine equations
    A Diophantine equation is a linear equations with solutions in integers.  This project should include a proof of the theorem which states when solutions to a Diophantine problem exist, as well as several examples.  It should also include some discussion of the history of Diophantine equations in different cultures.

Navigation and spherical trigonometry
    Spherical trigonometry is an extension of plane trigonometry to three dimensions.  Sailors have used it for centuries to navigate using the stars.  This project should briefly discuss the history of spherical trigonometry as used for navigation, and give some examples of how it can be used.