See schedule, below.

Each project will consist of a paper and a presentation, and should have both mathematical and historical content.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 may work alone or in a group of 2-3 people.  You should choose a topic and have a planning meeting with me by Tuesday, February 29.  At this time, you should let me know if you plan to use this paper as part of the writing requirement for the math major.  By March 30, you should have done some preliminary research and turn in an outline of what you plan to write about.  The paper should be about 5 pages long and is due at the beginning of class on Tuesday, April 25.  The presentation should be about 10-15 minutes long, and will take place during the last two weeks of class.

When writing your paper, be sure to include a bibliography and cite the sources you used.

Possible project topics:

Magic Squares
    A magic square is an arrangement of numbers into a square, like
 

Historical Methods in Algebra
    Throughout history, people have developed different methods of solving equations and doing large computations.  The "double false" method of solving a linear equation was popular through the late 1800's but is never taught today.  Large numbers can be multiplied fairly easily using a doubling method, which is sometimes called "Russian peasant multiplication."
    The historical goal of this project would be to research some of these old algebra techniques, and to describe when they were invented and who used them.  The mathematical goal would be to explain how and why they work.
    Some questions to think about:  1) Start by thinking about the modern method of multiplying large numbers, by taking each digit one at a time and shifting over.  How and why does it work?  2)  Try to find references to Russian peasant multiplication and the double false method, and see what else you can find.

Platonic Solids
    The ancient Greeks liked harmony and symmetry in geometry.  They were interested in polygons where the sides all had the same length and the angles between the sides were all the same (regular polygons).  For any number of sides, there is a regular polygon in the plane with that number of sides, so there are infinitely many.
    In three dimensions, however, the situation changes.  Regular polyhedra (the 3-dimensional analogues of regular polygons) must have all faces, edges, and angles the same.  Instead of infinitely many, there are only five, and you can prove this using some counting arguments.
    The main mathematical goal of this project would be to present this proof.  You should also summarize the history of regular polyhedra (why are they called Platonic?).  For your presentation, you will want to build models of the five polyhedra.  There are several further directions you can go:  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.
    Some questions to start thinking about:  1) What are the five regular polyhedra?  One of them you encounter frequently.  Keep in mind that the faces should all be regular pentagons.  Experiment with drawing several regular polygons of the same size, cutting them out, and taping them together in different ways.

Pythagorean Triples
   There are many sets of integers x, y, and z with the property that x² + y² = z².  Some examples:  3,4,5,  or 5,12,13.  These are called Pythagorean triples.  The mathematical goal of this project will be to find and classify all Pythagorean triples.
    Lists of Pythagorean triples are found among the oldest written documents from different parts of the world.  The historical goal of this project will be to find out about some of these ancient lists.  When and where do they come from?  What is known about their historical and mathematical contexts?
    Some questions to start thinking about:  1)  Find some more examples of Pythagorean triples.  Once you have one triple, how can you use it to find others?  2)  In the examples you found, which numbers are even?  Which are odd?

Fermat's Last Theorem, n=4 case.
    In contrast with the Pythagorean triples, Pierre de Fermat (1601-1665) once wrote in the margin of a book that there are no sets of integers x,y, and z with x³ + y³ = z³, or in general xⁿ + yⁿ = zⁿ for n > 2.  However, he did not give a proof.  This assertion, which is called Fermat's Last Theorem, was generally beleived to be true, but was not proven until 1995.
    The case where n=4 is the easiest to prove, and this would be the mathematical goal of this project.  You should also 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.
    Some questions to start thinking about:  The proof of the n=4 case of Fermat's last theorem relies on the classification of Pythagorean triples.  So, think about the problems listed for the Pythagorean Triples project.

The Chinese Remainder Theorem
    One version of this ancient Chinese mathematical problem is:  A woman is carrying a basket of eggs.  If she counts them by twos, she has one left over.  If she counts them by threes, she has one left over.  If she counts them by fives, she has one left over.  How many eggs are in the basket?
    The Chinese Remainder Theorem discusses when a problem of this kind has a solution, and how to find it.  The mathematical goal of the project will be to explain and prove this theorem.  The historical goal of the paper should be to discuss the origins of the problem, and its context in ancient Chinese mathematics.  How was ancient Chinese mathematics different from ancient Greek mathematics?
    Some questions to think about:  1) Find a solution to the above problem.  2)  Is there more than one solution?

Gaussian Integers
    The Gaussian integers are a subset of the complex numbers.  They are complex numbers a + bi, where a and b must be integers.  They have many properties in common with the integers.  One mathematical goal of this project will be to discuss some of these properties, including Gaussian primes, and factorization of Gaussian integers.  The historical goal of this project will be to discuss when and why they were invented by Gauss.  The other mathematical goal of the project will be to use them in solving the "two squares problem":  which integers can be written as the sum of two perfect squares?
    Some questions to think about:  1)  Practice adding, subtracting, and multiplying the Gaussian integers, using the fact that i² = -1.  2)  Can you write 2 as a product of two Gaussian integers?  What about 3?  3)  Can you write 2 as the sum of two perfect squares?  What about 3?

Quaternions
        The quaternions are a variation of the complex numbers.  Any complex number can be written as x + iy, where x and y are real numbers, and  i² = -1. The quaternions are an extension of this:  Any quaternion can be written as x +iy + jz +kw, where x, y, z, and w are real numbers, and  i² = j² = k² = -1.  The other multiplication rules are: ij = -ji = k, jk = -kj = i, and ki = -ik = j.   The multiplication defined this way is not commutative, since ij and ji are not equal!   The octonians, or Cayley numbers, are another extension of this idea, with seven square roots of -1, in which multiplication is not even associative (so (ab)c is not always the same as a(bc)).
    The historical goal of this project will be to discuss Hamilton and his  invention of the quaternions, including what problem he was working on at the time.  It should also include some background on the historical development of the complex numbers and the Cayley numbers.
    One mathematical goal of this project will be to understand and describe some of the basic properties of the quaternions.  From there, the project could be taken in two different directions.  One would be to study integral quaternions, in a manner similar to the Gaussian integers project, and applications to the "four squares problem":  which integers can be written as the sum of four squares?  Another possible direction is to study the geometry of the quaternions and Cayley numbers, in a manner similar to the Complex numbers and Plane Geometry project.
    Some questions to think about:  1)  Practice adding, subtracting, and multiplying quaternions.  Is (1 + 3i -4j +5k)(2 - 2i -j +k) = (2 - 2i -j +k)(1 + 3i -4j +5k)?   2)  The real numbers form a line, and the complex numbers form a plane.  What is the dimension of the space of quaternions?  What about the octonians (Cayley numbers)?  3)  Any complex number x + iy has a conjugate, x - iy.  Complex conjugation has two important properties:  a) if a number is equal to its conjugate, then it is real, and b) a number times its conjugate is always real.   What should the analogue of complex conjugation be for the quaternions?

Complex numbers and plane geometry
    Functions of one complex variable correspond to transformations of the complex plane.  For instance, the function z -> -z sends x + iy to -x -iy.  In the plane, this corresponds to sending a point to its reflection through the origin.
    The mathematical goal of this project is to discuss some functions of the form z -> (az +b)/(cz +d), and the corresponding transformations of the plane.  You should also discuss the history of the complex numbers and their role in geometry.
    Some questions to think about:  What are the plane transformations corresponding to 1) complex conjugation?  2) Multiplication by i?  3) Multiplication by 3?  4)  Adding a complex number?

Irrational numbers
    The square root of 2, pi, and e are all irrational numbers.   How are they alike?  How are they different?  The historical goal of this project will be to discuss the origins of the idea of an irrational number in ancient Greece, and to recount some historical proofs that the square roots of 2, 3, 5, etc. are irrational.  The mathematical goal of this project will be to discuss the difference between algebraic and transcendental numbers, and to prove that pi and e are irrational.
    Some questions to think about:  1)  Give some more examples of irrational numbers.  2) Why is the square root of 2 irrational?

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.  The historical goal of this project will be to describe methods of ancient navigation, and how spherical trigonometry was used.  The mathematical goal of the project will be to compare plane and spherical trigonometry, and give some examples of problems that can be solved using spherical trigonometry.
    Some questions to think about:  What does the position of the north star tell you about where you are on earth?

Fermat's Little Theorem and Euler's Theorem
    If a is any integer, a² - a is even (a multiple of 2), and a³ - a is a multiple of 3.  Is it true that a⁴ - a is a multiple of 4?  This question is related to Fermat's Little Theorem, and Euler's generalization of it.  The mathematical goal of this project will be to prove these two theorems.  The historical goal will be to discuss Fermat, Euler, and developments in number theory in the 17th and 18th centuries.
    Some questions to think about:  1)  Check and/or prove that the two statements I made are true!  2) Check some values to see if it holds for 4, and for other exponents.

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.
    The mathematical goals of this paper are to demonstrate the construction using origami and to prove that it in fact trisects the angle.  The project should include some discussion of the history behind ruler and compass constructions, and the trisection problem in particular.  You should also discuss the history of origami.
     Some questions to think about:  How do you bisect an angle using a ruler and compass?  How else could you do it if you are allowed to fold the paper?  What about other standard constructions, like a perpendicular bisector of a line segment or an equilateral triangle?

Permutation groups
    A permutation of a set is a way of changing the order of the elements.  For example, if the set is {a,b,c}, one permutation is to exchange the first two elements, yielding {b,a,c}.  The mathematical goal of this project is to describe the group of permutations of a set of n elements.  An important result that should be proven is that any permutation can be broken into steps, where at each step two elements are exchanged.  You may also discuss how to represent other finite groups as permutation groups, or how to represent permutations as matrices.
    The historical goal of this paper will be to discuss briefly how abstract group theory arose in the late 19th and early 20th centuries.
    Some questions to thinking about:  How many permutations are there of the set {a,b,c}?  What about the set {a,b,c,d}?  How can you express each one as described above, as a set of steps where at each step you exchange two elements?

The Perpetual Calendar
    On what day of the week were you born?  What day of the week was July 4, 1776?  The perpetual calendar is an algorithm that uses congruences to find out the day of the week given the date.  The mathematical goal of this project will be to understand and explain this algorithm and how it works.  The historical goal of the project will be to study some examples of ancient calendar systems, and how they were related to the development of ancient mathematics.
    Some questions to think about:  If your birthday falls on a Tuesday next year (2001), what day of the week will it be the following year (2002)?  What about 2003?  2004? 2005?  2010?  Don't forget that 2004 and 2008 will be leap years.

Continued Fractions
    There are many different ways to write the same number.  For example, we can write 2 7/8 as 2.875, or we can write it as 2 + 1/(1 + 1/7).  This last expression is called a continued fraction.  Any real number can be written in a similar form.  Just like terminating and repeating decimals, one can analyze what kind of numbers have terminating or repeating continued fractions.  The mathematical goal of this project will be to do this analysis.  The historical goal of the project will be to study how continued fractions were used historically.
    Some questions to think about:  1)  Check that the expression above is really equal to 2 7/8.  2)  Write the continued fraction 1 + 1/( 1 + 1/4) in a more usual form.  3) Write 16/9 as a continued fraction.

Tentative Schedule  (last modified 4-2-2000)

Tuesday, April 25:

Irrational Numbers	Jill, Jeff and Ryan
Platonic Solids	Al, Joseph and Claudine
Historical Methods	Carlington
Perpetual Calendar	Robyn and Michael C.

Thursday, April 27:

Pythagorean Triples	Dawn
Fermat's Last Theorem	Yvonne and Janice
Magic Squares	Bill
Navigation	Patricia

Tuesday, May 2:

Fermat's Little Theorem	Felix
Continued Fractions	Hemal
Origami Constructions	Damon and Sawi
Gaussian Integers	Sheila

Thursday, May 4:

Sets, Cardinality, and Cantor	Elizabeth and Roe
Platonic Solids	Shawn and Michael A.
Chinese Remainder Theorem	Natasha and Ernesto