**Sample Abstract**

Although there are those who still attempt
to solve this classical problem, it is now a well
established mathematical fact that it is impossible
to trisect an angle using only a compass and a
straightedge. But this doesn't mean that the ancient Greeks didn't know
how to trisect an angle. In this talk, we will discuss why the compass
and straightedge construction is impossible and show two simple methods
for trisecting an angle with additional equipment: a compass and ruler
construction known to Archimedes, and a more recent technique using
origami.

**Description**

The mathematics behind this talk comes from the discussion of
compass and straightedge constructions in
*What is Mathematics* by Courant and Robbins and from
the
discussion of origami constructions on
Thomas Hull's Origami Mathematics Page.
After sketching a proof of the impossibility of trisection
by compass and straightedge, we perform the Archimedes
compass-and-ruler
trisection and discuss the mathematics that makes trisection by
paper folding possible. At the end of the talk, each member of
the audience trisects an angle by origami.

**Level**

Advanced High School and up.

Most people will get something out of this talk, and anyone who can fold a piece of paper can perform the trisection at the end. However, to truly appreciate the talk, the listener should be comfortable with high-school algebra and know some plane geometry. Ideally, at least some of the audience will have seen basic compass-and-straightedge constructions.

An important feature of this talk is that it gives a sketch of
the classification of constructible numbers. For some groups of
students, it is a shame to give this away in an informal talk,
so *caveat emptor*.

**Mechanics**

While most of the talk is given on a chalkboard, I need a
small overhead (or computer) projector screen that can be
used in conjunction with
the board. (The projector displays the compass-and-straightedge axioms
for the first half of the talk and the origami axioms for the second
half.)
It is helpful if each member of the audience has a desktop to
fold on.

**Appearances**

- Amherst College MAA Student Chapter, February 2000
- Vassar College, April 2000
- Radical Pi, Ohio State University, May 2001