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.
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.
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.
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.