Crumpling Paper and the Brouwer Fixed Point Theorem

Sample Abstract
The Brouwer Fixed Point Theorem is a result in topology that has some startling and entertaining consequences. For instance, it implies that if you take a map of Maryland and spread it out on your desk, there is a point on the map that lies directly over the part of Maryland it represents. The speaker will attempt to demonstrate a (rather implausible) method of applying the theorem to convert crumpled paper into cash.

To follow this talk, the listener should have some notion of what it means for a function to be continuous.

The talk is presented as a wager. I take a sheet of paper, set it on a desktop, mark its position, and have an audience member crumple the sheet and place it within the marks. I then propose the following wager: I will bet any member of the audience that there is a point on the page that lies directly over its pre-crumple position. For the remainder of the talk, I convince the audience that I may collect on this bet.

This talk was inspired by a lecture given by Francis Su (creator of the Math Fun Facts web page) to the Harvard Graduate Student Seminar. His subject was applications of Sperner's Lemma, one of which is a straightforward proof of the Brouwer Fixed Point Theorem.


While all of the continuity arguments are supplemented with large doses of intuition, a listener who has not at least seen continuity of real, one-variable functions before will struggle with this talk. Knowledge of the epsilon-delta definition of continuity is useful but not necessary. There is also some uniform continuity hand-waving towards the end of the talk, my assumption being that much of the audience will not understand uniform continuity.

I need a desktop (though the floor might do) and an overhead projector. Ideally, I should have a co-conspirator to take the sucker bet.


Transparencies for Brown SUMS Talk (HTML)

First Talk

Back to Expository Talks

Back to my home page