Brouwer Fixed Point Theorem

One of the most useful theorems in mathematics is an amazing topological result known as the Brouwer Fixed Point Theorem.

Take two sheets of paper, one lying directly above the other. If you crumple the top sheet, and place it on top of the other sheet, then Brouwer’s theorem says that there must be at least one point on the top sheet that is directly above the corresponding point on the bottom sheet! Do you believe that?

In dimension three, Brouwer’s theorem says that if you take a cup of coffee, and slosh it around, then after the sloshing there must be some point in the coffee which is in the exact spot that it was before you did the sloshing (though it might have moved around in between). Moreover, if you tried to slosh that point out of its original position, you can’t help but slosh another point back into its original position!

More formally the theorem says that a continuous function from an N-ball into an N-ball must have a fixed point. Continuity of the function is essential (if you rip the paper or if you slosh discontinuously, then there may not be fixed point).

Presentation Suggestions:
Bring a coffee cup and 2 sheets of paper with you and demonstrate as you present the fun fact. Draw a grid on the paper, number the gridboxes, then xerox that sheet of paper. After you crumple the paper, you can say that at least one number is on top of the corresponding number on the lower sheet of paper. Alternatively, bring a map of Claremont (or whatever city you are in) to class and drop it on the floor—then there must be some point in the map lying directly over the point that it represents! A good follow-up Fun Fact is the Borsuk-Ulam Theorem.

The Math Behind the Fact:
Fixed point theorems are some of the most important theorems in all of mathematics. Among other applications, they are used to show the existence of solutions to differential equations, as well as the existence of equilibria in game theory. There are many proofs of the Brouwer fixed point theorem. The advanced student may wish to see if she can show the equivalence of this theorem with Sperner’s lemma, which yields a rather nice elementary proof.

How to Cite this Page: 
Su, Francis E., et al. “Brouwer Fixed Point Theorem.” Math Fun Facts. <https://www.math.hmc.edu/funfacts>.

References:
Any undergraduate text on .

Fun Fact suggested by:
Francis Su

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