Figure 1

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 Nball into an Nball 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 floorthen there must be some
point in the map lying directly over the point that it
represents!
A good followup Fun Fact is the BorsukUlam 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.
<http://www.math.hmc.edu/funfacts>.
