Figure 1

Another fun theorem from topology is the Hairy Ball
Theorem. It states that given a ball with hairs all over
it, it is impossible to comb the hairs continuously and have
all the hairs lay flat. Some hair must be sticking
straight up!
A more formal version says that any
continuous tangent vector field on the
sphere must have a point where the vector is zero.
Is the same true on a torus?
Presentation Suggestions:
Draw a picture of a sphere on the board, and have students
think together with you how trying to
draw a nonzero vector field
would cause "problem points", where the field is not
continuous.
The Math Behind the Fact:
If you've done the Fun Fact on the
Euler characteristic,
students will find it very surprising that the number of
"problem points" of a vector field on a surface
is related to the Euler characteristic of that
surface! Namely, every point has an "index" that describes
how many times the vector field rotates in a neighborhood
of the problem point. The sum of the indices of all the
vector fields will be the Euler characteristic. Since
the torus has Euler number 0, it is possible to have a
vector field on it without any "problem points".
A related Fun Fact is the Ham Sandwich Theorem.
How to Cite this Page:
Su, Francis E., et al. "Hairy Ball Theorem."
Math Fun Facts.
<http://www.math.hmc.edu/funfacts>.
