Hairy Ball Theorem

Another fun theorem from  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 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 non-zero 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 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. <>.

Fun Fact suggested by:
Francis Su

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