Hairy ball theorem - Everything2.com

You cannot smoothly comb the hair of a hairy ball without leaving a bald spot or making a parting.

The mathematical formulation of this theorem is:

Theorem If f:S²->S² is a continuous map then there exists a point where x and f(x) are not orthogonal as vectors in R³

To see how the two things are connected. Think of the surface of the hairy ball as the unit sphere S². When we comb the hair we get a continuous map f by associating to each x the direction vector of the hair at that point. Clearly x and f(x) are orthogonal.

One way that a proof of the theorem (and its higher-dimensional analogues) can be obtained is as a consequence of the calculation of the homology groups of the sphere.

The hairy ball theorem is usually attributed to Brouwer or Poincaré.

It is possible to comb the hair smoothly on a torus and that's why the magnetic containers in nuclear fusion are toroidal. (This time the hairs are the magnetic field lines.)

"true" story about Poincaré's baker	Hairy Ball	I had my balls shaved	torus
Words that sound dirty but really aren't	Why your balls seem to cringe sometimes	Borsuk-Ulam Theorem	Nodevertising
Gödel's incompleteness theorem	orthogonal	Some of the pitfalls inherent in removing one's pubic hair	Removing pubic hair
Wiener measure	σ algebra	Saddest thing a woman friend ever told me	Nefarious uses for a beard trimmer
Fundamental theorem of algebra	cat-shaving techniques	continuous