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.)