Hairy ball theorem
The hairy ball theorem of algebraic topology states that there is no nonvanishing continuous tangent vector field on even-dimensional n-spheres. For the ordinary sphere, or 2‑sphere, if f is a continuous function that assigns a vector in R3 to every point p on a sphere such that f(p) is always tangent to the sphere at p, then there is at least one p such that f(p) = 0. In other words, whenever one attempts to comb a hairy ball flat, there will always be at least one tuft of hair at one point on the ball. The theorem was first stated by Henri Poincaré in the late 19th century.
This is famously stated as "you can't comb a hairy ball flat without creating a cowlick", "you can't comb the hair on a coconut", or sometimes "every cow must have at least one cowlick." It can also be written as, "Every smooth vector field on a sphere has a singular point." It was first proved in 1912 by Brouwer.
From a more advanced point of view: every zero of a vector field has a (non-zero) "index", and it can be shown that the sum of all of the indices at all of the zeros must be two. (This is because the Euler characteristic of the 2-sphere is two.) Therefore, there must be at least one zero. This is a consequence of the Poincaré–Hopf theorem. In the case of the torus, the Euler characteristic is 0; and it is possible to "comb a hairy doughnut flat". In this regard, it follows that for any compact regular 2-dimensional manifold with non-zero Euler characteristic, any continuous tangent vector field has at least one zero.
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