What is "hairy ball theorem"

n. (context mathematics English) A theorem stating that there is no nonvanishing continuous tangent vector field on the sphere.

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 R 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.