Hairy ball theorem

The hairy ball theorem of algebraic topology (sometimes called the hedgehog theorem in Europe) 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. The theorem was first stated by Henri Poincaré in the late 19th century, and first proven in 1912 by Luitzen Egbertus Jan Brouwer. The hairy ball theorem of algebraic topology (sometimes called the hedgehog theorem in Europe) 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. The theorem was first stated by Henri Poincaré in the late 19th century, and first proven in 1912 by Luitzen Egbertus Jan Brouwer. The theorem has been expressed colloquially as 'you can't comb a hairy ball flat without creating a cowlick' or 'you can't comb the hair on a coconut'. 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, 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. A meteorological application of this theorem involves considering the wind as a vector defined at every point continuously over the surface of a planet with an atmosphere. As an idealisation, take wind to be a two-dimensional vector, so that any vertical motion is ignored. One scenario, in which there is absolutely no wind, corresponds to a field of zero-vectors. But in the case where there is at least some wind, the Hairy Ball Theorem dictates that at all times there must be at least one point on a planet with no wind at all and therefore a tuft. This corresponds to the above statement that there will always be p such that f(p) = 0. In a physical sense, this zero-wind point will be the center of a cyclone or anticyclone. Like the swirled hairs on the tennis ball, the wind will spiral around this zero-wind point - under the assumptions made, it cannot flow into or out of the point. Thus, given at least some wind on Earth, there must at all times be a cyclone or anticyclone somewhere. The center with zero wind can be arbitrarily large or small. Mathematical consistency dictates the wind forms a cyclonic wind pattern for at least one point on the planet, but this does not require the cyclone be a violent storm.