Hairy ball theorem
The hairy ball theorem of algebraic topology (sometimes called the hedgehog theorem in Europe)^{[1]} states that there is no nonvanishing continuous tangent vector field on evendimensional nspheres.^{[2]}^{[3]} For the ordinary sphere, or 2‑sphere, if f is a continuous function that assigns a vector in R^{3} 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,^{[citation needed]} and first proven in 1912 by Luitzen Egbertus Jan Brouwer.^{[4]}
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".
Contents
Counting zeros
Every zero of a vector field has a nonzero) "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 2sphere 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 2dimensional manifold with nonzero Euler characteristic, any continuous tangent vector field has at least one zero.
Cyclone consequences
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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 twodimensional vector, so that any vertical motion is ignored. One scenario, in which there is absolutely no wind, corresponds to a field of zerovectors. 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 zerowind point will be the center of a cyclone or anticyclone. Like the swirled hairs on the tennis ball, the wind will spiral around this zerowind 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.
The assumptions break down in reality, as the air above the earth has multiple layers, but for each layer there must be a point with zero horizontal wind speed.
Application to computer graphics
A common problem in computer graphics is to generate a nonzero vector in R^{3} that is orthogonal to a given nonzero one. There is no single continuous function that can do this for all nonzero vector inputs. This is a corollary of the hairy ball theorem. To see this, consider the given vector as the radius of a sphere and note that finding a nonzero vector orthogonal to the given one is equivalent to finding a nonzero vector that is tangent to the surface of that sphere where it touches the radius. However, the hairy ball theorem says there exists no continuous function that can do this for every point on the sphere (equivalently, for every given vector).
Lefschetz connection
There is a closely related argument from algebraic topology, using the Lefschetz fixedpoint theorem. Since the Betti numbers of a 2sphere are 1, 0, 1, 0, 0, ... the Lefschetz number (total trace on homology) of the identity mapping is 2. By integrating a vector field we get (at least a small part of) a oneparameter group of diffeomorphisms on the sphere; and all of the mappings in it are homotopic to the identity. Therefore, they all have Lefschetz number 2, also. Hence they have fixed points (since the Lefschetz number is nonzero). Some more work would be needed to show that this implies there must actually be a zero of the vector field. It does suggest the correct statement of the more general PoincaréHopf index theorem.
Corollary
A consequence of the hairy ball theorem is that any continuous function that maps an evendimensional sphere into itself has either a fixed point or a point that maps onto its own antipodal point. This can be seen by transforming the function into a tangential vector field as follows.
Let s be the function mapping the sphere to itself, and let v be the tangential vector function to be constructed. For each point p, construct the stereographic projection of s(p) with p as the point of tangency. Then v(p) is the displacement vector of this projected point relative to p. According to the hairy ball theorem, there is a p such that v(p) = 0, so that s(p) = p.
This argument breaks down only if there exists a point p for which s(p) is the antipodal point of p, since such a point is the only one that cannot be stereographically projected onto the tangent plane of p.
Higher dimensions
The connection with the Euler characteristic χ suggests the correct generalisation: the 2nsphere has no nonvanishing vector field for n ≥ 1. The difference between even and odd dimensions is that, because the only nonzero Betti numbers of the msphere are b_{0} and b_{m}, their alternating sum χ is 2 for m even, and 0 for m odd.
See also
Notes
 ^ Renteln, Paul (2013). Manifolds, Tensors, and Forms: An Introduction for Mathematicians and Physicists. Cambridge Univ. Press. p. 253. ISBN 1107659698.
 ^ Burns, Keith; Gidea, Marian (2005). Differential Geometry and Topology: With a View to Dynamical Systems. CRC Press. p. 77. ISBN 1584882530.
 ^ Schwartz, Richard Evan (2011). Mostly Surfaces. American Mathematical Society. pp. 113–114. ISBN 0821853686.
 ^ GeorgAugustUniversität Göttingen Archived 20060526 at the Wayback Machine
References
 Eisenberg, Murray; Guy, Robert (1979), "A Proof of the Hairy Ball Theorem", The American Mathematical Monthly, 86 (7): 571–574, doi:10.2307/2320587
Further reading
 Jarvis, Tyler; Tanton, James (20030723), The Hairy Ball Theorem via Sperner's Lemma (PDF)
 Reich, Henry (2011), OneMinute Math: Why you can't comb a hairy ball, New ScientistTV
 Richeson, David S. (2008), Euler's Gem: The Polyhedron Formula and the Birth of Topology, Princeton University Press, ISBN 0691126771. See Chapter 19, "Combing the Hair on a Coconut", pp. 202–218.
External links
 Weisstein, Eric W. "Hairy Ball Theorem". MathWorld.