 A failed attempt to comb a hairy ball flat, leaving two tufts at the top and bottom. The hairy ball theorem of algebraic topology states that there is no nonvanishing continuous tangent vector field on the 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. Image File history File links No higher resolution available. ...
Image File history File links No higher resolution available. ...
Algebraic topology is a branch of mathematics in which tools from abstract algebra are used to study topological spaces. ...
In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. ...
In differential geometry, one can attach to every point p of a differentiable manifold a tangent space, a real vector space which intuitively contains the possible directions in which one can pass through p. ...
Vector field given by vectors of the form (-y, x) In mathematics a vector field is a construction in vector calculus which associates a vector to every point in a Euclidean space. ...
In physics and in vector calculus, a spatial vector, or simply vector, is a concept characterized by a magnitude and a direction. ...
In mathematics, the word tangent has two distinct but etymologically-related meanings: one in geometry and one in trigonometry. ...
This is famously stated as "you can't comb a hairy ball flat". It was first proved in 1912 by Brouwer, see [1]. 1912 (MCMXII) was a leap year starting on Monday in the Gregorian calendar (or a leap year starting on Tuesday in the 13-day-slower Julian calendar). ...
Luitzen Egbertus Jan Brouwer (February 27, 1881 - December 2, 1966), usually cited as L. E. J. Brouwer, was a Dutch mathematician, a graduate of the University of Amsterdam, who worked in topology, set theory, measure theory and complex analysis. ...
 A hairy doughnut, on the other hand, is quite easily combable. In fact from a more advanced point of view it can be shown that the sum at the zeros of such a vector field of a certain 'index' must be 2, the Euler characteristic of the 2-sphere; and that therefore there must be at least some zero. This is a consequence of the Poincaré–Hopf theorem. In the case of the 2-torus, the Euler characteristic is 0; and it is possible to 'comb a hairy donut flat'. In this regard, it follows that for any compact regular 2-dimensional manifold with Euler characteristic any continuous tangent vector field has at least one zero. Image File history File links No higher resolution available. ...
Image File history File links No higher resolution available. ...
It has been suggested that Vertex/Face/Edge relation in a convex polyhedron be merged into this article or section. ...
In mathematics, the Poincaré–Hopf theorem (also known as the Poincaré–Hopf index formula, Poincaré–Hopf index theorem, or Hopf index theorem) is an important theorem in differential topology. ...
In geometry, a torus (pl. ...
Compact as a general noun can refer to: Look up Compact on Wiktionary, the free dictionary a diplomatic contract or covenant among parties, sometimes known as a pact, treaty, or an interstate compact; a British term for a newspaper format; In mathematics, it can refer to various concepts: Mostly commonly...
In ordinary English, regular is an adjective or noun used to mean in accordance with the usual customs, conventions, or rules, or frequent, periodic, or symmetric. ...
On a sphere, the sum of the angles of a triangle is not equal to 180°. A sphere is not a Euclidean space, but locally the laws of the Euclidean geometry are good approximations. ...
Cyclone consequences
A curious 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: suppose that relative to the planetary diameter of the Earth, its vertical (i.e., non-tangential) motion is negligible. Image File history File links atmospheric circulation diagram, showing the Hadley cell, the Ferrel cell, the Polar cell, and the various upwelling and subsidence zones between them. ...
Image File history File links atmospheric circulation diagram, showing the Hadley cell, the Ferrel cell, the Polar cell, and the various upwelling and subsidence zones between them. ...
Atmospheric circulation is the large-scale movement of air, and the means (together with the ocean circulation, which is smaller [1]) by which heat is distributed on the surface of the Earth. ...
One scenario, in which there is absolutely no wind (air movement), corresponds to a field of zero-vectors. This scenario is uninteresting from the point of view of this theorem, and physically unrealistic (there will always be wind). In the case where there is at least some wind, the Hairy Ball Theorem dictates that there must be at least one point on a planet at all times, with no wind at all. This corresponds to the above statement that there will be always be p such that f(p) = 0.
In a physical sense, this zero-wind point will be the eye of a cyclone or anticyclone. (Like the swirled hairs on the tennis ball, the wind will spiral around this zero-wind point - under our assumptions it cannot flow into or out of the point.) In brief, then, the Hairy Ball Theorem dictates that, given at least some wind on Earth, there must at all times be a cyclone somewhere. Note that the eye can be arbitrarily large or small and the magnitude of the wind surrounding it is irrelevant. Radar image of a tropical cyclone in the northern hemisphere. ...
Application to computer graphics A common subproblem in computer graphics is to generate a non-zero vector that is orthogonal to a specified one. For example the OpenGL GLU graphics library provides a gluLookAt function that provides this facility. By the Hairy Ball Theorem we know that this function must have discontinuities, or it must occasionally fail and produce zero vectors for non-zero input. OpenGL (Open Graphics Library) is a standard specification defining a cross-language cross-platform API for writing applications that produce 3D computer graphics (and 2D computer graphics as well). ...
Lefschetz connection There is a closely-related argument from algebraic topology, using the Lefschetz fixed point theorem. Since the Betti numbers of a 2-sphere 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 one-parameter 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 Poincaré-Hopf index theorem. Algebraic topology is a branch of mathematics in which tools from abstract algebra are used to study topological spaces. ...
In mathematics, the Lefschetz fixed-point theorem counts the number of fixed points of a mapping from a topological space X to itself (subject to some mild conditions on X), by means of traces of the induced mappings on the homology groups of X. The counting is subject to some...
In algebraic topology, the Betti number of a topological space is, in intuitive terms, a way of counting the maximum number of cuts that can be made without dividing the space into two pieces. ...
In mathematics, the Lefschetz fixed-point theorem counts the number of fixed points of a mapping from a topological space X to itself (subject to some mild conditions on X), by means of traces of the induced mappings on the homology groups of X. The counting is subject to some...
In mathematics (especially algebraic topology and abstract algebra), homology (in Greek homos = identical) is a certain general procedure to associate a sequence of abelian groups or modules to a given mathematical object (such as a topological space or a group). ...
An identity function f is a function which doesnt have any effect: it always returns the same value that was used as its argument. ...
Vector field given by vectors of the form (-y, x) In mathematics a vector field is a construction in vector calculus which associates a vector to every point in a Euclidean space. ...
In mathematics, a one-parameter group or one-parameter subgroup usually means a continuous group homomorphism φ : R → G from the real line R (as an additive group) to some other topological group G. That means that it is not in fact a group, strictly speaking; if φ is...
In mathematics, a diffeomorphism is a kind of isomorphism of smooth manifolds. ...
An illustration of a homotopy between the two bold paths In topology, two continuous functions from one topological space to another are called homotopic (Greek homeos = identical and topos = place) if one can be continuously deformed into the other, such a deformation being called a homotopy between the two functions. ...
In mathematics, the Poincaré-Hopf Theorem (also known as the Poincaré-Hopf index formula, Poincaré-Hopf index theorem, or Hopf index theorem) states: Let M be a compact differentiable manifold. ...
Corollary A consequence of the hairy ball theorem is that any continuous function that maps a 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. Partial plot of a function f. ...
In mathematics, a fixed point (sometimes shortened to fixpoint) of a function is a point that is mapped to itself by the function. ...
In mathematics, the antipodal point of a point on the surface of a sphere is the point which is diametrically opposite it — so situated that a line drawn from the one to the other passes through the centre of the sphere and forms a true diameter. ...
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. Stereographic projection of a circle of radius R onto the x axis. ...
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 2n-sphere has no non-vanishing vector field for n ≥ 1. The difference in even and odd dimension is that the Betti numbers of the m-sphere are 0 except in dimensions 0 and m. Therefore their alternating sum χ is 2 for m even, and 0 for m odd. It has been suggested that Vertex/Face/Edge relation in a convex polyhedron be merged into this article or section. ...
A sphere is a perfectly symmetrical geometrical object. ...
In algebraic topology, the Betti number of a topological space is, in intuitive terms, a way of counting the maximum number of cuts that can be made without dividing the space into two pieces. ...
In mathematics, an alternating series is an infinite series of the form with an ≥ 0. ...
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