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The hairy ball theorem of
algebraic topology Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify ...
(sometimes called the hedgehog theorem in Europe) states that there is no nonvanishing
continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous g ...
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 Vector most often refers to: *Euclidean vector, a quantity with a magnitude and a direction *Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathematic ...
in R3 to every point ''p'' on a sphere such that ''f''(''p'') is always
tangent In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. More ...
to the sphere at ''p'', then there is at least one pole, a point where the field vanishes (a ''p'' such that ''f''(''p'') = 0). The theorem was first proved by
Henri Poincaré Jules Henri Poincaré ( S: stress final syllable ; 29 April 1854 – 17 July 1912) was a French mathematician, theoretical physicist, engineer, and philosopher of science. He is often described as a polymath, and in mathematics as "The ...
for the 2-sphere in 1885, and extended to higher dimensions in 1912 by
Luitzen Egbertus Jan Brouwer Luitzen Egbertus Jan Brouwer (; ; 27 February 1881 – 2 December 1966), usually cited as L. E. J. Brouwer but known to his friends as Bertus, was a Dutch mathematician and philosopher, who worked in topology, set theory, measure theory and c ...
. 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".


Counting zeros

Every zero of a vector field has a (non-zero) "
index Index (or its plural form indices) may refer to: Arts, entertainment, and media Fictional entities * Index (''A Certain Magical Index''), a character in the light novel series ''A Certain Magical Index'' * The Index, an item on a Halo megastru ...
", 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 In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological space' ...
of the 2-sphere is two. Therefore, there must be at least one zero. This is a consequence of the
Poincaré–Hopf theorem 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 that is used in differential topology. It is named after Henri Poincaré ...
. In the case of the
torus In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle. If the axis of revolution does not tou ...
, the Euler characteristic is 0; and it is possible to "comb a hairy doughnut flat". In this regard, it follows that for any
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in Briti ...
regular 2-dimensional
manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a ...
with non-zero Euler characteristic, any continuous tangent vector field has at least one zero.


Application to computer graphics

A common problem in computer graphics is to generate a non-zero vector in R3 that is orthogonal to a given non-zero vector. There is no single continuous function that can do this for all non-zero 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 non-zero vector orthogonal to the given one is equivalent to finding a non-zero 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 Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify ...
, using the
Lefschetz fixed-point theorem In mathematics, the Lefschetz fixed-point theorem is a formula that counts the fixed points of a continuous mapping from a compact topological space X to itself by means of traces of the induced mappings on the homology groups of X. It is name ...
. Since the
Betti number In algebraic topology, the Betti numbers are used to distinguish topological spaces based on the connectivity of ''n''-dimensional simplicial complexes. For the most reasonable finite-dimensional spaces (such as compact manifolds, finite simplicia ...
s of a 2-sphere are 1, 0, 1, 0, 0, ... the ''
Lefschetz number In mathematics, the Lefschetz fixed-point theorem is a formula that counts the fixed points of a continuous mapping from a compact topological space X to itself by means of traces of the induced mappings on the homology groups of X. It is name ...
'' (total trace on homology) of the
identity mapping Graph of the identity function on the real numbers In mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function that always returns the value that was used as its argument, unch ...
is 2. By integrating a vector field we get (at least a small part of) a
one-parameter group In mathematics, a one-parameter group or one-parameter subgroup usually means a continuous group homomorphism :\varphi : \mathbb \rightarrow G from the real line \mathbb (as an additive group) to some other topological group G. If \varphi is i ...
of
diffeomorphism In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are differentiable. Definition Given two ...
s on the sphere; and all of the mappings in it are
homotopic In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a deform ...
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.


Corollary

A consequence of the hairy ball theorem is that any continuous function that maps an even-dimensional 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 In mathematics, a stereographic projection is a perspective projection of the sphere, through a specific point on the sphere (the ''pole'' or ''center of projection''), onto a plane (the ''projection plane'') perpendicular to the diameter thro ...
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''. A further corollary is that any even-dimensional
projective space In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a Euclidean space, or, more generall ...
has the fixed-point property. This follows from the previous result by lifting continuous functions of \mathbb^ into itself to functions of S^ into itself.


Higher dimensions

The connection with the
Euler characteristic In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological space' ...
χ suggests the correct generalisation: the 2''n''-sphere has no non-vanishing vector field for . The difference between even and odd dimensions is that, because the only nonzero
Betti number In algebraic topology, the Betti numbers are used to distinguish topological spaces based on the connectivity of ''n''-dimensional simplicial complexes. For the most reasonable finite-dimensional spaces (such as compact manifolds, finite simplicia ...
s of the ''m''-sphere are b0 and bm, their alternating sum χ is 2 for ''m'' even, and 0 for ''m'' odd. Indeed it is easy to see that an odd-dimensional sphere admits a non-vanishing tangent vector field through a simple process of considering coordinates of the ambient even-dimensional Euclidean space \mathbb^ in pairs. Namely, one may define a tangent vector field to S^ by specifying a vector field v: \mathbb^ \to \mathbb^ given by : v(x_1,\dots,x_) = (x_2, -x_1,\dots,x_,-x_). In order for this vector field to restrict to a tangent vector field to the unit sphere S^\subset \mathbb^ it is enough to verify that the
dot product In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an algeb ...
with a unit vector of the form x=(x_1,\dots,x_) satisfying \, x\, =1 vanishes. Due to the pairing of coordinates, one sees : v(x_1,\dots,x_) \bullet (x_1,\dots,x_) = (x_2 x_1 - x_1 x_2) + \cdots + (x_ x_ - x_ x_) = 0. For a 2''n''-sphere, the ambient Euclidean space is \mathbb^ which is odd-dimensional, and so this simple process of pairing coordinates is not possible. Whilst this does not preclude the possibility that there may still exist a tangent vector field to the even-dimensional sphere which does not vanish, the hairy ball theorem demonstrates that in fact there is no way of constructing such a vector field.


Physical exemplifications

The hairy ball theorem has numerous physical exemplifications. For example, rotation of a rigid ball around its fixed axis gives rise to a continuous tangential vector field of velocities of the points located on its surface. This field has two zero-velocity points, which disappear after drilling the ball completely through its center, thereby converting the ball into the topological equivalent of a torus, a body to which the “hairy ball” theorem does not apply. The hairy ball theorem may be successfully applied for the analysis of the propagation of
electromagnetic waves In physics, electromagnetic radiation (EMR) consists of waves of the electromagnetic (EM) field, which propagate through space and carry momentum and electromagnetic radiant energy. It includes radio waves, microwaves, infrared, (visible) ligh ...
, in the case when the wave-front forms a surface, topologically equivalent to a sphere (the surface possessing the Euler characteristic χ = 2). At least one point on the surface at which vectors of electric and magnetic fields equal zero will necessarily appear.


See also

*
Fixed-point theorem In mathematics, a fixed-point theorem is a result saying that a function ''F'' will have at least one fixed point (a point ''x'' for which ''F''(''x'') = ''x''), under some conditions on ''F'' that can be stated in general terms. Some authors clai ...
*
Intermediate value theorem In mathematical analysis, the intermediate value theorem states that if f is a continuous function whose domain contains the interval , then it takes on any given value between f(a) and f(b) at some point within the interval. This has two impor ...
* Vector fields on spheres


Notes


References

*


Further reading

* *


External links

*{{MathWorld, title=Hairy Ball Theorem, id=HairyBallTheorem Theorems in differential topology Fixed points (mathematics) Theorems in algebraic topology