Editing Hairy ball theorem

{{Short description|Theorem in differential topology}}
[[File:Hairy ball.png|thumb|A failed attempt to comb a hairy 3-ball (2-sphere), leaving a tuft at each pole]]
[[File:Hairy doughnut.png|thumb|A hairy doughnut (2-torus), on the other hand, is quite easily combable.]]
[[File:Hairy ball one pole.jpg|thumb|A continuous tangent vector field on a 2-sphere with only one pole, in this case a [[dipole]] field with index 2. See also an [[:File:Hairy ball one pole animated.gif|animated version of this graphic]].]]
[[File:Baby hairy head DSCN2483.jpg|thumb|A [[hair whorl]]]]

The '''hairy ball theorem''' of [[algebraic topology]] (sometimes called the '''hedgehog theorem''' in Europe)<ref name="Renteln">{{cite book| last1  = Renteln| first1 = Paul| title  = Manifolds, Tensors, and Forms: An Introduction for Mathematicians and Physicists| publisher = Cambridge Univ. Press| date   = 2013
 | page  = 253| url    = https://books.google.com/books?id=uJWGAgAAQBAJ&q=hairy+ball+theorem&pg=PA253|isbn   = 978-1107659698 }}</ref> states that there is no nonvanishing [[continuous function|continuous]] tangent [[vector field]] on even-dimensional [[n‑sphere|''n''-spheres]].<ref name="Burns">{{cite book
 | last1  = Burns
 | first1 = Keith
 | last2  = Gidea
 | first2 = Marian
 | title  = Differential Geometry and Topology: With a View to Dynamical Systems
 | publisher = CRC Press
 | date   = 2005
 | location = 
 | pages  = 77
 | language = 
 | url    = https://books.google.com/books?id=tV9sTDnaf40C&q=hairy+ball+theorem&pg=PA77
 | doi    = 
 | id     = 
 | isbn   = 1584882530
 }}</ref><ref name="Schwartz">{{cite book
 | last1  = Schwartz
 | first1 = Richard Evan 
 | title  = Mostly Surfaces
 | publisher = American Mathematical Society
 | date   = 2011
 | location = 
 | pages  = 113–114
 | language = 
 | url    = https://books.google.com/books?id=sS2IAwAAQBAJ&q=hairy+ball+theorem&pg=PA113
 | doi    = 
 | id     = 
 | isbn   = 978-0821853689
 }}</ref> For the ordinary sphere, or 2‑sphere, if ''f'' is a continuous function that assigns a [[Vector (geometric)|vector]] in {{math|ℝ<sup>3</sup>}} 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 pole, a point where the field vanishes (a ''p'' such that ''f''(''p'') = '''[[Null vector|0]]''').

The theorem was first proved by [[Henri Poincaré]] for the 2-sphere in 1885,<ref>{{citation |last=Poincaré |first=H. |title=Sur les courbes définies par les équations différentielles |journal=Journal de Mathématiques Pures et Appliquées |volume=4 |pages=167–244 |year=1885}}</ref> and extended to higher even dimensions in 1912 by [[Luitzen Egbertus Jan Brouwer]].<ref>[http://dz-srv1.sub.uni-goettingen.de/sub/digbib/loader?ht=VIEW&did=D28661 Georg-August-Universität Göttingen] {{webarchive|url=https://web.archive.org/web/20060526145611/http://dz-srv1.sub.uni-goettingen.de/sub/digbib/loader?ht=VIEW&did=D28661 |date=2006-05-26 }} - [https://eudml.org/doc/158520 L.E.J. Brouwer. Über Abbildung von Mannigfaltigkeiten / Mathematische Annalen (1912) Volume: 71, page 97-115; ISSN: 0025-5831; 1432-1807/e], [https://gdz.sub.uni-goettingen.de/id/PPN235181684_0071?tify=%7B%22view%22:%22info%22,%22pages%22:%5B103%5D%7D full text]</ref>

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".<ref>{{cite book |last1=Richeson |first1=David S. |title=Euler's gem : the polyhedron formula and the birth of topology |date=23 July 2019 |location=Princeton |isbn=978-0691191997 |pages=5 |edition=New Princeton science library }}</ref>

==Counting zeros==
Every zero of a vector field has a (non-zero) "[[Vector field#Index of a vector field|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 space|compact]] [[Irregularity of a surface|regular]] 2-dimensional [[manifold]] 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 {{math|ℝ<sup>3</sup>}} that is [[Orthogonality|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]], using the [[Lefschetz fixed-point theorem]]. Since the [[Betti number]]s of a 2-sphere are 1, 0, 1, 0, 0, ... the ''[[Lefschetz number]]'' (total trace on [[homology (mathematics)|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 [[diffeomorphism]]s 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]].

==Corollary==
A consequence of the hairy ball theorem is that any continuous [[Functions (mathematics)|function]] that maps an even-dimensional sphere [[Endomorphism|into itself]] has either a [[fixed point (mathematics)|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''.

A further corollary is that any even-dimensional [[Real projective space|projective space]] has the [[fixed-point property]]. This follows from the previous result by [[Covering space#Lifting property|lifting]] continuous functions of <math>\mathbb{RP}^{2n}</math> into itself to functions of <math>S^{2n}</math> into itself.

==Higher dimensions==
The connection with the [[Euler characteristic]] χ suggests the correct generalisation: the [[N-sphere|2''n''-sphere]] has no non-vanishing vector field for {{nowrap|''n'' ≥ 1}}. The difference between even and odd dimensions is that, because the only nonzero [[Betti number]]s of the ''m''-sphere are b<sub>0</sub> and b<sub>m</sub>, 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]] <math>\mathbb{R}^{2n}</math> in pairs. Namely, one may define a tangent vector field to <math>S^{2n-1}</math> by specifying a vector field <math>v: \mathbb{R}^{2n} \to \mathbb{R}^{2n}</math> given by
:<math> v(x_1,\dots,x_{2n}) = (x_2, -x_1,\dots,x_{2n},-x_{2n-1}).</math>
In order for this vector field to restrict to a tangent vector field to the unit sphere <math>S^{2n-1}\subset \mathbb{R}^{2n}</math> it is enough to verify that the [[dot product]] with a [[unit vector]] of the form <math>x=(x_1,\dots,x_{2n})</math> satisfying <math>\|x\|=1</math> vanishes. Due to the pairing of coordinates, one sees
:<math> v(x_1,\dots,x_{2n}) \bullet (x_1,\dots,x_{2n}) = (x_2 x_1 - x_1 x_2) + \cdots + (x_{2n} x_{2n-1} - x_{2n-1} x_{2n}) = 0.</math>
For a 2''n''-sphere, the ambient Euclidean space is <math>\mathbb{R}^{2n+1}</math> 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.<ref>{{Cite journal|last1=Bormashenko|first1=Edward|last2=Kazachkov|first2=Alexander|date=June 2017|title=Rotating and rolling rigid bodies and the "hairy ball" theorem|url=https://doi.org/10.1119/1.4979343|journal=American Journal of Physics|language=en|volume=85|issue=6|pages=447–453|doi=10.1119/1.4979343|bibcode=2017AmJPh..85..447B|issn=0002-9505}}</ref> The hairy ball theorem may be successfully applied for the analysis of the propagation of [[Electromagnetic radiation|electromagnetic waves]], 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.<ref>{{Cite journal|last=Bormashenko|first=Edward|date=2016-05-23|title=Obstructions imposed by the Poincaré–Brouwer ("hairy ball") theorem on the propagation of electromagnetic waves|url=https://www.tandfonline.com/doi/full/10.1080/09205071.2016.1169226|journal=Journal of Electromagnetic Waves and Applications|language=en|volume=30|issue=8|pages=1049–1053|doi=10.1080/09205071.2016.1169226|bibcode=2016JEWA...30.1049B |s2cid=124221302|issn=0920-5071}}</ref> On certain 2-spheres of [[parameter space]] for electromagnetic waves in plasmas (or other complex media), these type of "cowlicks" or "bald points" also appear, which indicates that there exists topological excitation, i.e., robust waves that are immune to scattering and reflections, in the systems.<ref>{{Cite journal |last1=Qin |first1=Hong |last2=Fu |first2=Yichen |date=2023-03-31 |title=Topological Langmuir-cyclotron wave |journal=Science Advances |language=en |volume=9 |issue=13 |pages=eadd8041 |doi=10.1126/sciadv.add8041 |issn=2375-2548 |pmc=10065437 |pmid=37000869|arxiv=2205.02381 |bibcode=2023SciA....9D8041Q }}</ref>
If one idealizes the wind in the Earth's atmosphere as a tangent-vector field, then the hairy ball theorem implies that given any wind at all on the surface of the Earth, there must at all times be a [[cyclone]] somewhere. Note, however, that wind can move vertically in the atmosphere, so the idealized case is not [[Meteorology|meteorologically]] sound. (What ''is'' true is that for every "shell" of atmosphere around the Earth, there must be a point on the shell where the wind is not moving horizontally.) The theorem also has implications in [[Computer simulation|computer modeling]] (including [[video game design]]), in which a common problem is to compute a non-zero <abbr>3-D</abbr> vector that is orthogonal (i.e., perpendicular) to a given one; the hairy ball theorem implies that there is no single continuous function that accomplishes this task.<ref>{{Cite web |last=Kohulák |first=Rudolf |date=2016-09-02 |title=Hairy balls, cyclones and computer graphics |url=https://chalkdustmagazine.com/blog/hairy-balls-cyclones-computer-graphics/ |access-date=2023-08-14 |website=Chalkdust |language=en-GB}}</ref>  

==See also==
*[[Fixed-point theorem]]
*[[Intermediate value theorem]]
*[[Vector fields on spheres]]

==References==
{{reflist}}

==Further reading==
* {{Citation |first1=Murray |last1=Eisenberg |first2=Robert |last2=Guy |title=A Proof of the Hairy Ball Theorem |journal=The American Mathematical Monthly |volume=86 |issue=7 |year=1979 |pages=571–574 |doi=10.2307/2320587|jstor=2320587 }}

* {{Citation|title=The Hairy Ball Theorem via Sperner's Lemma|last1=Jarvis|first1=Tyler|last2=Tanton|first2=James |journal=[[American Mathematical Monthly]] |volume=111 |issue=7 |year=2004 |pages=599–603 |jstor=4145162 |doi=10.1080/00029890.2004.11920120|s2cid=29784803}}
* {{Citation|last=Richeson|first=David S.|authorlink= David Richeson |title=Euler's Gem: The Polyhedron Formula and the Birth of Topology|publisher=Princeton University Press|year=2008|isbn=978-0-691-12677-7 |chapter=Combing the Hair on a Coconut |pages=202–218|title-link= Euler's Gem }}

==External links==
*{{MathWorld|title=Hairy Ball Theorem|id=HairyBallTheorem}}

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