Editing Hairy ball theorem (section)

{{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>