page_title: Hairy ball theorem
hierarchical_section_title: Hairy ball theorem
caption_attribution_description: English: A failed attempt to comb a hairy ball flat. (I, anonymous author of this work, claim no author's rights.)
context_page_description: The hairy ball theorem of algebraic topology states that there is no nonvanishing continuous 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 in R³ to every point p on a sphere such that f is always tangent to the sphere at p, then there is at least one p such that f = 0. The theorem was first proved by Henri Poincaré for the 2-sphere in 1885, and extended to higher dimensions in 1912 by Luitzen Egbertus Jan Brouwer.
  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".
context_section_description: The hairy ball theorem of algebraic topology (sometimes called the hedgehog theorem in Europe) states that there is no nonvanishing continuous 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 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. The theorem was first proved by Henri Poincaré for the 2-sphere in 1885, and extended to higher dimensions in 1912 by Luitzen Egbertus Jan Brouwer.
  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".
caption: A failed attempt to comb a hairy 3-ball (2-sphere), leaving a tuft at each pole
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