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hairy ball theorem

Definitions

from Wiktionary, Creative Commons Attribution/Share-Alike License

  • n. A theorem stating that, given a sphere covered in fur, one cannot brush all the hairs flat without creating at least two whirls.
  • n. A theorem stating that there is no nonvanishing continuous tangent vector field on the sphere.

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  • Were you looking for hairy Ball theorem?

    No, I was not. And I speak as someone who actually used a functional analytical version of the Brouwer Fixed Point Theorem to prove one of the major* results in my dissertation.

    *: well, it was major to me. And it seemed to impress the committee members.

    May 9, 2011

  • I was going to say something about Chuck Norris and hairy balls but am now disinclined to do so.

    November 19, 2008

  • 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.

    This is famously stated as "you can't comb a hairy ball flat without creating a cowlick", or sometimes, "you can't comb the hair on a billiard ball". It was first proved in 1912 by Brouwer, and is also known as Brouwer's fixed point theorem.



    A failed attempt to comb a hairy ball flat, leaving a tuft at each pole.

    October 27, 2008