# The Hairy Ball Theorem

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The hairy ball theorem states that for a sphere or any surface homeomorphic to a sphere, there is no continuous, non-vanishing tangent vector field. In other words, you cannot comb a hairy ball flat without at least one part or cowlick.

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Contributed by: Seán Flaherty (July 2015)

(*Mathematica* Summer Camp 2014)

Open content licensed under CC BY-NC-SA

## Snapshots

## Details

Snapshot 1: Initial settings of the Demonstration. The vector field shown on the sphere is a cross-product of the variable "comb" vector field and the surface normal vector field, yielding a vector field that is always tangent to the surface.

Snapshot 2: Torus with a "combed" vector field. The smallest vector has a magnitude of 1; since the surface local minimum is not zero, the vector field is non-vanishing and the torus can be considered perfectly combed.

Snapshot 3: Torus with a vector field that is "not combed". Although a continuous non-vanishing tangent vector field can exist on a genus 1 torus, this case shows a vector field that is not perfectly combed.

## Permanent Citation

"The Hairy Ball Theorem"

http://demonstrations.wolfram.com/TheHairyBallTheorem/

Wolfram Demonstrations Project

Published: July 24 2015