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WOLFRAM NOTEBOOK
The Hairy Ball Theorem
The Hairy Ball Theorem
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.
In this Demonstration, you vary a vector field to "comb" a tangent vector field on a sphere or torus, showing the point with the local minimum vector norm on that surface as a small sphere. If the smallest vector norm is zero, the hairy surface has a vanishing point, or "part", and is considered to be "not combed". On a sphere, this minimum point is always zero for a tangent vector field (indicating a part), while a hairy torus can, in fact, be combed flat. The minimum point can be moved using the Locator.