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.



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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.
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