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If a sphere is covered in hair then the hair cannot be combed to be smooth all over.

IMG:hair1.png

Somewhere on the sphere must be a point or region to which hair is brushed creating a tuft or from which the hair is brushed away creating a bald patch.

For example:

IMG:hair2.png IMG:hair3.png IMG:hair4.png

More exactly the theorem states that on a sphere there does not exist an everywhere nonzero tangent vector field.

A result of this is that somewhere on the surface of the Earth, there is always somewhere a point with zero horizontal wind velocity.

It is however possible to comb the hair on a torus to be smooth all over.

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http://en.wikipedia.org/wiki/Hairy_ball_theorem

http://mathworld.wolfram.com/HairyBallTheorem.html

* http://en.wikipedia.org/wiki/Hairy_ball_theorem

* http://mathworld.wolfram.com/HairyBallTheorem.html

* http://en.wikipedia.org/wiki/Vector_field