Edit made on February 23, 2009 by derekcouzens at 18:36:24
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If a sphere is covered in hair then the hair cannot be combed to be smooth all over.
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.
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.