Theorem

Hairy Ball Theorem

Topology

There is no nonvanishing continuous tangent vector field on an even-dimensional sphere, popularly summarized as 'you cannot comb a hairy ball flat without a cowlick.' It is a classical application of algebraic topology to a vivid, intuitive claim, and implies that at any time somewhere on Earth's surface the wind is not blowing.

Facts

Statement

There is no non-vanishing continuous tangent vector field on an even-dimensional sphere; on the ordinary 2-sphere, every continuous assignment of a tangent vector to each point must be zero somewhere. 1

Proof Year

1885 1

Classification

Statement Form

Impossibility Theorem 1

Connections

Sources

1. Hairy Ball Theorem (Wikipedia)

Wikimedia Foundation

lead paragraph, theorem statement sentence
there is no non-vanishing continuous tangent vector field on even-dimensional n-spheres
Lead section
The theorem was first proven by Henri Poincaré for the 2-sphere in 1885, and extended to higher even dimensions in 1912 by L.
In Group: Algebraic Topology, Lead paragraph
The hairy ball theorem of algebraic topology (formally, the Sphere Vector Field Theory, sometimes called the hedgehog theorem) states that there is no non-vanishing continuous tangent vector field on even-dimensional n-spheres.

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Hairy Ball Theorem

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