Hairy Ball Theorem

You can't comb a hairy ball flat — a tangent field on S^2 must vanish.

The idea

Theorem (Hairy Ball Theorem).

Every continuous tangent vector field on the sphere $S^{2}$ vanishes at some point.

Comb a hairy ball however you please, and some hair stands up: there is always a cowlick.

A tangent vector field assigns to each point of the surface an arrow lying flat against it, and does so continuously, so nearby points receive nearly equal arrows. The field vanishes at a point when it assigns the zero vector there — an arrow of no length, and hence of no direction. The theorem does not say the field is zero everywhere; it says a nowhere-zero field cannot exist.

The obstruction is not roundness or size but the Euler characteristic $\chi$, the invariant that $V - E + F$ computes from any decomposition of the surface. Poincaré and Hopf showed that the zeros of a field can be counted with signs — each zero carries an integer index, recording how the surrounding arrows turn as you walk around it — and that on a closed surface the indices sum to $\chi$ (the Poincaré–Hopf theorem). On the sphere $\chi(S^{2}) = 2 \neq 0$, and a nonzero sum cannot be a sum over no zeros at all, so every field vanishes somewhere. The same count shows where the theorem stops: a closed surface with $\chi = 0$, such as the torus, can be combed flat.

Ways to work on it

• Walkthrough. Why a sphere can't be combed, via the Euler characteristic.
• Practice. Decide whether a given surface can be combed.
• Hardest. Constrain a vector field's zeros by the sum of their indices.

Not sure where to start? Take the ten-question placement test.