The Hairy Ball Theorem Explained

I know what you are thinking — weird name, right?

But there’s an interesting analogy-connection behind it. I’m sure you must have played with a tennis ball at least once in your life. Let’s try and understand what this theorem is with the tennis ball analogy. Let’s imagine you have a perfectly smooth sphere — like a tennis ball — covered in tiny hairs (like most of them usually have if you zoom in enough). Now try to comb all those hairs so that they lie flat against the ball’s surface, with none sticking straight up or forming a whorl. The theorem says that no matter how you try to comb those hairs flat so each one lies smoothly against the surface, you will always end up with at least one point where the hair sticks straight up (or becomes a “cowlick”).

Here’s the formal topological definition of the hairy ball theorem:

That there is no continuous, non-zero tangent vector field on the 2-sphere (the surface of a ball).

Equivalently: any continuous assignment of an arrow (vector) that’s always tangent to the sphere must vanish at at least one point.

And that’s pretty much it.

If you’re wondering why such a weird name then it comes from the very playful visualisation that we used earlier.

• Hair → the tangent vectors on the sphere.
• Ball → the 2-sphere (surface of a ball).