The Hairy Ball Theorem

At our organizational Math Club meeting, we talked about the “Greatest Hits” of mathematics. I found the following list of the 100 Greatest Theorems in Mathematics. compiled by Paul and Jack Abad (I must confess I don’t know anything about their background in math). Just like the 100 Greatest Films of all time by the American Film Institute, any such list is bound to cause a lot of arguments (Forest Gump in the top 100?!? You’ve got to be kidding me!).

A Hairy Ball

One big omission from the list is the theorem somebody gave as an example of a “Greatest Hit”: the Hairy Ball Theorem. Heck, the name alone makes it worth checking out! We’ll talk about it below.

So what’s the Hairy Ball Theorem?

Theorem: If S is the sphere in three dimensional space, then every continuous vector field on surface of S must vanish at some point.

What does that mean? Well, first of all, a vector field on the surface of S is a function which assigns a vector (ie. an arrow) in $\mathbb{R}^{3}$ for each point on the sphere. And recall that a function is continuous if a small change in the input causes only a small change in the output (so you don’t have any jumps or breaks in your function). So the Hairy Ball Theorem says that if you start with any such function, then for at least one point on the sphere it must give you the 0 vector.

What does that have to do with Hairy Balls? Well the sphere S is really a ball, and you can imagine that the vector/arrow at each point on the surface of the ball is a single hair. Then the continuous function is simply pointing each of these arrows in some direction. That is, the function is just combing the hair on the ball in a continuous way.

The Hairy Ball Theorem then says that no matter how you combed the hair, as long as it was done continuously, at the end you have a hair which sticks straight up (that’s what “vanishes” becomes in the hairy ball scenario).

The Proof: The Hairy Ball Theorem was first proved by Brouwer in 1912. You can see the original paper here. Look for Satz 2 on page 112. In fact, if you read closely, you’ll notice that actually Brouwer proves the Hairy Ball Theorem for any sphere which as an even dimension. So dimension 2 is just one case of the more general result. The proofs most people see involves some heavy duty algebraic topology and most people don’t learn that until graduate school. However, it turns out that there was a proof given by Milnor in the 1970’s which uses only clever undergraduate analysis. A nice explanation of it is given here.

The Applications: One real world application (if hairy balls isn’t real world enough!), is that the Hairy Ball Theorem proves that at any moment there is someplace in the world where the wind is perfectly still. If we imagine the Earth is a sphere, and at each point we make an arrow pointing in the direction the wind blows at that one moment (think about why it’s reasonable to believe this is a continuous function). Then the Hairy Ball Theorem proves that somewhere on the Earth we must have the 0 arrow. That is, at that spot wind is blowing in no direction, so it’s not blowing at all!

For a mathematical application, one can use the Hairy Ball Theorem to show that if you have a continuous function from the sphere to itself, $f: S \to S$ then there is a point p on the sphere where one of two things happens:

1. $f(p) =p$ (ie. p is a fixed point).

2. $f(p)$ is the point you get by starting at p and going straight through the center to the other side (ie. the antipodal point).

Finally, there is something special about the sphere in the Hairy Ball Theorem. It is possible to comb the Hairy Dougnut!

A Hairy Doughnut