The Hairy Ball Theorem

These days, whenever I look at the back of my beloved 7-month-old baby's head, this little swirl of tiny hairs reminds me of one of the most ridiculously named facts in math, the hairy ball theorem. I promise this is a genuinely serious bit of math, where informally the statement is that if you have a ball that's covered in hair and you try to comb it down, there is no way to do it without having the hair stick up at at least one point. For example, let's say you try to comb it all counterclockwise around some axis. Then at the top and the bottom, you end up with these little swirls, and the hair at the centermost point of those swirls would have nowhere to go.

It's forced to stick up. It's actually very fun to play around with this in your mind where no matter how you try to flatten out the hair, it is a mathematical guarantee that you will be left with at least one tufted like this. In fact, even getting it down to just a single problem point as opposed to two is a bit of a challenge. It is possible, and if you like puzzles, I encourage you to try thinking of how it could work.

Later on in this video, I'm going to show you at least one way you can think about doing it. For the moment, though, I imagine there's a more burning question, which is that you might be wondering why a mathematician would care about combing fluffy spheres like this. And of course, the answer is they don't. The name and the informal statement are a bit tongue-in-cheek.

I will of course share the more formal statement and in fact my real reason for making this video is to share an unusually elegant way to prove it. [music] One that I think will delight any math lovers. But before any of that, let's motivate things with an example of the kind of situation where these fluffy spheres naturally arise in practice in a context that initially seems completely unrelated. Okay, so imagine that you are a game developer and you're programming some game where you have a 3D model of an airplane and what you want is to be able to take an arbitrary trajectory for this plane to fly along, presumably something userdefined and your ...