I’ve been thinking about the hairy ball theorem a lot recently. Because I
have the sophisticated sense of humor of a twelve-year-old just got a haircut, and I am newly reminded of my cowlicks.
At this length, I’m much more aware of my hairs as vectors than I was when they were longer. A vector is basically a straight line that points in some direction and has some length, like a strand of hair sticking out from someone’s head. And the hairy ball theorem is all about vectors.
The hairy ball theorem says that you can’t comb a hairy ball. In technical terms, if you have a tangent vector at every point on the surface of a sphere, you can’t make them all line up continuously with their neighbors without having some point where the tangent vector is zero. Or, if all your vectors are nonzero, you end up with a point where the vectors change direction abruptly or stick straight up. In other words, a cowlick. If you were trying to comb a rambutan, there would be one spot where you couldn’t get the hair to lie flat. Your rambutan would have a cowlick.
I would like to say that I have cowlicks because of math, but in truth, my head doesn’t satisfy the hypotheses of the hairy ball theorem. For one, the hair on the top of my head sticks straight up, rather than lying tangent to my scalp. But even when it grows in more and lies flat, the theorem still won’t apply.
The problem isn’t that my head has lumps and bumps like ears and a nose. The hairy ball theorem comes from the field of topology, the branch of mathematics that pretends all objects are made of infinitely stretchy rubber or putty. The theorem applies to any object that can be squished around into the shape of a sphere without making any tears or pinching a hole closed, no matter how much squishing that requires.
The reason my head doesn’t satisfy the hypotheses of the hairy ball theorem is that I don’t have enough hair. The hairy ball theorem requires the hair to cover the entire ball. As soon as you remove just one point from the ball, the hairy ball theorem breaks. An infinitely stretchy sphere with a pinprick in it can be stretched out to look like a circle in the normal 2-dimensional plane, where it’s pretty easy to comb hair. (Think shag carpet.) My head hair only covers part of my head. Even if I count the rest of the hair on my body, I still have eyes and palms and other hairless parts.
According to my 9th grade biology teacher, the lining of my digestive tract, from my mouth all the way to the other end, is an extension of my skin. If you look at it like that, I violate the assumptions of the hairy ball theorem in another way: I’m not topologically equivalent to a sphere at all, but to a torus, the surface of a donut. (I’m not really sure where my ears and sinuses fit into this. I’m not that kind of doctor.) You can comb a hairy donut. The way to see this is to imagine a rectangle covered in hair, maybe a carpet square or some sod. You can definitely comb that, and you can attach opposite sides to make a torus with a well-behaved hairstyle. Although if I had a hairy digestive tract, I don’t think I’d be worried about cowlicks.
I don’t know why I have cowlicks, but sadly, it’s not because of topology. For more on cowlicks, fingerprint whorls, and other natural singularities, check out Steven Strogatz’s New York Times article from last September about Singular Sensations. And for a nice video introduction to the hairy ball theorem, watch this Minute Physics video.