If you have the type of not too short hair that completely succumbs to gravity, you will have a cowlick somewhere on your head.** It’s a direct consequence of a mathematical theorem called the hairy ball theorem. Yes, it’s called that and it’s a bona fide mathematical theorem with a real but difficult proof that requires you to get a fancy-schmancy degree in math to really understand it.
But if we just want to feel like we understand it, we can certainly do that. It’s the Internet, after all.
Let’s comb through some informal ways of describing this theorem. One is
- You can’t comb a hairy ball flat without creating a cowlick.
or closely related
- You can’t comb the hair on a coconut.
A perhaps more useful, less frivolous description of the theorem is:
- A cyclonic system must always exist on a planet with an atmosphere.
In other words, somewhere on Earth, at any given moment, there will be a cyclone (in the most general sense).*** I should note that, in certain ways, this is a supremely useless theorem because it doesn’t tell you where the hell the cyclone is. But it is 100% absolutely sure that there is one.
The reason that people built analogies with hair and wind is that the “hairs” on the ball correspond to what are called vector fields for systems that have stuff that moves like a fluid, like wind or piss streaming over someone’s hairy ball. The hair is mostly a visualization tool for particles that move continuously in a closed environment, like the surface of a sphere.
Imagine you could track some representative particles in a fluid. Say, particularly big balls of dust are floating around. Or fireflies are caught in a dust devil at night. And you snap a picture of this with not too short exposure. The balls of dust or the fireflies would come out as blurry streaks that move in thrall to the air. The blurry streaks would correspond approximately to vector fields. So a fairly accurate way of looking at vector fields is to understand it as a concise description of the motion of particles at every single point at a given moment in whatever system you are looking at.
And what do cowlicks and cyclones have to do with this? Basically, the CENTER of the cowlicks/cyclones are points where there is no motion. It’s the eye of the storm you always hear about where everything is calm. These are called the zeros of the vector fields. So if you were to drop something lightweight at the exact center of where the cyclone occurs, that thing you dropped wouldn’t move because there’s nothing moving around it.****
If you’re looking at a stream or a river, you see eddies, the watery analog of cyclones, all the time. But would you still see these if the water were flowing in a completely smooth, straight, artificial indoor stream? Probably not. But now let’s say there’s a sphere and there’s water coursing over the surface. And imagine that it’s completely self-contained. No water drips from it and no water drops on to it. The kicker in this hypothetical situation is this: there will be at least one eddy (i.e. a “cyclone” in our previous parlance or a “zero” in our more technical one) no matter how smooth we make the surface of the sphere. The mathematics of the hairy ball theorem guarantees this.
If you’re curious how the hairy ball theorem is stated in technical terms, it goes like this:
- There does not exist an everywhere nonzero tangent vector field on the 2-sphere.
**A technical caveat. This doesn’t apply if you’re wearing your hair slicked back or in a ponytail or in some way that makes it defy gravity. It’s mostly applicable if you’ve let your hair loose so that it looks like it’s cascading down your head.
***This ties in with something called the Poincare-Hopf theorem which is even more general than the hairy ball theorem. Because a cyclone is a “cowlick” (i.e. not just any zero, but a special type of zero called a sink of a vector field) with index one, there must be a second cyclone. This more general theorem predicts not one but two cyclones.
****Not in reality, of course. It’s a thought experiment, so bear with me.