For so many of us, mathematics can be a daunting field. Math often seems like an insurmountable obstacle, which we may feel will never prove useful or relevant in our lives. However, math, like other fields, is a topic that is not merely understandable, but can often be quite interesting and even amusing.
If you are skeptical (perhaps rightly so) that math can be amusing, then I would like to present to you a theorem in a field of math called “algebraic topology.” (Don’t worry about that name too much, its just nice to know where we’re coming from). The theorem is called: The Hairy Ball Theorem.
Yes, you read that correctly. And no, that is not the theorem attributed to two individuals surnamed Hairy and Ball. This theorem really and truly does concern hairy balls. The theorem goes as follows: If you have a ball covered in hair, you cannot comb the hair on that ball without creating a cowlick. Alternatively, the theorem is sometimes stated as, “you can’t comb the hair on a coconut.”
What does this mean? Well suppose we have a sphere with little hairs sticking up at regular points on its surface. We wish to flatten this hair so that hairs don’t overlap, and so that all of the hairs are flat on the surface. The thrust of the Hairy Ball Theorem is that this task is impossible; we will eventually end up with a tuft of hair somewhere on the sphere.
At this point, its eminently reasonable to think, “that’s nice (and kind of funny), but why is this useful?” Well, there is a quite good application of this theorem: wind. Wind systems can be complex, but it is not a huge leap to simplify wind as short lines that lie flat on the surface of the planet. The direction of the line gives the direction of the wind and the length of the line gives the strength of the wind at that point on the globe. For a good example of thinking of wind this way, check out this wind map which gives real time data on wind across the entire US. Here we see that wind is visualized as short lines that are either thicker or thinner (based on strength) and point in the direction of the wind emanating from all points on the map.
This way of thinking about of wind allows us to imagine the entire globe (as you may suspect) as a hairy ball. Thus, by the Hairy Ball Theorem, we know that at all times there must be at least one point on the globe where there is no wind at all (a tuft of hair). Furthermore, this explains something fundamental about cyclones and hurricanes. In cyclones, high speed wind swirls around an eye. The eye of the storm is a point in the center of the storm with no wind at all. Effectively, when trying to comb the wind around in circle to form the hurricane, we get stuck at the middle, making a cowlick, and there is no wind. Tada!
The Hairy Ball Theorem is a great example about how math doesn’t have to be a brick wall, it can both be funny and explain a lot of what we see in the physical world. And if you think the Hairy Ball Theorem is funny enough, then you should check out, and perhaps send your sympathy to, the former mayor of Fort Wayne, Indiana, Harry Baals.
By: Jonah Scheinerman
(Photo by kakela under a Creative Commons license)