Monday morning / Friday afternoon thought inspiring blog

Picture yourself a ball with hairs attached to the surface. You are given the task to comb the hairs on the surface in such a manner that there are no “bald spots” or “opposing combings”, that means that no two adjacent hairs should point in opposite or “radically different” directions. Can you do it?

Too bad for those who thought that they can: mathematics tell us that this is impossible. This result is popularly known as the hairy ball theorem. The details of the mathmatical statement are naturally more intricate than the explanation I gave in the previous paragraph. In more mathematical terms is says that a smoothly varying vector field over the surface of a three-dimensional sphere, has to have at least one point where the field is 0. In terms of the hairs on the ball, the only solution to have a ‘smooth combing‘, thus without places where the hairs have radical different directions, is to have some place where the hair lenghts are zero.

Another consequence of this theorem applies to wind blowing over the surface of the earth. Either there have to be places where there is no wind or at some places the wind would be blowing in opposite directions, directly adjacent to each other. This corresponds nicely to what we know about hurricanes and tornados, that in their centre — the eye — there is no wind. Even if we were to imagine one big flow of wind going over the earth’s surface in a direction along the equator, then still there would be areas without wind at the poles.

So what about the hairs on a four-dimensional sphere? It turns out that those can be combed smoothly over the entire surface. That means that owners of four-dimensional pets have a much easier time preparing their furry companion for the pet show than the people on earth with their three-dimensional pets.

Note that a pet ‘without holes’ is mathematically equivalent to a sphere, thus the result also applies to objects that are topologically the same as a sphere. Topologically equivalent means that the object can be elastically deformed from one object into the other without connecting the surface with itself or breaking the object. The most popular example of topological equivalence is that of a coffee cup and a doughnut. Now you can laugh at the joke: A topologist is one who doesn’t know the difference between a doughnut and a coffee cup.

This entry was posted on Friday, March 9th, 2007 at 9:46 am and is filed under Mathematics. You can follow any responses to this entry through the RSS 2.0 feed. Both comments and pings are currently closed.