This strangely-named theorem has a rarefied theoretical background, arising from a perfectly round ball covered with perfectly fine hair. Despite that, it has real-world applications, including the ability to build gold nanowires — and finding the one spot on earth where there is no wind.
Top image: BryceGuy72 on Deviant Art.
The Hairy Ball Theorem springs from an early 20th century mathematician's thoughts about vector fields applied to ideal shapes. I know, it sounds fascinating. But Luitzen Brouwer took that ridiculously abstract idea and gave it an easily understood form - especially since Koosh balls were invented.
According to Brouwer, to understand this, people should picture a perfectly round ball covered with perfectly fine hair all over. Next, imagine taking a brush and brushing the hair on the ball in continuous strokes, trying to flatten it to the body of the ball. After the brushing, the hair is essentially a vector field. Each hair shows the motion of the brush, since each hair has been tugged in whatever direction the brush was moving as it tugged the hair into place. From the point of view of someone standing on the surface of the ball, the hairs drawn perpendicular to the surface of the ball received the most vigorous motion, while those standing up and down, just the way they were before the brush moved through, show no motion at all. They are 'zero vectors.'
Well, fine, but what does this tell us? Brouwer figured out that, with the brush moving continuously over the surface of the ball, there was always one point with a 'zero vector.' In other words, there was no way to brush all the hair so that it lies flat on a sphere. One point will always have a hair standing up, perpendicular to the surface. This is not so on other shapes, like a square or a donut shape.
That sounds mathematically interesting, but not the least little bit useful to anyone. Amazingly, though, The Hairy Ball Theorem can be applied to nanotechnology. The Hairy Ball Theorem in 2007 was employed by an engineering team. Using a gold nanoparticle as the ball, they covered it in sulfurous 'hairs' that stood out and created defects in the surface of the ball that could be used to attach it to other balls. Because of the theorem, they knew that some hairs would stand out from the surface of the ball, no matter what. Even under the most ideal conditions, one hair at least could be used to attach one nanoparticle to another, and build up a chain of them that would make a gold wire.
The Hairy Ball Theorem also tells us something about the earth that seems almost mystical. Winds move in continuous strokes over the surface of the earth. They could be considered as a vector field. Since the earth is basically a sphere, and the winds move over it, there has to be one point at which the vector is zero. There has to be one point on earth, all the time, where there is no wind, even though there is air flow all around it. That spot won't stay in place all the time. It would be interesting though, to see if there were a way of finding that one spot, at one point in time. The Hairy Ball theorem could, then, be an inspiration for people wishing to chase a dream. As well as for people interested in the mechanics of having their balls brushed. (I'm sorry. I had to.)
Image: K Tempest Bradford