The Hairy Ball Theorem

Baby hairy head DSCN2483Last week I was at a conference at the Isaac Newton Institute at Cambridge. The subject was what you might call liquids with small self-propelled things in them. This may sound a bit odd, but the inside of our cells is more-or-less liquid and they are lots of self-propelled things in them. Our cells are a storm of a little machines in constant motion. As you probably know, our genes are in the form of large molecules of DNA, and DNA needs to be copied to make more cells. Tiny machines zip around inside the cells making DNA copies and then dragging them round inside cells.

When enough of these machines zip around they can create flows of the liquid. This brings me to what is known as The Hairy Ball Theorem of mathematics. This is a real theorem from formal mathematics, and that is its real name. Don’t believe me? The Wikipedia page is here, and a lovely 1 min YouTube clip taking you through what it means is here.

The Hairy Ball Theorem is excellent: it has an amusing and memorable name, and is genuinely useful. It is useful as it tells you (roughly speaking) that if you have flow of a liquid on the surface of the sphere, then this there must be points where the flow is zero. It cannot be non-zero everywhere, there must be points where there is no flow. Smart mathematicians have shown that somewhere on the surface of the sphere, the liquid is not going anywhere. It is impossible to avoid this.

Incidentally, it applies to flow on the surface of a sphere or similar object of any size. So it applies both to the surface of a cell, and to wind (which is flow of air) on the surface of the Earth. Somewhere on the surface of the Earth there can be no wind. Mathematicians have proved this. It must be true. They don’t know where this is of course. This is true of a lot of maths results. They are are useful, and you can’t argue with them, but they don’t tell you all you need to know. Here they leave the weather forecasters with work to do.

In fact, the theorem does not only apply to the flow of liquids and gases, it applies to anything with a direction (i.e., a vector) on the surface of a roughly spherical object. Another example is hair on our heads. At the top is a picture of a baby’s head. The baby’s hair forms a swirl or vortex. For most of the swirl it is clear what is the direction of the hairs, but right at the centre you can’t say where the hairs are pointing. The Hairy Ball Theorem tells us that the presence of such a point is impossible to avoid.

Incidentally, the theorem does not apply to objects with a hole them, there is no Hairy Donut Theorem.