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The Hairy Ball Theorem

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Mathematicians delight in making childish jokes, so they like to come up with interesting names for theorems. One of the best-named must surely be the 'Hairy Ball Theorem'. This states that if you have a ball that is completely covered in hair, you can't comb it so that all the hair lies flat and neat on the ball. You'll always end up with at least one point where the hair stands up in a tuft, or where there's a gaping hole and you can see the scalp.

You've probably seen a number of different hairstyles in your day. There's the combed-straight-back look, the central parting, and then there's the 'small boy' look with all the hair pointing downwards, with a point near the top and slightly to the back which all the hair radiates from. This looks like a 'hole' in the hair, where we can see through to the scalp. You may have also seen that other 'small boy' feature, the 'tuft', where all the hair comes together at one point and sticks out.

In all of these instances, the hair only covers about half of the head. What would happen if the head were disembodied and the hair went all the way around? Well, we'll find that no matter how we arrange it, there will always be a special point, which may be a 'hole' from where all the hairs radiate, or a 'tuft' where all the hairs meet and protrude.

Of course mathematicians are never content to state things as simply as that. That's because they have to make it completely clear to other mathematicians exactly what they are saying. Unfortunately, this means that nobody but another mathematician will understand what they're saying. The mathematical version of the Hairy Ball Theorem was proved by LEJ Brouwer1 in 1912 and goes as follows:

Given a tangential vector field on the surface of a sphere in three-dimensional space, there must be at least one point where the field is zero.

Wind Patterns

While we don't encounter balls entirely covered in hair every day, the Hairy Ball Theorem can be applied to other situations, such as wind patterns. At each point on the Earth's surface, we can measure the wind speed and direction, and this is just like the direction of a hair combed flat on the ball. If the wind is blowing in some direction at every point on the Earth's surface, then there must be some point where the wind speed is zero  — that is, the wind is not blowing and it is calm.

This assumes that wind is purely a horizontal phenomenon. In fact, air is capable of moving up or down as well as across, so that at the zero-point whose existence is guaranteed by the theorem, all we can say is that the horizontal wind speed is zero. The wind may go straight up or straight down. Such an effect happens in a tornado, where objects may be lifted off the ground with catastrophic effect.

Other Hairy Shapes

Of course, heads are not spherical and not all balls are, either. The rugby ball and the American football both are elongated, but this does not affect the theorem. Even if we flatten the ball to a dinner-plate shape, we can still apply the Hairy Ball Theorem, although now it would be called the Hairy Dinner Plate Theorem.

But topology gives us lots of other shapes to play around with. A more complex shape than the sphere is the torus, or doughnut shape. This is the American doughnut with a hole in the middle rather than the British doughnut which has jam in the middle. So, if we have a hairy doughnut, it will be extremely unpleasant to eat, but we can think about whether it is possible to comb the hair on it so that it lies flat everywhere. It is easy enough to see that this is possible. All we have to do is comb the hair 'around the hole', so that every hair lies along a circle which has its centre on the main axis of the doughnut, the line that runs straight through the hole. This isn't the only solution, either.

So while mathematicians are thinking about hairy balls, they can eat their hairless doughnuts, without having to think about them being hairy too.

1Luitzen Egbertus Jan Brouwer, 1881 - 1966.

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