The Hairy Ball Theorem

By Anupum Pant

At any point in time, there’s at least one such place on earth where the wind isn’t blowing. It sounds like a deep thing to think about. And you might even endeavour to prove it wrong, but you shouldn’t try. That’s because a theorem in algebraic topology called the hairy ball theorem proves that there is no nonvanishing continuous tangent vector field on even-dimensional n-spheres. Oh wait, that’s a bit too much jargon.

In simple words, if you have a hairy ball and tried to comb the hair on it such that the hair is neatly folded everywhere on the surface of that ball, well, that wouldn’t happen. It’s impossible to do it in 3D spheres (not donuts). Not just spheres, you won’t even be able to do it on a banana shaped hairy ball. Nor would you accomplish it on any 3D shape that can be squished into a ball shaped, to picture it.

So how is it even related to the earth and the wind? Well, if you think about it, say earth is that ball and the hair on that ball is the wind (with a direction and a specific length – magnitude). According to the hairy ball theorem, there’d be at least one point on earth where the tangential vector (hair) would be zero – or in other words, standing up.

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