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There's an advanced subfield of mathematics called algebraic topology, and in that subfield, there's a thing called the Hairy Ball Theorem. Like most of algebraic topology it's of little or no practical importance*, but I thought I'd write about it because, at least in its simplest form, it's easy to understand, and of course it's got a great name and it allows me to include a cute animal picture. What's not to like?
What the theorem says is that there is no nonvanishing continuous tangent vector field on even-dimensional n-spheres. No, I don't understand that either. More colloquially, it says that you can't comb a hairy ball (or a coconut) flat without creating a cowlick - in contrast to a hairy doughnut, for which it's easy. Try it and see next time any of these objects comes your way. You can also try it with a hedgehog (it's called the Hedgehog Theorem in some places) but do make sure you wear thick gloves.
I like to give proofs of the theorems I talk about, but as a rule of thumb, if a theorem hadn't been proved by about 1820, it's because it's really, really hard to prove, and this one missed the cutoff date by over fifty years. It was eventually proved by Henri Poincare, one of a very large number of amazing French mathematicians, who was called the "last universalist" because after him and largely because of him, maths became so big and so complicated that no single person could hope to master all of it.
Fortunately it's only maths as a whole that's too much for one person to handle, not the various A-level courses on offer, which is why tutoring can be so helpful. So get in touch if you'd like to discuss some.
*If you are an algebraic topologist and are for some reason offended by my sweeping and possibly underinformed dismissal of your entire life's work, you too are welcome to drop me a line.
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