I got a haircut, and it got me thinking about the hairy ball theorem from algebraic topology, which believe it or not is about vectors and cowlicks. Sadly, my head does not satisfy the hypotheses of the hairy ball theorem, so my cowlicks are not made of math.
According to my 9th grade biology teacher, the lining of my digestive tract, from my mouth all the way to the other end, is an extension of my skin. If you look at it like that, I violate the assumptions of the hairy ball theorem in another way: I’m not topologically equivalent to a sphere at all, but to a torus, the surface of a donut. (I’m not really sure where my ears and sinuses fit into this. I’m not that kind of doctor.) You can comb a hairy donut. The way to see this is to imagine a rectangle covered in hair, maybe a carpet square or some sod. You can definitely comb that, and you can attach opposite sides to make a torus with a well-behaved hairstyle. Although if I had a hairy digestive tract, I don’t think I’d be worried about cowlicks.
Read the full post at Roots of Unity.
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