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Think of it like this. Imagine a ball that has hair sprouting out of it. Each of these hairs will be "combed" down, i.e., not mapped to themselves, but mapped to some other point on the sphere. But if each of these hairs is combed down, there must exist one hair that stands completeley straight up: that is mapped onto itself. The other "combed" hairs start at their origin but end up at a different point on the sphere.
Basically, if you look at your own head of hair, it is equivalent to the (at least) one hair on your head that is sticking up: because it it mapped to no other "point" on your head but the point that it originates from.
I'm not totally sure that I explained that correctly, but I'm pretty sure. Brouwer's fixed point theorem is referred to as the "hairy ball" theorem in L.C. Thomas' Games, Theory and Applications. It's a very good introductory game theory text (a very good balance of quantitative and qualitative aspects) that uses this theorem as a leading up to Nash's theorem about equilibria in n-person games.
Like I saie, I'm not totally sure that Brouwer's hairy ball theorem fits in, but that's what jumped at me when I first saw your thread.
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