A theorem in topology says:

For every continuous map f: S^n \rightarrow R^n there exists a pair of anitpodal points x and - x in S^n such that f(x) = f(-x).

Specializing to the case n = 2 , one might conclude that at any point of time there are two antipodal points on Earth’s surface (which is homeomorphic to S^2) having same, say, pressure P and temperature T (which together constitute R^2 of the theorem) .

Question: Isn’t it that the presence of \textit{two} polar caps (The Arctic and The Antarctica) where the day/night variation in temperature is quite low, may then be looked upon as a kind of `consequence’ of this theorem? (ofcourse the theorem doesn’t say where x is located but at almost all other regions on Earth’s surface, the antipodal points are expected to have quite different temperatures due to day/night variation.)

ps1: One might worry about the validity of the assumption of continuity of P and T since there are wild local fluctuations; but I feel once you coarse-grain things out, this is a reasonable assumption.

ps2: For proof of the theorem, see http://www.mi.ras.ru/~scepin/elem-proof-reduct.pdf

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