A theorem in topology says:
For every continuous map there exists a pair of anitpodal points and in such that .
Specializing to the case , one might conclude that at any point of time there are two antipodal points on Earth’s surface (which is homeomorphic to ) having same, say, pressure and temperature (which together constitute of the theorem) .
Question: Isn’t it that the presence of 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 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 and 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