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