A corollary of the Borsuk-Ulam theorem tells us that at any point in time there exists two antipodal points on the earth's surface which have precisely the same temperature and pressure. Pretty surprising! Here's the statement.

Borsuk-Ulam Theorem There is no continuous map f:S2->S1 such that f(-x)=-f(x), for all x.

Here S2 is the unit sphere and S1 the unit circle. The proof uses some algebraic topology.

Here's how to deduce the physical interpretation I mentioned at the beginning from the theorem. First of all the earth is a sphere! So in mathematical terms the result will follow if we can show that for any continuous map h:S2->R2 there exists x with h(x)=h(-x).

We give a reductio ad absurdum proof of this. Suppose that there is no such x. Then we can define a new function g:S2->S1 by g(x)=(h(x)-h(-x))/||h(x)-h(-x)||. By the hypothesis on h this function is well-defined, and it is obviously continuous. Note that g(-x)=-g(x), for all x. This contradicts the Borsuk-Ulam theorem, proving our result.

For another corollary of the theorem see the ham sandwich theorem.