of the Borsuk-Ulam theorem
tells us that at any point in time
there exists two antipodal
points on the earth
which have precisely the same temperature
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.