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:S*^{2}->S^{1} such that *f(-x)=-f(x)*,
for all *x*.

Here *S*^{2} is the unit sphere and *S*^{1} 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:S*^{2}->**R**^{2}
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:S*^{2}->S^{1}
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.