Borsuk-Ulam Theorem (idea) by Noether

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.

ham sandwich theorem	When not to break up with your girlfriend	geoid	Why my co-worker, "Bob," is an idiot
proof of the ham sandwich theorem	Homeomorphism	Antipodal	Fundamental theorem of algebra
How to compliment a female coworker	reductio ad absurdum	star convex	Hairy ball theorem
For all its insanity, I like my job and my co-workers	continuous	topology	Corners on the Curving Sky
All of my coworkers spit into urinals	algebraic geometry	Intermediate Value Theorem	Co-worker: pretzel-smoking man
Stanislaw Ulam	Corollary	pressure