Borsuk-Ulam Theorem

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.

But it's not. The Earth is slightly squashed like a flattened tennis ball, bulging outwards at the equator due to the spin of magma. In addition, there are the small problems of geography, coastal dynamics, and ablative heat transform (I don't know what that means, but it sounds impressive).

The Earth's shape is also affected (minimally) by the position of the moon, and the other solar bodies.

So unless this is true for any vaguely sphere like surface, the theorem doesn't apply.

Although the Earth is not a perfect sphere, a mathematical object representing its surface (a geoid) is homeomorphic to S². Therefore, the theorem applies.

Execpt for the fact that, after we've located¹ the aforementioned two points, and after transforming back to the geoid, the points probably won't be exactly antipodal anymore.

ariels says "One way to do it is to show that the Earth is star convex around its centre. Define a map R^3\{0} -> R^3 by x -> x/|x|. Then the map transforms the Earth says to a sphere, while preserving antipodality."

¹located, somehow. We were given an existence proof, not a construction.