The Borsuk–Ulam theorem

The mathematical tidbit for today is the Borsuk–Ulam theorem, which states that every continuous function f from the n-dimensional sphere to n-dimensional space must satisfy f(p) = f(−p) for some point p.

In particular, every continuous function f from a 2-dimensional sphere (say, the Earth’s surface) to the plane must satisfy f(p) = f(q) for some antipodal pair of points p and q.

Thus, if we can describe weather by a pair of numbers (say, temperature and rainfall), there must be an antipodal pair of points p and q with the same weather (because two numbers specify a point in the plane).

The maps above (for average maximum temperature) and below (for rainfall) show July weather at various places on Earth, and a pair of points with the same weather is highlighted.

It’s a miracle that it works in this case, of course, because the maps only define temperature and rainfall on the land; I would not have been able to recognise a suitable antipodal pair of points if one or both were at sea.

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