Borsuk-Ulam Theorem

The Borsuk-Ulam Theorem states that every continuous function from a n-sphere into Euclidean n-space maps maps some pair of antipodal points onto the same point.

A nice illustration of this (and the example that made this theorem interesting to me), is that at any point in time, there are two points exactly opposite each other (antipodal) on the earths surface that have exactly the same temperature and atmospheric pressure.

Wikipedia