Borsuk-Ulam Theorem

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

The case n = 2 is often illustrated by saying that at any moment there is always a pair of antipodal points on the Earth's surface with equal temperature and equal barometric pressure. This assumes that temperature and barometric pressure vary continuously.

The Borsuk-Ulam Theorem was first conjectured by Stanislaw Ulam. It was proved by Karol Borsuk in 1933.