Main Page | See live article | Alphabetical index

Borsuk-Ulam Theorem

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. (Two points on a sphere are called antipodal if they sit on directly opposite sides of the sphere's center.)

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.