The Borsuk-Ulam Theorem
states that any continuous function
from an n
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.