Borsuk–Ulam theorem

Informally, the Borsuk–Ulam theorem states that, for a "balloon animal" (or any arbitrarily distorted shape) made out of a spherical balloon, and then squashed into a plane (letting the air out somehow), at least one pair of points that were opposite each other on the original sphere will be squashed onto the same point of the plane.

More formally, every continuous map from the sphere to the plane maps some pair of antipodal points to the same point. An analogous statement is true in higher dimensions (see below).