Brouwer fixed-point theorem

This article describes a theorem about spheres

Statement

In the language of spheres

Any continuous map from a sphere to itself must have a fixed point. In other words, if $S^{n}$ denotes the $n$-dimensional sphere, any continuous map $f:S^{n}\to S^{n}$ must have a point $x$ such that $f(x)=x$.

In the language of simplices

Any continuous map from the boundary of the standard $n$-simplex, to itself, has a fixed point.