Brouwer fixed-point theorem

This article describes a theorem about spheres

Statement

In the language of spheres

Any continuous map from a disc to itself must have a fixed point. In other words, if $D^{n}$ denotes the spherical disc in $\mathbb {R} ^{n}$, any continuous map $f:D^{n}\to D^{n}$ must have a point $x$ such that $f(x)=x$.

In the language of simplices

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