Regular simplices and periodic billiard orbits

A simplex is the convex hull of $n+1$ points in ${\mathbb{R}}^n$ which form an affine basis. A regular simplex ${\mathrm{\Delta }}^n$ is a simplex with sides of the same length. We consider the billiard flow inside a regular simplex of ${\mathbb{R}}^n$. We show the existence of two types of periodic trajectories. One has period $n+1$ and hits once each face. The other one has period $2n$ and hits $n$ times one of the faces while hitting once any other face. In both cases we determine the exact coordinates for the points where the trajectory hits the boundary of the simplex.