Regular simplices and periodic billiard orbits

TL;DR

This paper investigates periodic billiard trajectories inside regular simplices in n-dimensional space, identifying two types of such trajectories and precisely locating their boundary contact points.

Contribution

It demonstrates the existence of two distinct periodic billiard trajectories in regular simplices and provides exact boundary contact coordinates for each.

Findings

Existence of a period (n+1) trajectory hitting each face once

Existence of a period (2n) trajectory hitting one face n times

Explicit coordinates for boundary contact points

Abstract

A simplex is the convex hull of $n+1$ points in $\mathbb{R}^{n}$ which form an affine basis. A regular simplex $\Delta^n$ 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.