Periodic orbits for square and rectangular billiards

TL;DR

This paper classifies all periodic orbits in square and rectangular billiards, revealing that such orbits cannot have an odd number of collisions and linking orbit classes to Euler's totient function.

Contribution

It provides a complete classification of periodic orbits in square and rectangular billiards, including the impossibility of odd collisions and the connection to number theory.

Findings

Periodic orbits cannot have an odd number of collisions.

The number of orbit classes relates to Euler's totient function.

Methods to construct orbits with prescribed even collisions.

Abstract

Mathematical billiards is much like the real game: a point mass, representing the ball, rolls in a straight line on a (perfectly friction-less) table, striking the sides according to the law of reflection. A billiard trajectory is then completely characterised by the number of elastic collisions. The rules of mathematical billiards may be simple, but the possible behaviours of billiard trajectories are endless. In fact, several fundamental theory questions in mathematics can be recast as billiards problems. A billiard trajectory is called a periodic orbit if the number of distinct collisions in the trajectory is finite. We classify all possible periodic orbits on square and rectangular tables. We show that periodic orbits on such billiard tables cannot have an odd number of distinct collisions. We also present a connection between the number of different classes of periodic orbits and Euler's totient function, which for any integer $N$ counts how many integers smaller than $N$ share no common divisor with $N$ other than $1$. We explore how to construct periodic orbits with a prescribed (even) number of distinct collisions, and investigate properties of inadmissible (singular) trajectories, which are trajectories that eventually terminate at a vertex (a table corner).