Periodic paths on the pentagon, double pentagon and golden L

TL;DR

This paper introduces a tree structure for periodic billiard trajectories on the golden L, pentagon, and double pentagon, providing new insights into their periods and properties, supported by computer experiments and conjectures.

Contribution

It establishes a novel tree structure on periodic directions, enabling the determination of periods and revealing intricate trajectory patterns.

Findings

Tree structure on periodic directions established

Periods of trajectories explicitly determined

Examples of aesthetically striking trajectories provided

Abstract

We give a tree structure on the set of all periodic directions on the golden L, which gives an associated tree structure on the set of periodic directions for the pentagon billiard table and double pentagon surface. We use this to give the periods of periodic directions on the pentagon and double pentagon. We also show examples of many periodic billiard trajectories on the pentagon, which are strikingly beautiful, and we describe some of their properties. Finally, we give conjectures and future directions based on experimental computer evidence.