Digit sequences of rational billiards on tables which tile $\mathbb{R}^2$

TL;DR

This paper classifies the periodic digit sequences generated by billiard orbits on specific tiling polygons in the plane, solving a problem posed by Baxter and Umble.

Contribution

It provides a complete classification of digit strings for billiard orbits on four types of tiling convex polygons in the plane.

Findings

Classified digit sequences for equilateral triangle billiards

Classified digit sequences for 45-45-90 triangle billiards

Classified digit sequences for 30-60-90 triangle billiards and rectangles

Abstract

We classify the periodic digit strings which arise from periodic billiard orbits on the four convex $n$-gons $\Delta$ which tile $\mathbb{R}^2$ under reflection, answering problem a posed by Baxter and Umble. $\Delta$ is either an equilateral triangle, a $45-45-90$ triangle a $30-60-90$, triangle or a rectangle.