Hyperbolic Staircases: Periodic Paths on $2g+1$-gons

TL;DR

This paper unifies two approaches to studying periodic billiard trajectories in regular polygons and generalizes the results from the pentagon to all odd-sided polygons of the form 2g+1.

Contribution

It connects hyperbolic and translation surface methods to analyze periodic paths and extends these techniques to arbitrary odd-sided regular polygons.

Findings

Unified two different methods for analyzing periodic billiard trajectories.

Generalized results from pentagon to all 2g+1-sided polygons.

Provided a framework for understanding periodic paths in complex polygons.

Abstract

The study of polygonal billiards, particularly those in the regular pentagon, has been the subject of two recent papers. One of these papers approaches the problem of discovering the periodic trajectories on the pentagon by identifying slopes of periodic directions with points in the Poincar\'e disk generated by hyperbolic isometric transformations. The other approach, coming from the other paper, transforms the double pentagon into a rectilinear translation surface called the 'golden L', where periodic directions are generated by a set of matrices associated with this surface in a special way. We connect and unify these two approaches, and use our unification of these results to generalize them to arbitrary $2g+1$-sided regular polygons.