Long and Short Periodic Billiard Trajectories in the Regular Pentagon

TL;DR

This paper develops an algorithm to classify periodic billiard trajectories in a regular pentagon as long, short, or saddle connections, specifically for trajectories starting at midpoints, advancing understanding of billiard path properties.

Contribution

It introduces a novel algorithm to determine the length classification of periodic trajectories from midpoints in the regular pentagon billiard.

Findings

Algorithm successfully classifies trajectories as long, short, or saddle.

Provides a systematic method for analyzing billiard paths in the pentagon.

Advances understanding of geometric and combinatorial properties of billiard trajectories.

Abstract

In any periodic direction on the regular pentagon billiard table, there exists two combinatorially different billiard paths, with one longer than the other. For each periodic direction, McMullen asked if one could determine whether the periodic trajectory through a given point is long, short, or a saddle connection. In this paper we present an algorithm resolving this question for trajectories emanating from the midpoints of the pentagon.