Polygons in billiard orbits

TL;DR

This paper investigates the geometric properties of billiard orbits in rectangles, establishing a sharp upper bound of thirteen for the number of distinct polygonal areas formed by truncated billiard trajectories.

Contribution

It introduces a new bound on the number of different polygon areas generated by billiard orbits in rectangular tables.

Findings

Thirteen is the maximum number of distinct polygon areas in such billiard partitions.

The study provides a geometric framework for analyzing billiard orbits.

Results improve understanding of billiard dynamics and polygonal partitions.

Abstract

We study the geometry of billiard orbits on rectangular billiards. A truncated billiard orbit induces a partition of the rectangle into polygons. We prove that thirteen is a sharp upper bound for the number of different areas of these polygons.