The maximum number of cycles in a triangular-grid billiards system with a given perimeter

TL;DR

This paper establishes a sharp inequality relating the perimeter of a grid triangle polygon to the maximum number of billiard trajectories inside it, resolving a conjecture and characterizing cases of equality.

Contribution

It proves a conjectured upper bound on the number of billiard cycles in grid polygons based on perimeter, and characterizes when equality holds.

Findings

Proved the inequality: cyc(P) ≤ (perim(P) + 2)/4.

Resolved a conjecture by Defant and Jiradilok.

Characterized all polygons where equality is achieved.

Abstract

Given a (simple) grid polygon $P$ in a grid of equilateral triangles, Defant and Jiradilok considered a billiards system where beams of light bounce around inside of $P$. We study the relationship between the perimeter $\operatorname{perim}(P)$ of $P$ and the number of different trajectories $\operatorname{cyc}(P)$ that the billiards system has. Resolving a conjecture of Defant and Jiradilok, we prove the sharp inequality $\operatorname{cyc}(P) \leq (\operatorname{perim}(P) + 2)/4$ and characterize the equality cases.