Triangular-Grid Billiards and Plabic Graphs

TL;DR

This paper explores a billiards system within polygons on a triangular grid, establishing bounds relating the polygon's area and perimeter to the number of light beam trajectories, and connects these results to plabic graph theory.

Contribution

It introduces a novel billiards model on triangular grids and links geometric properties to combinatorial structures in plabic graphs, providing tight bounds and characterizations.

Findings

Area of polygon bounds by number of light beam cycles

Perimeter bounds by number of light beam cycles

Characterization of polygons with minimal area for given cycles

Abstract

Given a polygon $P$ in the triangular grid, we obtain a permutation $\pi_P$ via a natural billiards system in which beams of light bounce around inside of $P$. The different cycles in $\pi_P$ correspond to the different trajectories of light beams. We prove that \[\text{area}(P)\geq 6\text{cyc}(P)-6\quad\text{and}\quad\text{perim}(P)\geq\frac{7}{2}\text{cyc}(P)-\frac{3}{2},\] where $\text{area}(P)$ and $\text{perim}(P)$ are the (appropriately normalized) area and perimeter of $P$, respectively, and $\text{cyc}(P)$ is the number of cycles in $\pi_P$. The inequality concerning $\text{area}(P)$ is tight, and we characterize the polygons $P$ satisfying $\text{area}(P)=6\text{cyc}(P)-6$. These results can be reformulated in the language of Postnikov's plabic graphs as follows. Let $G$ be a connected reduced plabic graph with essential dimension $2$. Suppose $G$ has $n$ marked boundary points and $v$ (internal) vertices, and let $c$ be the number of cycles in the trip permutation of $G$. Then we have \[v\geq 6c-6\quad\text{and}\quad n\geq\frac{7}{2}c-\frac{3}{2}.\]