Open Access Publications from the University of California

## Published Web Location

https://doi.org/10.5070/C63160422
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 $\operatorname{area}(P)\geq 6\operatorname{cyc}(P)-6\quad\text{and}\quad\operatorname{perim}(P)\geq\frac{7}{2}\operatorname{cyc}(P)-\frac{3}{2},$ where $$\operatorname{area}(P)$$ and $$\operatorname{perim}(P)$$ are the (appropriately normalized) area and perimeter of $$P$$, respectively, and $$\operatorname{cyc}(P)$$ is the number of cycles in $$\pi_P$$. The inequality concerning $$\operatorname{area}(P)$$ is tight, and we characterize the polygons $$P$$ satisfying $$\operatorname{area}(P)=6\operatorname{cyc}(P)-6$$. 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 $v\geq 6c-6\quad\text{and}\quad n\geq\frac{7}{2}c-\frac{3}{2}.$

Mathematics Subject Classifications: 05D99, 51M04

Keywords: Triangular grid, billiards, plabic graph, membrane