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 \[v\geq 6c-6\quad\text{ and}\quad n\geq\frac{ 7}{ 2}c-\frac{ 3}{ 2}.\]
Categories: math.CO