Triangular-Grid Billiards and Plabic Graphs
Colin Defant, Pakawut Jiradilok

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 in the triangular grid, we obtain a permutation via a natural billiards system in which beams of light bounce around inside of . The different cycles in 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 and are the (appropriately normalized) area and perimeter of , respectively, and is the number of cycles in . The inequality concerning is tight, and we characterize the polygons satisfying . These results can be reformulated in the language of Postnikov's plabic graphs as follows. Let be a connected reduced plabic graph with essential dimension . Suppose has marked boundary points…
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Taxonomy
TopicsCellular Automata and Applications · Mathematical Dynamics and Fractals · Stochastic processes and statistical mechanics
