Colin R Defant, Princeton University

Abstract

Given a polygon P in the triangular grid, we define a rigid billiards system by allowing beams of light to bounce around inside of P so that each beam always meets the midpoint of a unit-length boundary segment at a 60-degree angle. This basic setup seems ripe for investigation; we will focus on some natural extremal questions concerning how large P must be relative to the number c of light-beam trajectories. Our main theorem states that the (appropriately normalized) area of P is at least 6c-6 and that the (appropriately normalized) perimeter of P is at least (7c-3)/2. This talk will sketch some of the geometric aspects of the proof of this result. We will also briefly discuss how one can reformulate these inequalities in the language of Postnikov's plabic graphs. This talk is based on joint work with Pakawut Jiradilok.