Periodic Trajectories in the Regular Pentagon, II

Abstract

In our recent paper [1], we studied periodic billiard trajectories in a regular pentagon and in the isosceles triangle with with the angles (π/5, π/5, 3π/5). We provided a full computation of the lengths of these trajectories, both geometric and combinatoric, and formulated some conjectures concerning symbolic periodic trajectories. The main goal of this article is to prove two of these conjectures. Technically, the study of billiard trajectories in a regular pentagon is essentially equivalent to the study of geodesics in the “double pentagon,” a translation surface obtained from two centrally symmetric copies of a regular pentagon by pairwise pasting the parallel sides. The result is a surface of genus 2 that has a flat structure inherited from the plane and a conical singularity. See [2, 3, 4, 5, 6, 7] for surveys of flat surfaces and rational polygonal billiards. Let us describe the relevant results from [1]. First of all, a periodic linear trajectory is always included into a parallel family of such trajectories, and when we talk about the period, length, symbolic orbit, etc., we always mean these parallel families. See Figure 1. Second, the double pentagon has an involution, the central symmetry that exchanges the two copies of the regular pentagon. This involution interchanges the linear trajectories that have the opposite directions. For this reason, we identify the opposite directions, so the set of directions is the real projective line RP. We identify this projective line with the circle at infinity of the hyperbolic plane in the Poincare disc model. It is clear from Figure 1 that the directions of the sides of the pentagons are periodic: every linear trajectory in this direction is closed. In fact, the socalled Veech dichotomy applies to the double pentagon: if there exists a periodic trajectory in some direction then all parallel trajectories are also periodic (and they form two strips, longer – shaded in Figure 1, and shorter – left unshaded,