Periodic trajectories in the regular pentagon, II

TL;DR

This paper proves two conjectures about periodic billiard trajectories and geodesics in the regular pentagon and related translation surface, advancing understanding of their symbolic dynamics.

Contribution

It provides rigorous proofs for two previously conjectured properties of periodic trajectories in the regular pentagon.

Findings

Proved two conjectures on symbolic periodic trajectories.

Enhanced understanding of billiard dynamics in regular polygons.

Contributed to the mathematical theory of translation surfaces.

Abstract

In our recent paper, we studied periodic billiard trajectories in the regular pentagon and closed geodesic on the double pentagon, a translation surface of genus two. In particular, we made a number of conjectures concerning symbolic periodic trajectories. In this paper, we prove two of these conjectures.