Periodicity and ergodicity in the trihexagonal tiling

TL;DR

This paper studies light ray behavior in a trihexagonal tiling with specific refraction properties, revealing dense trajectories, and classifying initial conditions for periodic and drift-periodic rays.

Contribution

It provides a detailed analysis of light dynamics in the trihexagonal tiling, including density results and classification of periodic trajectories.

Findings

Almost all light rays are dense in certain infinite regions.

Complete description of initial conditions for periodic and drift-periodic rays.

Regions of density have infinite area with periodic triangular holes.

Abstract

We consider the dynamics of light rays in the trihexagonal tiling where triangles and hexagons are transparent and have equal but opposite indices of refraction. We find that almost every ray of light is dense in a region of a particular form: the regions have infinite area and consist of the plane with a periodic family of triangles removed. We also completely describe initial conditions for periodic and drift-periodic light rays.