General Behavior of Brick Tiling Billiards

Abstract

This paper studies tiling billiard trajectories on brick tilings using folding and interval map techniques. Inspired by triangle tiling billiards, where researchers used folding to understand reflections and dynamics, we first review how folding works in the triangle case and how trajectories can be represented as chords on a circle. Then we move to square and brick tiling. In brick tiling with rational offset p/q, we define a folding map and prove that every nonsingular trajectory is either periodic or drift periodic, with period at most 4|2q-1|. In the special case where the offset is 1/q}, we use an interval map that tracks how the trajectory moves along the edges of the squares, and we prove a sharper upper bound of 4q on the period