We introduce a new dynamical system that we call tiling billiards, where trajectories refract through planar tilings. This system is motivated by a recent discovery of physical substances with negative indices of refraction. We investigate several special cases where the planar tiling is created by dividing the plane by lines, and we describe the results of computer experiments.

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Consider all geodesics between two given points on a polyhedron. On the regular tetrahedron, we describe all the geodesics from a vertex to a point, which could be another vertex. Using the Stern-Brocot tree to explore the recursive structure of geodesics between vertices on a cube, we prove, in some precise sense, that there are twice as many geodesics between certain pairs of vertices than other pairs. We also obtain the fact that there are no geodesics that start and end at the same vertex on the regular tetrahedron or the cube.

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. arXiv:1609.00772v2 [math.MG] 18 Jul 2018Theorem 1.2 (Ergodic directions). If θ ∈ E then the flows T θ,+ and T θ,− are ergodic when the domains X θ,+ and X θ,− are equipped with their natural Lebesgue measures.By remarks above this implies that the set of non-ergodic directions has Hausdorff dimension less than 1. Example 1.3 (θ = π 4 ). The angle θ = π 4 lies in E. To see this observe that θ = θ + π 3 = 7π 12 ∈ [ π 3 , 2π 3 ]. By definition u θ is the unit tangent vector based at i pointed into ∆ at angle 5π 6 from the vertical. The billiard trajectory is periodic and is depicted in Figure 2. This billiard trajectory repeatedly travels above the line where y = 1 √ 3 and so by the Theorem above the flows T θ,+ and T θ,− are ergodic. A trajectory of T θ,+ is shown on the left side of Figure 2. We do not know if this trajectory equidistributes.

We describe the cutting sequences associated to geodesic flow on regular polygons, in terms of a combinatorial process called derivation. This work is an extension of some of the ideas and results in Smillie and Ulcigrai's recent paper, where the analysis was made for the regular octagon. It turns out that the main structural properties of the octagon generalize in a natural way.