Symplectic tiling billiards, planar linkages, and hyperbolic geometry

In this paper I will unite two games, symplectic billiards and tiling billiards. The new game is called symplectic tiling billiards. I will prove a result about periodic orbits of symplectic tiling billiards in a very special case and then show how this result combines with the construction in Thurston’s paper <italic>Shapes of Polyhedra</italic> to give hyperbolic structures on moduli spaces of planar equilateral polygons.