Periodic paths on the pentagon, double pentagon and golden L

We give a tree structure on the set of all periodic directions on the golden L, which gives an associated tree structure on the set of periodic directions for the pentagon billiard table and double pentagon surface. We use this to give the periods of periodic directions on the pentagon and double pentagon. We also show examples of many periodic billiard trajectories on the pentagon, which are strikingly beautiful, and we describe some of their properties. Finally, we give conjectures and future directions based on experimental computer evidence.