Periodic paths on the pentagon, double pentagon and golden L

Abstract

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.

A short introduction to translation surfaces, Veech surfaces, and Teichmueller dynamics

“…The corner of coordinates (± cot π 2n+1 , y), is the image by Ψ, when n = 2, of the golden L (see [8]), and when n > 2, of stair-shaped polygons (see [26]). Those polygons are invariant by S, which acts as the involution z → − 1 z on the hyperbolic plane, this is why the corner becomes a conical point with angle π in the Teichműller curve.…”

“…Consider the surface L(a, 1) on the left of Figure 17, where a = 1+ √ d 2 . For d = 5, this surface lies in the Teichműller curve of the double pentagon, it is usually called the Golden L (see [7,8]). For d = 2, it can be shown to lie in the Teichműller curve of the regular octagon (see 47).…”