Outer billiards outside a regular dodecagon

The existence of an aperiodic orbit for an outer billiard outside a regular dodecagon is proved. It is shown that almost all orbits of such an outer billiard are periodic, and all possible periods are explicitly listed. The proofs of the theorems make use of computer calculations.

The Nature of the Compactification

This chapter looks more closely at Theorem 15.1 and gives more information about the PET that appears in that result. The basic idea of the proof is to remove from the torus Ŝ the singular set, i.e., the places where the PET is not defined. What is left over is isometric to the interior of a convex parallelotope. Section 16.2 analyzes the singular set and Section 16.3 constructs X1. Section 16.4 constructs the second parallelotope based on the action of the PET from Theorem 15.1. The proof of Theorem 16.1 finishes at the end of Section 16.4. Section 16.5 restates the case of Theorems 15.1 and 16.1 that apply to the pinwheel map associated to outer billiards on a polygon without parallel sides. The result is Theorem 16.9. Finally, Section 16.7 shows how Theorems 0.4 and 16.9 match up.

Graph Master Picture Theorem

This chapter aims to prove Theorem 0.4, the Graph Master Picture Theorem. Theorem 0.4 is proven in two different ways, the first proof is discussed here; it deduces Theorem 0.4 from Theorem 13.2, which is a restatement of [S1, Master Picture Theorem] with minor cosmetic changes. The chapter is organized as follows. Section 13.2 discusses the special outer billiards orbits on kites. Section 13.3 defines the arithmetic graph, which is an arithmetical way of encoding the behavior of a certain first return map of the special orbits. Section 13.4 states Theorem 13.2, a slightly modified and simplified version of [S1, Master Picture Theorem]. Section 13.5 deduces Theorem 0.4 from Theorem 13.2 and one extra piece of information. Finally, Section 13.6 lists the polytopes comprising the partition associated to Theorems 13.2 and 0.4.

Pinwheels and Quarter Turns

This chapter is the first of three that will prove a generalization of the Graph Master Picture Theorem which works for any convex polygon P without parallel sides. The final result is Theorem 16.9, though Theorems 15.1 and 16.1 are even more general. Let θ‎ denote the second iterate of the outer billiards map defined on R 2 − P. Section 14.2 generalizes a construction in [S1] and defines a map closely related to θ‎, called the pinwheel map. Section 14.3 shows that, for the purposes of studying unbounded orbits, the pinwheel map carries all the information contained in θ‎. Section 14.4 will defines another dynamical system called a quarter turn composition. A QTC is a certain kind of piecewise affine map of the infinite strip S of width 1 centered on the X-axis. Section 14.5 shows that the pinwheel map naturally gives rise to a QTC and indeed the pinwheel map and the QTC are conjugate. Section 14.6 explains how this all works for kites.

The Plaid Model

Outer billiards provides a toy model for planetary motion and exhibits intricate and mysterious behavior even for seemingly simple examples. It is a dynamical system in which a particle in the plane moves around the outside of a convex shape according to a scheme that is reminiscent of ordinary billiards. This book provides a combinatorial model for orbits of outer billiards on kites. The book relates these orbits to such topics as polytope exchange transformations, renormalization, continued fractions, corner percolation, and the Truchet tile system. The combinatorial model, called “the plaid model,” has a self-similar structure that blends geometry and elementary number theory. The results were discovered through computer experimentation and it seems that the conclusions would be extremely difficult to reach through traditional mathematics. The book includes an extensive computer program that allows readers to explore the materials interactively and each theorem is accompanied by a computer demonstration.