Interactive Guide
to the Monograph

by Rich Schwartz

Billiard King homepage


The purpose of this guide is to explain the main ideas in my monograph using Java applets. I discovered all the results in the monograph using my Java program, Billiard King. Any potential reader of the monograph would greatly benefit from playing with Billiard King, but Billiard King is pretty complicated. The applets in this guide break Billiard King into small pieces that are easy to understand and operate. Before reading further, you should read these basic definitions. Note: In a certain sense, the monograph comes in 2 halves. Parts I-IV comprise the first half, and parts V-VI comprise the second half. The second half has much sharper results. (I added the second half in the last 6 months.) This exposition really only deals with the first half.

The Main Results

Here are the main results of the (first half of) the monograph. 1. Erratic Orbits Theorem Outer billiards on any irrational kite has uncountably many erratic special orbits. 2. Dichotomy Theorem Relative to any irrational kite, every special orbit is periodic or else unbounded in both directions. 3. Density Theorem Relative to any irrational kite, the set of periodic special orbits is dense in the set of all special orbits. 4. Hexagrid Theorem (See below for the statement.) 5. Master Picture Theorem (See below for the statement.) The Erratic Orbits Theorem gives a robust positive solution to the Moser-Neumann problem , posed around 1960, which asks if one can have unbounded orbits in an outer billiards system. The other results are answers to unasked questions. This guide will focus on the Erratic Orbits Theorem. We will also discuss the Hexagrid Theorem and the Master Picture Theorem, because these results are needed for the Erratic Orbits Theorem. We hope to later expand the guide so that it also discusses the Dichotomy Theorem and the Density Theorem.

The Arithmetic Graph

The key idea in my monograph is to study an object that I call the arithmetic graph . The arithmetic graph is a graph in the plane, with integer coordinate vertices, that encodes some of the dynamics of outer billiards on a kite. Before you read anything else about the monograph, you need to understand the arithmetic graph. Click here to learn about it.

Ingredients for the EOT

The proof of the Erratic Orbits Theorem has 3 main ingredients. 1. Box Theorem 2. Period Copying Theorem 3. Geometric limits You should read about these in order.

The Hexagrid Theorem

The Box Theorem is a consequence of the Hexagrid Theorem . Roughly speaking, the H.T. says that the (extended) arithmetic graph is controlled by 6 infinite families of parallel lines. This result reflects a similar kind of quasi-periodicity one sees in DeBruijn's pentagrid method for generating the Penrose tiling.

Master Picture Theorem

All the results mentioned above are derived from our main structural result, the Master Picture Theorem . This result gives a "formula" for the arithmetic graph, in terms of certain 4-dimensional integer lattice polytope partitions. The Master Picture Theorem seems likely to generalize to outer billiards on polygons, though I have not worked this out.

The Penrose Kite

Before I proved the Erratic Orbits Theorem in general, I proved it for the single example of the Penrose kite, the kite that arises in the Penrose tiling To learn about the E.O.T. for this example, see the Penrose kite demo . You can understand half the demo before knowing about the arithmetic graph, and the whole demo after knowing about the arithmetic graph.

Modular Limits

I discovered another neat property of outer billiards on kites, which I call the modular limit phenomenon . This is a certain connection between outer billiards and the modular group related to renormalization. Here is the modular limit demo .