Outer Billiards

Outer billiards, also known as dual billiards, is a dynamical system where a point in the plane outside a convex shape jumps around the shape according to precise geometric rules. Instead of reflecting inside a boundary like standard inner billiards, the exterior point moves along a tangent ray by a distance equal to the length from the starting point to the contact vertex. Depending on the shape of the table, trajectories form intricate periodic cycles, self-similar fractal patterns, or chaotic boundary webs. When tracking cycle lengths across the entire plane, distinct regions emerge with specific periodic behaviors, creating colorful tiling patterns across Euclidean space. While regular polygon tables typically generate bounded periodic orbits, certain irregular tables create unbounded orbits that escape to infinity. This phenomenon, proved by mathematician Richard Schwartz in 2007 using Penrose kite structures, demonstrates how simple deterministic rules produce deep geometric complexity.