Outer billiards is a simple dynamical system in which a point “orbits” around a convex shape in the plane. This system was introduced by B.H. Neumann in the 1950s and then popularized by J. Moser in the 1970s as a toy model for celestial mechanics. All along, one of the central questions has been whether or not one can find a shape and a starting point, so that this point “escapes to infinity” as it orbits around. This question is vaguely related to the question of whether or not the orbit of the Earth around the sun is stable: will the Earth eventually wander away from the sun?

In this talk I will explain how outer billiards works. I will also explain my recent solution to the basic problem I mentioned. My solution relates the problem to certain kinds of self-similar tilings. During the talk I will demonstrate my program, Billiard King, the graphical user interface I built and used to solve the problem of unbounded orbits.