  Outer billiards is a basic dynamical system, defined relative to a planar
convex shape. This system was introduced in the 1950's by B.H. Neumann and
later popularized in the 1970's by J. Moser. All along, one of the central
questions has been: is there an outer billiards system with an unbounded orbit.
We answer this question by proving that outer billiards defined relative to the
Penrose Kite has an unbounded orbit. The Penrose kite is the quadrilateral that
appears in the famous Penrose tiling. We also analyze some of the finer orbit
structure of outer billiards on the penrose kite. This analysis shows that
there is an uncountable set of unbounded orbits. Our method of proof relates
the problem to self-similar tilings, polygon exchange maps, and arithmetic
dynamics.
