Outer billiards is a simple geometric dynamical system, defined
relative to a planar convex shape, that serves
as a toy model for celestial mechanics.  It was introduced in the
1950s by B.H. Neumann and later popularized by J. Moser.  One of
the central questions all along has been:  Can an outer billiards
system have any unbounded orbits?  This question is at least
vaguely related to questions about the stability of the solar system.
I'll prove that you can have unbounded orbits for outer billiards
when it is defined relative to the Penrose kite, the convex
quadrilateral that appears in the Penrose tiling.  I will relate
the solution of this problem to things like self-similar tilings.
polygon exchange maps, and arithmetic dynamics.