  In this long paper we give a fairly complete analysis of outer billiards on
the Penrose kite. Our analysis reveals that this 2-dimensional non-compact
system has a 3-dimensional compactification, a certain polyhedron exchange map,
and that this compactification has a renormalization scheme. These two features
allow us to make some sharp statements concerning the distribution, large-scale
geometry, fine-scale geometry, and hidden algebraic symmetries of the orbits.
For instance, one of our results is that the union of the unbounded orbits has
Hausdorff dimension 1. We give a computer-aided proof of the results concerning
the compactification and the renormalization. This proof involves finitely many
calculations done with exact integer arithmetic.
