  We analyze the general problem of determining optimally dense packings, in a
Euclidean or hyperbolic space, of congruent copies of some fixed finite set of
bodies. We are strongly guided by examples of aperiodic tilings in Euclidean
space and a detailed analysis of a new family of examples in the hyperbolic
plane. Our goal is to understand qualitative features of such optimum density
problems, in particular the appropriate meaning of the uniqueness of solutions,
and the role of symmetry in classfying optimally dense packings.
