259 research outputs found
Periodicity and Circle Packing in the Hyperbolic Plane
We prove that given a fixed radius , the set of isometry-invariant
probability measures supported on ``periodic'' radius -circle packings of
the hyperbolic plane is dense in the space of all isometry-invariant
probability measures on the space of radius -circle packings. By a periodic
packing, we mean one with cofinite symmetry group. As a corollary, we prove the
maximum density achieved by isometry-invariant probability measures on a space
of radius -packings of the hyperbolic plane is the supremum of densities of
periodic packings. We also show that the maximum density function varies
continuously with radius.Comment: 25 page
Optimally dense packings of hyperbolic space
In previous work a probabilistic approach to controlling difficulties of
density in hyperbolic space led to a workable notion of optimal density for
packings of bodies. In this paper we extend an ergodic theorem of Nevo to
provide an appropriate definition of optimal dense packings. Examples are given
to illustrate various aspects of the density problem, in particular the shift
in emphasis from the analysis of individual packings to spaces of packings.Comment: 27 pages, 11 figure
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