[Paper Review] Periodicity and Circle Packing in the Hyperbolic Plane

Lewis Bowen|ArXiv.org|Apr 22, 2003

Quasicrystal Structures and Properties13 references3 citations

TL;DR

This paper establishes that periodic circle packings—those with cofinite symmetry groups—are dense in the space of isometry-invariant probability measures on radius $ r $-circle packings in the hyperbolic plane. The key result is that the supremum density of such packings equals the maximum density achievable by any isometry-invariant measure, and this maximum density function varies continuously with radius.

ABSTRACT

We prove that given a fixed radius $r$, the set of isometry-invariant probability measures supported on ``periodic'' radius $r$-circle packings of the hyperbolic plane is dense in the space of all isometry-invariant probability measures on the space of radius $r$-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 $r$-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.

Motivation & Objective

• To resolve foundational issues in hyperbolic circle packing by addressing the lack of well-defined density and existence of optimal packings.
• To show that isometry-invariant probability measures on radius $ r $-circle packings in the hyperbolic plane admit a dense subset of periodic packings.
• To establish that the maximum density over all invariant measures is achieved as the supremum of densities of periodic packings.
• To prove the maximum density function is continuous in the radius $ r $, extending results from Euclidean to hyperbolic geometry.

Proposed method

• Uses an ergodic-theoretic framework to analyze isometry-invariant probability measures on the space of radius $ r $-circle packings in the hyperbolic plane.
• Applies Nevo and Stein's ergodic theory results to show that for $ \mu $-almost every packing, the density exists and is independent of origin.
• Defines the density of a measure $ \mu $ as the $ \mu $-measure of packings covering the origin, ensuring consistency across the space.
• Leverages the compactness of the space of isometry-invariant measures under the weak* topology to guarantee existence of an optimally dense measure.
• Utilizes the topology of uniform convergence on compact subsets to analyze convergence and density behavior of sequences of packings.
• Applies results from hyperbolic geometry, including the existence of complete finite-volume hyperbolic manifolds and branched covers, to extend continuity and density arguments.

Experimental results

Research questions

• RQ1 Is the set of periodic radius $ r $-circle packings dense in the space of isometry-invariant probability measures on the hyperbolic plane?
• RQ2 Does the maximum density over all isometry-invariant measures on radius $ r $-circle packings equal the supremum of densities of periodic packings?
• RQ3 Is the maximum density function $ D(r) $ continuous with respect to the radius $ r $ in the hyperbolic plane?
• RQ4 Can the optimal density in the hyperbolic plane be realized as a limit of periodic packings, even when no single periodic packing achieves the maximum?
• RQ5 How does the behavior of density in the hyperbolic plane compare to that in Euclidean space, particularly regarding homothety and invariance?

Key findings

• The set of isometry-invariant probability measures supported on periodic radius $ r $-circle packings is dense in the space of all such measures on the hyperbolic plane.
• The maximum density achievable by any isometry-invariant measure on radius $ r $-circle packings is equal to the supremum of densities of periodic packings.
• The maximum density function $ D(r) $ varies continuously with respect to the radius $ r $, ensuring stability of optimal packing density across radius changes.
• For every $ r $, there exists a sequence of periodic packings whose densities converge to the optimal density, even if no single periodic packing achieves it.
• The density of a packing is well-defined and independent of origin for $ \mu $-almost every packing when $ \mu $ is an isometry-invariant probability measure.
• The framework extends to horoball packings, and continuity of $ D(r) $ at infinity (i.e., as $ r \to \infty $) is established via convergence to Euclidean sphere packing density.