The Apollonian metric is a generalization of the hyperbolic metric introduced by Beardon [2]. It is defined in arbitrary domains in Rn and is Möbius invariant. Another advantage over the well-known quasihyperbolic metric [8] is that it is simpler to evaluate. On the downside, points cannot generally be connected by geodesics of the Apollonian metric. This paper is the last in a series of four p...

In this paper we study the integral properties of Apollonian-3 circle packings, which are variants of the standard Apollonian circle packings. Specifically, we study the reduction theory, formulate a local-global conjecture, and prove a density one version of this conjecture. Along the way, we prove a uniform spectral gap for the congruence towers of the symmetry group.

We introduce the apollonian metric in Carnot groups using capacity. Extending Beardon’s result for euclidean space, we give an equivalent definition using the cross ratio in Iwasawa groups. We also show that the apollonian metric is bounded above by twice the quasihyperbolic metric in domains in Iwasawa groups.

The fractal dimension of the Apollonian sphere packing has been computed numerically up to six trusty decimal digits. Based on the 31 944 875 541 924 spheres of radius greater than 2-19 contained in the Apollonian packing of the unit sphere, we obtained an estimate of 2.4739465, where the last digit is questionable. Two fundamentally different algorithms have been employed. Outlines of both alg...

Two computer models were developed to realize the algorithms of random Apollonian and Apollonian packings of the spherical particles in a unit cube. A genetic algorithm was developed for the latter packing model. The upper and lower bounds of a fractal dimension were computed using various numerical methods. It was observed that for both models the fractal properties were not essentially affect...

Apollonian gaskets are formed by repeatedly filling the gaps between three mutually tangent circles with further tangent circles. In this paper we give explicit formulas for the the limiting pair correlation and the limiting nearest neighbor spacing of centers of circles from a fixed Apollonian gasket. These are corollaries of the convergence of moments that we prove. The input from ergodic the...

An Apollonian configuration of circles is a collection of circles in the plane with disjoint interiors such that the complement of the interiors of the circles consists of curvilinear triangles. One well studied method of forming an Apollonian configuration is to start with three mutually tangent circles and fill a curvilinear triangle with a new circle, then repeat with each newly created curv...

The Apollonian networks display the remarkable power-law and small-world properties as observed in most realistic networked systems. Their dual graphs are extended Tower of Hanoi graphs, which are obtained from the Tower of Hanoi graphs by adding a special vertex linked to all its three extreme vertices. In this paper, we study analytically maximum matchings and minimum dominating sets in Apoll...

In the last decade tremendous effort has been put in the study of the Apollonian circle packings. Given the great variety of mathematics it exhibits, this topic has attracted experts from different fields: number theory, homogeneous dynamics, expander graphs, group theory, to name a few. The principle investigator (PI) contributed to this program in his PhD studies. The scenery along the way fo...

Apollonian circle packings arise by repeatedly filling the interstices between mutually tangent circles with further tangent circles. It is possible for every circle in such a packing to have integer radius of curvature, and we call such a packing an integral Apollonian circle packing. This paper studies number-theoretic properties of the set of integer curvatures appearing in such packings. Ea...