Generalized Apollonian packings

Abstract

In this paper we generalize the classical two-dimensional Apollonian packing of circles to the case where the circles are no more tangent. We introduce two elements ofSL(2,ℂ) as generators:R andT that are hyperbolic rotations of 2π/3 and 2π/N (N=2,3,4....), around two distinct points. The limit set of the discrete group generated byR andT provides, forN=7,8,9,.... a generalization of the Apollonian packing (which is itself recovered forN=∞). The valuesN=2,3,4,5 produce a very different result, giving rise to the rotation groups of the cube forN=2 and 4, and the icosahedron forN=3 and 5. ForN=6 the group is no longer discrete. To further analyze this structure forN≥7, we move to the Minkowski space in which the group acts on a one sheeted hyperboloid. The circles are now represented by points on this variety and generate a crystal on it.