A game of packings

Abstract

In this note, we investigate an infinite one parameter family of circle packings, each with a set of three mutually tangent circles. We use these to generate an infinite set of circle packings with the Apollonian property. That is, every circle in the packing is a member of a cluster of four mutually tangent circles.

A geometric study of circle packings and ideal class groups

“…As any prime ideal containing ∆ equals its conjugate, we see that D also divides (i Baragar and Lautzenheiser recently discovered a one-parameter family of circle packings that generalized the classical Apollonian strip packing [1]. They begin with four circles, say C 1 , C 2 , C 3 , and C 4 , oriented so as to have disjoint interiors.…”

“…Two are circles of radius 1/2 with centers distanced √ D apart for some D ∈ N, each circle tangent to both lines. It is then proved in [1] that circles from the lattice generated by v C1 , v C2 , v C3 , and v C4 are dense in C and only intersect tangentially. Among positively oriented circles from this lattice, they keep those which lie between the two lines and are not contained in the interior of another positively oriented circle.…”

“…Most notably, S 1 always contains (and equals in Figure 1 as 19 is prime) Stange's Schmidt arrangement [21], which motivated this paper. Also, S 1 for Q(i) is in [10]; S 2 for Q(i √ 2) is in [11]; S D for Q(i √ D) contains circle packings in [3], [15], and [13] (D = 2), [20] (D = 3), [13] (D = 6), and [1] (all D ∈ N); S 24 for Q(i √ 30) contains a circle packing in [12]; and S D for D = (∆ 2 + 14∆ + 1)/16 when 4 ∆ and D = (∆ 2 + 12∆)/16 when 4 | ∆ are Stange's ghost circles [21].…”

“…Vectors of the form v C generate a four-dimensional subspace of C 2 × R 2 , which we identify with Minkowski space via the Hermitian form of signature (1,3),…”

“…where n is even if ∆ is. By Proposition 2.2, C and C intersect if and onlyif v C , v C Q ∈ [−1,1], in which case the angle of intersection is arccos(n/2D).For the second claim, suppose C and C intersect (but do not coincide). Since PSL2 (O) preserves S D , intersection angles, and K, we may assume without loss of generality that C and C have nonzero curvature r and r -if not replace them by M C and M C for almost any choice of M ∈ PSL 2 (O).…”