Apollonian Circle Packings: Geometry and Group Theory I. The Apollonian Group

TL;DR

This paper explores the geometry and group theory underlying Apollonian circle packings, focusing on the algebraic and group actions that explain their integrality properties and geometric structures.

Contribution

It introduces the Apollonian group and related groups, demonstrating their hyperbolic Coxeter group structure and explaining integrality properties of certain circle packings.

Findings

The Apollonian group is a finitely generated discrete group of integer matrices.

Describes the space of Descartes configurations using specific quadratic forms.

Identifies the Apollonian, dual, and super-Apollonian groups as hyperbolic Coxeter groups.

Abstract

Apollonian circle packings arise by repeatedly filling the interstices between four mutually tangent circles with further tangent circles. We observe that there exist Apollonian packings which have strong integrality properties, in which all circles in the packing have integer curvatures and rational centers such that (curvature)$\times$(center) is an integer vector. This series of papers explain such properties. A {\em Descartes configuration} is a set of four mutually tangent circles with disjoint interiors. We describe the space of all Descartes configurations using a coordinate system $\sM_\DD$ consisting of those $4 \times 4$ real matrices $\bW$ with $\bW^T \bQ_{D} \bW = \bQ_{W}$ where $\bQ_D$ is the matrix of the Descartes quadratic form $Q_D= x_1^2 + x_2^2+ x_3^2 + x_4^2 -{1/2}(x_1 +x_2 +x_3 + x_4)^2$ and $\bQ_W$ of the quadratic form $Q_W = -8x_1x_2 + 2x_3^2 + 2x_4^2$. There are natural group actions on the parameter space $\sM_\DD$. We observe that the Descartes configurations in each Apollonian packing form an orbit under a certain finitely generated discrete group, the {\em Apollonian group}. This group consists of $4 \times 4$ integer matrices, and its integrality properties lead to the integrality properties observed in some Apollonian circle packings. We introduce two more related finitely generated groups, the dual Apollonian group and the super-Apollonian group, which have nice geometrically interpretations. We show these groups are hyperbolic Coxeter groups.