Apollonian Circle Packings : Geometry and Group Theory

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)×(center) is an integer vector. This series of papers explain such properties. A Descartes configuration is a set of four mutually tangent circles with disjoint interiors. An Apollonian circle packing can be described in terms of the Descartes configuration it contains. We describe the space of all ordered, oriented Descartes configurations using a coordinate system MD consisting of those 4 × 4 real matrices W with WT QDW = QW where QD is the matrix of the Descartes quadratic form Q D = x2 1 + x2 2 + x2 3 + x2 4 − 12 (x1 + x2 + x3 + x4) and QW of the quadratic form QW = −8x1x2 +2x2 3 +2x2 4 . On the parameter ∗ Ronald L. Graham was partially supported by NSF Grant CCR-0310991. Catherine H. Yan was partially supported by NSF Grants DMS-0070574, DMS-0245526 and a Sloan Fellowship. She is also affiliated with Nankai University, China. 548 R. L. Graham, J. C. Lagarias, C. L. Mallows, A. R. Wilks, and C. H. Yan space MD the group Aut(Q D) acts on the left, and Aut(QW ) acts on the right, giving two different “geometric” actions. Both these groups are isomorphic to the Lorentz group O(3, 1). The right action of Aut(QW ) (essentially) corresponds to Möbius transformations acting on the underlying Euclidean space R2 while the left action of Aut(Q D) is defined only on the parameter space. We observe that the Descartes configurations in each Apollonian packing form an orbit of a single Descartes configuration under a certain finitely generated discrete subgroup of Aut(Q D), which we call the Apollonian group. This group consists of 4×4 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 in Aut(Q D), the dual Apollonian group produced from the Apollonian group by a “duality” conjugation, and the superApollonian group which is the group generated by the Apollonian and dual Apollonian groups together. These groups also consist of integer 4× 4 matrices. We show these groups are hyperbolic Coxeter groups.