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

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 WQDW = QW where QD is the matrix of the Descartes quadratic form QD = x 2 1 + x 2 2 + x 2 3 + x 2 4 − 12(x1 + x2 + x3 + x4) and QW of the quadratic form QW = −8x1x2 +2x3 +2x4. On the parameter space MD the group Aut(QD) 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