Apollonian Circle Packings: Geometry and Group Theory I. The Apollonian Group
R.L. Graham, J.C. Lagarias, C.L. Mallows, A.R. Wilks, C.H. Yan

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)(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 consisting of those real matrices with where is the matrix of the Descartes quadratic form and of the quadratic form . There are…
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Taxonomy
TopicsQuasicrystal Structures and Properties · Mathematical Dynamics and Fractals · Advanced Materials and Mechanics
