A new look at Apollonian circle packings

Abstract

We define an abstract Apollonian supergasket using the solution set of a certain Diophantine equation, showing that the solutions are in bijective correspondence with the circles of any concrete supergasket. Properties of the solution set translate directly to geometric and algebraic properties of Apollonian gaskets, facilitating their study. In particular, curvatures of individual circles are explored and geometric relationships among multiple circles are given simple algebraic expressions. All results can be applied to a concrete gasket using the curvaturecenter coordinates of its four defining circles. These techniques can also be applied to other types of circle packings and higher-dimensional analogs. An Apollonian gasket is a type of circle packing in the plane generated recursively starting from a set of four mutually tangent circles. The curvatures of any four such circles are related by an equation discovered by Descartes, and every circle in a gasket generated by four circles with integer curvatures will have integer curvature. While these gaskets have been fascinating to mathematicians for some time — the use of group theory in their study was initiated by Keith Hirst [1967] and they even inspired a poem 1 — it was only relatively recently that Jeffrey Lagarias, Colin Mallows, and Allan Wilks [Lagarias et al. 2002] gave an algebraic characterization of Descartes configurations. One question in particular has inspired much work but resisted a complete answer: given the four original integer curvatures, which other curvatures can or will occur, and how frequently? Peter Sarnak [2011], Elena Fuchs [2013], and Hee Oh [2014] have recent surveys on this topic, which has seen significant progress in the past five years [Bourgain 2012; Bourgain and Kontorovich 2014; Bourgain and Fuchs 2011; Fuchs and Sanden 2011]. In this paper, inspired by recent work of Sam Northshield [2015], we provide a four-dimensional label to each circle that does not depend on the location of the circle but refers instead to its geometric relationship to the original four circles. Since we consider only the process of generating the gasket, the labels provide an