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 curvature-center coordinates of its four defining circles. These techniques can also be applied to other types of circle packings and higher-dimensional analogs.