Vol. 10, No. 2, 2017

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A new look at Apollonian circle packings

Isabel Corona, Carolynn Johnson, Lon Mitchell and Dylan O’Connell

Vol. 10 (2017), No. 2, 345–360

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.

Apollonian circle packing, Apollonian gasket, Apollonian supergasket
Mathematical Subject Classification 2010
Primary: 52C26
Secondary: 11D09
Received: 16 February 2016
Revised: 16 March 2016
Accepted: 19 March 2016
Published: 10 November 2016

Communicated by Scott T. Chapman
Isabel Corona
Department of Mathematics
University of Colorado Boulder
Boulder, CO 80309
United States
Carolynn Johnson
Department of Mathematics
Middlebury College
14 Old Chapel Road
Middlebury, VT 05753
United States
Lon Mitchell
Mathematical Reviews
American Mathematical Society
416 4th Street
Ann Arbor, MI 48103
United States
Dylan O’Connell
Mathematics Department
Haverford College
370 Lancaster Avenue
Haverford, PA 19041
United States