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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
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Abstract |
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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. |
Keywords
Apollonian circle packing, Apollonian gasket, Apollonian
supergasket
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Mathematical Subject Classification 2010
Primary: 52C26
Secondary: 11D09
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Milestones
Received: 16 February 2016
Revised: 16 March 2016
Accepted: 19 March 2016
Published: 10 November 2016
Communicated by Scott T. Chapman
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Authors |
| Isabel Corona | |
| Department of Mathematics University of Colorado Boulder Boulder, CO 80309 %-0395 United States |
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| Carolynn Johnson | |
| Department of Mathematics Middlebury College 14 Old Chapel Road Middlebury, VT 05753 United States |
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| Lon Mitchell | |
| Mathematical Reviews American Mathematical Society 416 4th Street Ann Arbor, MI 48103 United States |
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| Dylan O’Connell | |
| Mathematics Department Haverford College 370 Lancaster Avenue Haverford, PA 19041 United States |
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