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2017 A new look at Apollonian circle packings
Isabel Corona, Carolynn Johnson, Lon Mitchell, Dylan O’Connell
Involve 10(2): 345-360 (2017). DOI: 10.2140/involve.2017.10.345


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.


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Isabel Corona. Carolynn Johnson. Lon Mitchell. Dylan O’Connell. "A new look at Apollonian circle packings." Involve 10 (2) 345 - 360, 2017.


Received: 16 February 2016; Revised: 16 March 2016; Accepted: 19 March 2016; Published: 2017
First available in Project Euclid: 13 December 2017

zbMATH: 1353.52020
MathSciNet: MR3574305
Digital Object Identifier: 10.2140/involve.2017.10.345

Primary: 52C26
Secondary: 11D09

Keywords: Apollonian circle packing , Apollonian gasket , Apollonian supergasket

Rights: Copyright © 2017 Mathematical Sciences Publishers


Vol.10 • No. 2 • 2017
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