Involve: A Journal of Mathematics

  • Involve
  • Volume 10, Number 2 (2017), 345-360.

A new look at Apollonian circle packings

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

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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.

Article information

Source
Involve, Volume 10, Number 2 (2017), 345-360.

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

Permanent link to this document
https://projecteuclid.org/euclid.involve/1513135635

Digital Object Identifier
doi:10.2140/involve.2017.10.345

Mathematical Reviews number (MathSciNet)
MR3574305

Zentralblatt MATH identifier
1353.52020

Subjects
Primary: 52C26: Circle packings and discrete conformal geometry
Secondary: 11D09: Quadratic and bilinear equations

Keywords
Apollonian circle packing Apollonian gasket Apollonian supergasket

Citation

Corona, Isabel; Johnson, Carolynn; Mitchell, Lon; O’Connell, Dylan. A new look at Apollonian circle packings. Involve 10 (2017), no. 2, 345--360. doi:10.2140/involve.2017.10.345. https://projecteuclid.org/euclid.involve/1513135635


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