Discrete & Computational Geometry

, Volume 44, Issue 3, pp 487–507

Irreducible Apollonian Configurations and Packings

  • Steve Butler
  • Ron Graham
  • Gerhard Guettler
  • Colin Mallows
Open AccessArticle

DOI: 10.1007/s00454-009-9216-9

Cite this article as:
Butler, S., Graham, R., Guettler, G. et al. Discrete Comput Geom (2010) 44: 487. doi:10.1007/s00454-009-9216-9


An Apollonian configuration of circles is a collection of circles in the plane with disjoint interiors such that the complement of the interiors of the circles consists of curvilinear triangles. One well-studied method of forming an Apollonian configuration is to start with three mutually tangent circles and fill a curvilinear triangle with a new circle, then repeat with each newly created curvilinear triangle. More generally, we can start with three mutually tangent circles and a rule (or rules) for how to fill a curvilinear triangle with circles.

In this paper we consider the basic building blocks of these rules, irreducible Apollonian configurations. Our main result is to show how to find a small field that can realize such a configuration and also give a method to relate the bends of the new circles to the bends of the circles forming the curvilinear triangle.


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© The Author(s) 2009

Authors and Affiliations

  • Steve Butler
    • 1
  • Ron Graham
    • 2
  • Gerhard Guettler
    • 3
  • Colin Mallows
    • 4
  1. 1.UCLALos AngelesUSA
  2. 2.UCSDSan DiegoUSA
  3. 3.University of Applied Sciences Giessen FriedbergGiessenGermany
  4. 4.Avaya LabsBasking RidgeUSA