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Irreducible Apollonian Configurations and Packings

Irreducible Apollonian Configurations and Packings,10.1007/s00454-009-9216-9,Discrete & Computational Geometry,Steve Butler,Ronald L. Graham,Gerhard G

Irreducible Apollonian Configurations and Packings  
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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.
Journal: Discrete & Computational Geometry - DCG , vol. 44, no. 3, pp. 487-507, 2010
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