Irreducible Apollonian Configurations and Packings

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.