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.