What Type of Apollonian Circle Packing Will Appear?

Abstract

An Apollonian circle packing is created by starting with three pairwise tangent circles, adding the two circles—or circle and line—tangent to the first three, then repeating the process forever by successively adding new circles and lines tangent to every new tangent threesome of circles and/or lines. The resulting packing is one of four types: bounded (enclosed by one of the circles), half-plane (with one line), strip (with two lines), or full-plane. Given three starting circles, what type of Apollonian circle packing will appear? This article gives an answer in the form of a picture, i.e., a plot in the parameter space of relative sizes of the starting circles, indicating the types of packings. The result is a fractal, which is then further explored.