Apollonian Circle Packings

Abstract

Circle packings are a particularly elegant and simple way to construct quite complicated and elaborate sets in the plane. One systematically constructs a countable family of tangent circles whose radii tend to zero. Although there are many problems in understanding all of the individual values of their radii, there is a particularly simple asymptotic formula for the radii of the circles, originally due to Kontorovich and Oh. In this partly expository note we will discuss the history of this problem, explain the asymptotic result and present an alternative approach.