Apollonian Circle Packings: Geometry and Groups

Catherine Yan of Texas A&M University

Apollonian circle packings arise by repeatedly filling the interslices between four mutually tangent circles with further tangent circles. The curvature of any four mutually tangent circles are described by the Descartes Equation. We generalize the Descartes Equation to the curvature-center coordinates, and describe various geometric and group theoretic properties of such packings.