Circle Packings and Conformal Maps

By Sanjay Ramassamy

April 16, 2014

Abstract

This talk will give an overview of circle packings and how they related to complex analysis. A circle packing is a pattern of circles that may be tangent to one another, but whose interiors never overlap. The tangency graph of a circle packing is obtained by putting a vertex at the center of each circle, and connecting two vertices by an edge if the corresponding circles are tangent. The Koebe-Andreev-Thurston theorem states that any triangulation of the sphere is the tangency graph of some circle packing, and that packing is unique up to Möbius transformations. In the 1980s, Thurston conjectured (and Rodin-Sullivan proved) that circle packings could be used to approximate the conformal maps given by the Riemann mapping theorem between some simply connected domain (different from the whole plane) and the unit disc.