### 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.