Circle Packing Theorem - Proofs of The Theorem

Proofs of The Theorem

There are many known proofs of the circle packing theorem. Paul Koebe's original proof is based on his conformal uniformization theorem saying that a finitely connected planar domain is conformally equivalent to a circle domain. There are several different topological proofs that are known. Thurston's proof is based on Brouwer's fixed point theorem. There is also a proof using a discrete variant of Perron's method of constructing solutions to the Dirichlet problem. Yves Colin de Verdière proved the existence of the circle packing as a minimizer of a convex function on a certain configuration space.