Packing Disks by Flipping and Flowing

Abstract

We provide a new type of proof for the Koebe–Andreev–Thurston (KAT) planar circle packing theorem based on combinatorial edge-flips. In particular, we show that starting from a disk packing with a maximal planar contact graph G, one can remove any flippable edge $$e^-$$ of this graph and then continuously flow the disks in the plane, so that at the end of the flow, one obtains a new disk packing whose contact graph is the graph resulting from flipping the edge $$e^-$$ in G. This flow is parameterized by a single inversive distance.