Density of Binary Disc Packings: The Nine Compact Packings

Abstract

A disc packing in the plane is compact if its contact graph is a triangulation. There are nine values of r such that a compact packing by discs of radii 1 and r exists. For each of these nine values, we prove that the maximal density over all the packings by discs of radii 1 and r is reached for a compact packing. We describe such a packing and give its density.