Density of Binary Disc Packings: The 9 Compact Packings

TL;DR

This paper identifies the nine specific radii ratios for which compact binary disc packings exist in the plane and proves that the densest packings for each ratio are achieved by these compact configurations, providing explicit examples and densities.

Contribution

The paper proves that for each of the nine radii ratios allowing compact packings, the maximum density is attained by a specific compact packing, and it explicitly describes these packings and their densities.

Findings

Nine specific radii ratios allow compact packings.

Maximum density packings are achieved by these compact configurations.

Explicit examples and densities are provided for each case.

Abstract

A disc packing in the plane is compact if its contact graph is a triangulation. There are $9$ values of $r$ such that a compact packing by discs of radii $1$ and $r$ exists. We prove, for each of these $9$ values, that the maximal density over all the packings by discs of radii $1$ and $r$ is reached for a compact packing (we give it as well as its density).