Density of triangulated ternary disc packings

TL;DR

This paper investigates the densest arrangements of three different-sized discs in the plane, identifying optimal packings for various radius combinations and classifying cases based on triangulation and density.

Contribution

It develops enhanced methods to determine maximal-density ternary disc packings and classifies cases where triangulated packings are optimal or not.

Findings

16 pairs where triangulated packings maximize density

15 pairs where two-size packings are optimal

40 pairs with non-triangulated denser packings

Abstract

We consider ternary disc packings of the plane, i.e. the packings using discs of three different radii. Packings in which each ''hole'' is bounded by three pairwise tangent discs are called triangulated. There are 164 pairs $(r,s)$, $1{>}r{>}s$, allowing triangulated packings by discs of radii 1, $r$ and $s$. In this paper, we enhance existing methods of dealing with maximal-density packings in order to find ternary triangulated packings which maximize the density among all the packings with the same disc radii. We showed for 16 pairs that the density is maximized by a triangulated ternary packing; for 15 other pairs, we proved the density to be maximized by a triangulated packing using only two sizes of discs; for 40 pairs, we found non-triangulated packings strictly denser than any triangulated one; finally, we classified the remaining cases where our methods are not applicable.