Density of binary disc packings: lower and upper bounds

TL;DR

This paper establishes new bounds on the maximum density of mixed-radius disc packings in the plane, improving previous upper bounds for certain radius ratios and narrowing down the possible densities.

Contribution

It provides the first comprehensive bounds for all radius ratios between 0 and 1, with improved upper bounds for many cases, advancing understanding of binary disc packing densities.

Findings

Improved upper bounds for radius ratios in [0.11,0.74]

New intervals where hexagonal packing is optimal

Enhanced understanding of maximum packing densities

Abstract

We provide, for any $r\in (0,1)$, lower and upper bounds on the maximal density of a packing in the Euclidean plane of discs of radius $1$ and $r$. The lower bounds are mostly folk, but the upper bounds improve the best previously known ones for any $r\in[0.11,0.74]$. For many values of $r$, this gives a fairly good idea of the exact maximum density. In particular, we get new intervals for $r$ which does not allow any packing more dense that the hexagonal packing of equal discs.