Density of Binary Disc Packings: Lower and Upper Bounds

Abstract

We provide, for any r ∈ ( 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 ∈ [ 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.