Rigorous results

Thue's proof that the densest packing with a single size disc is to put the centers on a triangular lattice is a century old [Thu1892,Thu10]. L. Fejes Toth observed that for r slightly less than 1, one cannot do any better than this single size packing [Toth53]. Florian showed this is the case for r greater than 0.906 [Flo63]. G. Blind extended this result to r greater than 0.742 [Bli66,Bli69].

For six values of r for which there are compact packings, Heppes has proved that a compact packing is optimal [Hep00,Hep03]. The values of r are 0.637...,0.533..., 0.414..., 0.349..., 0.280...,0.154.... In [Hep03], Heppes claims that the method introduced can be used to prove that the density has strict local maxima at these values of r and can also be used to prove that at r=0.6 and r=0.67 one cannot do any better than the single size packing density. Kennedy has shown that there are only nine values of r<1 such that compact packings exist using discs of radius 1 and r [Ken04].

Any packing gives a lower bound on the packing density, so the blue curve in the plot of the density is a lower bound. Florian showed that the density is no greater than the packing density in a triangle formed by the centers of two discs of radius r and one of radius 1 which are tangent to each other [Flo60]. So the green curve is an upper bound on the density.