fraction φ ∞ ∗ of nonoverlapping spheres in$_{R}$ $^{d}$. The maximal local packing fraction ˆ φ$_{∗}$ ( N$_{∗}$ ) is therefore an upper bound on the maximal infinite-volume packing fraction φ ∞ ∗ , or Figure 5 plots for d = 2 and for selected values of N$_{∗}$ ∈$_{N}$ up to N$_{∗}$ = 348 the putative upper bound defined by ˆ φ$_{∗}$ ( N$_{∗}$ ) alongside the proved maximal packing fraction, φ ∞ ∗ = π/ √ 12. While the maximal infinite-volume packing fractions for d -dimensional identical nonover- lapping spheres are known with analytical rigor for d = 1 − 3, d = 8 and d = 24 [26], this method could be used to improve upon the upper bounds on maximal packing fraction φ ∞ ∗ in dimensions where a value for φ ∞ ∗ has not not yet been proved. It is important to reiterate though that to be rigorous, the upper bound as defined requires a DLP radius R$_{min}$ ( N ) that is proved optimal. The packing of nonoverlapping disks that uniquely achieves the highest infinite-volume packing fraction φ ∞ ∗ = π/ √ 12 for d = 2 is well-known to be the packing such that each disk is in contact with exactly six others, with centers arranged on the sites of the triangular lattice. As DLP optimal packings are packing fraction-maximizing arrangements of N disks centered on a (additional) disk, one might expect that all or a major subset of disks in any given DLP optimal packing will always sit on the sites of the triangular lattice. Over the range of N studied, however, this is in fact very infrequently the case. We find that at only three values of N greater than 6, at N = 12, 30 and 54, are triangular lattice configurations also DLP optimal configurations, while most DLP optimal packings are significantly more locally dense than a packing of N disks around a central disk with centers on the sites of the triangular lattice, as is illustrated in Fig. 2 for N = 1 − 109. In general, we find wide variation in the symmetries and other characteristics of DLP optimal packings. For the majority of cases, there appear to be an uncountably infinite number (a continuum) of optimal configurations of N sphere centers at optimal R$_{min}$ ( N ), with the continuum attributable to the presence of rattlers. A rattler in a packing of spheres in$_{R}$ d is a sphere that is positioned such that it may be individually moved in at least one 13 IV. OPTIMAL PACKINGS IN TWO DIMENSIONS ˆ φ$_{∗}$ ( N$_{∗}$ ) ≥ φ ∞ ∗ , N$_{∗}$ ∈$_{N}$ . (9)