sponding infinite packing maximal packing fractions. 

Our work has direct application to packaging, particularly to problems involving identical nonoverlapping disks within a circular boundary. Further, as is discussed in Appendix A, the algorithm we have employed to find the d = 2 putative DLP optimal packings presented in this work may be modified to study dense packings of d -dimensional differentiated objects of various shape within different boundaries. Additionally, our work has implications for nucleation theory and surface physics, particularly in terms of the effects of imposing a circular boundary upon a packing of spheres with centers initially placed on the sites of a triangular lattice. In future work, we expect to investigate these implications and others in more depth. 

In a sequel to this paper, we will present and analyze DLP optimal packings and their corresponding R$_{min}$ ( N ) for three-dimensional spheres over a larger range of N . We will catalogue optimal packings of particularly high packing fraction and of unusual symmetry, and we will investigate the possibility of d = 3 extensions to special d = 2 classes of packings such as the curved hexagonal and wedge hexagonal packings. We will compare R$_{min}$ ( N ) values to shell distances in Barlow packings [20], where the Barlow packings, of which the best-known are face centered cubic (FCC) and hexagonal close packed (HCP) arrangements, all individually achieve the maximal infinite-volume packing fraction for d = 3, φ ∞ ∗ = π/ √ 18 = 0 . 740481 ... . 

Packing in$_{R}$ 3 is intrinsically more complicated than in$_{R}$ $^{2}$. In$_{R}$$^{3}$, there is a wider range of possibilities for contact coordination, particularly for twelve contacting spheres around a central sphere (with K$_{3}$ = 12) where there is an infinity of possible configurations, as compared to just one in$_{R}$ $^{2}$. Additionally, there is a single optimal infinite-volume packing configuration in R $^{2}$, i.e., the triangular lattice, whereas there is an infinite number in R $^{3}$, i.e., the Barlow packings. 

Preliminary findings indicate that there is less symmetry (as quantified by point groups) present for the majority of DLP optimal packings in$_{R}$ 3 and more variation with N . Further, observations suggest that for the same value of N that there are significantly more locally jammed packing configurations in$_{R}$ 3 with radius R > R$_{min}$ ( N ) than is the case in$_{R}$ $^{2}$. This finding has implications for the dynamics of nucleation occuring in pure supersaturated liquids. If a nucleus in a supersaturated liquid of identical nonoverlapping spheres in$_{R}$ 3 is approximated as a group of N densely-packed spheres with centers within distance R of a 

