time constraints, we were unfortunately not able to quantitatively verify this result at much larger N ; however, should the observed trend continue, the width of the surface zone would continue to grow as the bulk grows, eventually becoming infinitely large as N → ∞ . This does not imply that at very large N the surface zone would represent a substantial fraction of the total packing; as N → ∞ , the ratio of space covered by the surface zone to the space covered by the bulk zone will still be zero. For certain optimal packings exhibiting this surface effect, we also find that the optimal radius R$_{min}$ ( N ) is significantly smaller than the radius of the smallest disk centered on a fixed central disk that can enclose (an additional) N disks with centers arranged on the sites of the triangular lattice. Figure 16 depicts three optimal packings for which this difference is relatively large, at N = 168, 264 and 270, where the R$_{min}$ ( N ) differ by 0 . 248190, 0 . 128616 and 0 . 162510 from the respective radii for the triangular lattice configuration, 7, √ 73 and √ 75. Each of the packings in Fig. 16 also displays its own interesting features, including a close resemblance on the left of the image of the N = 168 packing to the (3 , 4 , 1) wedge hexagonal packing, and imperfect three-fold rotational symmetry for the N = 264 and N = 270 packings. The DLP problem is a local packing problem that in certain limits encompasses both the infinite sphere packing and kissing number problems. DLP optimal packings exhibit a wide variety of symmetries, vary significantly from packings with disk centers placed on the sites of a triangular lattice, and often vary significantly with consecutive N . Two local packing classes, the curved hexagonal and wedge hexagonal packings, lead to the densest or very dense packings of disks over the range of N studied. The optimal radii R$_{min}$ ( N ) corresponding to a packing of N spheres in$_{R}$ d form a realiz- ability condition on functions that are candidates to be the pair correlation function g$_{2}$ ( r ) of a statistically homogeneous and isotropic packing of spheres. This realizability condition incorporates more information than is included in the structure factor and pair correlation function nonnegativity conditions alone. Though the condition discussed here only applies to packings of identical spheres, equivalent Z$_{max}$ ( R ) functions for packings of differentiated non-spherical objects can be found and corresponding realizability conditions imposed. 27 V. CONCLUSIONS AND DISCUSSION