differ for each wedge hexagonal packing, but from our observations produce packings where the backbone maintains inversion symmetry through the origin, where the backbone of a packing is defined as the packing excluding the rattlers. Such deviations can be seen in the branches and wedges in the DLP optimal packings for N = 198 and N = 312 in Fig. 9, which correspond to the (slightly altered) wedge hexagonal packings with ( k, p, a ) = (5 , 3 , 3) and (7 , 3 , 3), respectively. Many DLP optimal packings exhibit symmetries other than inversion symmetry through the origin as exhibited by the altered wedge hexagonal packings shown in Figs. 8 and 9. These symmetries include perfect bond orientational order, invariance under rotation D. DLP optimal packings with high symmetry 19 FIG. 9: DLP optimal packings that are minor deviations from the “wedge hexagonal” packings. (a) N = 198, d = 2. R$_{min}$ (198) = 7 . 201130 . . . , ( k, p, a ) = (5 , 3 , 3), point group C$_{i}$ . (b) N = 312, R$_{min}$ (312) = 9 . 141107 . . . , ( k, p, a ) = (7 , 3 , 3), point group C$_{i}$ . (a) (b)