Perfect five-fold symmetry, disallowed to regular infinite crystals, is exhibited by three of the optimal packings studied. Five-fold rotational symmetry is present in the N = 15 (Fig. 1), N = 10 (bottom panel of Fig. 6) and N = 25 (top panel of Fig. 10) packings. The N = 25 optimal packing also has perfect five-fold bond orientational order, evident in that all nearest-neighbor disk pairs are at one of five angles relative to any fixed coordinate system. Additionally, it is of note that the N = 25 packing may be tiled by 15 identical rhombuses of acute angle 72 ◦ with vertices placed at disk centers, where the rhombus of acute angle 72 ◦ is known to be the ‘thicker’ of the two types of rhombus present in a Penrose tiling [29]. The top panel of Fig. 14, a diagram of the contact network for the N = 25 optimal packing, depicts these rhombuses. Two other optimal packings incorporating interesting symmetry elements are the N = 11 and N = 32 packings (center and bottom panels of Fig. 10, respectively). The packing depicted for N = 11 belongs to symmetry group C$_{2}$$_{v}$ and appears to be the unique DLP optimal packing of 11 disks. The packing for N = 32 is one of an infinity of possible packings due to the presence of two rattlers; however, the backbone of the packing has reflection symmetry across two axes and inversion symmetry through the origin and belongs to symmetry group D$_{2}$$_{h}$ . Though five-fold symmetry may be limited to packings with small N , high symmetry in general is not. For example, the backbone of the optimal packing with the largest N presented here, N = 348, has six-fold rotation symmetry and belongs to point group D$_{6}$$_{h}$ . Figure 11 depicts the N = 348 optimal packing, which contains 24 rattlers. FIG. 11: DLP optimal packing for N = 348, R$_{min}$ (348) = 9 . 620709 . . . , belonging to point group D$_{6}$$_{h}$ . 21