(a) (b) (c) FIG. 6: Three of a finite number of optimal cases for N = 10, R$_{min}$ (10) = τ = 1 . 618204 . . . formed by radially translating disks whose centers lie on a circle of radius R$_{min}$ (10) = τ to be in contact with the fixed central disk. (a) point group C$_{2}$$_{v}$ . (b) point group C$_{2}$$_{v}$ . (c) point group D$_{5}$$_{h}$ . for k = 1 . . . 5. The characteristics of this class include that each packing has a disk fixed at the center of the encompassing disk, and that all curved hexagonal packings belong to point group C$_{6}$$_{h}$ , meaning that they are invariant under 60 ◦ rotation or inversion through the origin. Further, for k ≥ 4, for each k there are a finite number of degenerate packings of equal density (for 1 ≤ k ≤ 3, there is a unique densest packing). The degenerate packings 15