direction without resulting overlap of any other sphere within the packing or the packing boundary (in this case, the encompassing sphere of radius R$_{min}$ ( N ) + 0 . 5), i.e., a rattler is a sphere that is not locally jammed [27, 28]. The rattlers present in the following figures are indicated by a lighter shading, while the fixed central spheres (disks) are indicated by an open circle. In two dimensions for N ≤ 10, we proved [11] that optimal packings are those in which R$_{min}$ ( N ) is equal to the radius of the minimal-radius circle onto which the centers of N disks may be packed. These R$_{min}$ ( N ) may be analytically calculated via simple trigonometry, yielding R$_{min}$ ( N ) = 1 for N ≤ 6, and For 6 ≤ N ≤ 9, DLP optimal packings with R$_{min}$ ( N ) as defined in Eq. (10) are unique up to rotations and correspond to configurations where all sphere centers lie on the circle of radius R$_{min}$ ( N ) at distance unity from two adjacent centers. In general, for many N there is a unique (up to rotations) optimal packing, and in a smaller number of cases such as for N = 10, 60, 90 and 126, there are a finite number of degenerate optimal packings. Figure 6 depicts three of a finite number of optimal packings for N = 10, d = 2, where R$_{min}$ (10) = τ ; these packings are formed by radially translating up to five disks with centers on the circle at radius R$_{min}$ (10) = τ , the golden ratio, to be in contact with the fixed central disk at distance unity. For N > 10, we know of no packings that have been proved optimal. However, for certain N , previous studies have found packings that we conjecture to be DLP optimal packings. For N + 1 equal to hexagonal number 3 k ( k + 1) + 1, k ≥ 1 an integer, Lubachevsky and Graham [19] found a class of packings called curved hexagonal packings that they conjectured to be the densest packings of N + 1 identical nonoverlapping disks within an encompassing disk 14 B. Curved hexagonal packings A. Packings that are proved optimal R$_{min}$ ( N ) = 1 $^{√}$2(1 − cos 2 π/N ) , 6 ≤ N ≤ 10 . (10)