(a) (b) (c) FIG. 16: Three packings for which R$_{min}$ ( N ) is significantly less than the radius of a disk enclosing the centers of N +1 disks arranged on the sites of the triangular lattice. (a) N = 168, R$_{min}$ (168) = 6 . 680013, point group C$_{1}$ . (b) N = 264, R$_{min}$ (264) = 8 . 417769, point group C$_{1}$ . (c) N = 270, R$_{min}$ (270) = 8 . 497744, point group C$_{1}$ . The function Z$_{max}$ ( R ) can also be employed in the two ways discussed to bound from above the packing fraction of an infinite sphere packing in dimension d . Similarly, Z$_{max}$ ( R ) for differentiated non-spherical objects can be employed to form upper bounds on corre- 28