continued from previous page Number of disks Figure R$_{min}$ ( N ) Tri. latt. shell Point group 162 8 (bottom) 6.539939 6.557439 C$_{i}$ 168 16 (top) 6.680013 6.928203 C$_{1}$ 198 9 (top) 7.201130 7.211103 C$_{i}$ 264 16 (center) 8.417769 8.544004 C$_{1}$ 270 16 (bottom) 8.497744 8.660254 C$_{1}$ 312 9 (bottom) 9.141107 9.165151 C$_{i}$ 348 11 9.620709 9.643651 D$_{6}$$_{h}$ [1] P. M. Chaikin and T. C. Lubensky, Principles of Condensed Matter Physics (Cambridge, 1995). [2] J. P. Hansen and I. R. McDonald, Theory of Simple Liquids, 3rd ed. (Academic, 2006). [3] R. Zallen, The Physics of Amorphous Solids (John Wiley and Sons, 1983). [4] C. H. Davis, H. Nie, and N. V. Dokholyan, Phys. Rev. E 75 , 051921 (2007). [5] N. V. Dokholyan, Curr. Opin. Struct. Bio. 16 , 79 (2006). [6] F. Ding and N. V. Dokholyan, Trends in Biotech. 23 , 450 (2005). [7] J. H. Conway and N. J. A. Sloane, Disc. Comp. Geo. 13 , 383 (1995). [8] T. C. Hales, Ann. Math. 162 , 1065 (2005). [9] J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups (Springer, 1999). [10] O. R. Musin, Ann. Math. 168 , 1 (2008). [11] A. B. Hopkins, F. H. Stillinger, and S. Torquato, J. Math. Phys. (2009). [12] F. A. Cotton, Chemical Applications of Group Theory, 3rd ed. (Wiley, 1990). [13] For all putative optimal packings presented in this paper for N up to N = 168, accuracy is to at least 10 − 8 diameters, while for some N > 168, an accuracy to at least 10 − 6 diameters was used to reduce computation time. These accuracy values apply to all R$_{min}$ ( N ), the contact distance between disks in contact and the tolerance used in identifying symmetry elements in order to assign point groups. Additionally, we have used the convention of placing dots after a R$_{min}$ ( N ) value, such as with R$_{min}$ (15) = 1 . 873123 ... , to imply that the complete R$_{min}$ ( N ) value may be calculated analytically using trigonmetry (though we do not present 34