these calculations here). This is the case for every packing presented with point group other than C$_{1}$ , as exploitation of existing symmetries in symmetric optimal packings allows explicit calculation of R$_{min}$ ( N ). 

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[21] The supplementary materials can be found at <http://cherrypit.princeton.edu> . 

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[23] H. Cohn, A. Kumar, and S. Torquato, unpublished. 

[24] In practice, for d = 2 we find that the only N > 6 for which R$_{min}$ ( N ) = R$_{min}$ ( N − 1) is N = 18. 

[25] Note that this definition allows for a number of spheres to have only a fraction of their volume included in the window. 

[26] H. Cohn and A. Kumar, To appear in Ann. Math. (2010), <http://arxiv.org/abs/math. MG/0403263> . 

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[30] Images created using Jmol: an open-source Java viewer for chemical structures. <http:// www.jmol.org/> . 

[31] A. Ruszczynski, Nonlinear Optimization (Princeton University Press, 2006). 

[32] Although for this work we sought DLP optimal configurations only for packings of identical nonoverlapping spheres, the two step process is versatile and can be adapted to attempt solutions to any packing problem seeking the maximum packing fraction of a finite number of identical or differentiated nonoverlapping objects. 

