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central sphere, then there are more packing configurations available to a nucleus of radius R , with R confined to a small finite range R > R$_{min}$ ( N ), in$_{R}$ 3 than in$_{R}$ $^{2}$. 

## ACKNOWLEDGEMENTS: 

S.T. thanks the Institute for Advanced Study for its hospitality during his stay there. This work was supported by the Division of Mathematical Sciences at the National Sci- ence Foundation under Award Number DMS-0804431 and by the MRSEC Program of the National Science Foundation under Award Number DMR-0820341. 

## Appendix A: Description of the algorithm 

The algorithm employed to find the DLP optimal packings presented operates in two steps repeated iteratively in succession. In the first step, we use a method in nonlinear programming often called an augmented Lagrangian method [31] to find a local minimum of a specially formulated problem in Nd + 1 variables, where the first Nd variables correspond to coordinates of the centers of N spheres in d dimensions. In the second step, the configu- ration of N spheres found in the first step is spatially repositioned using a random number generator. Keeping track of the least local minimum, the two steps are repeated iteratively until (presumably) no further improvement on the least minimum can be achieved [32]. 

The nonlinear programming algorithm used in the first step seeks to solve a problem posed for Nd + 1 variables with 2 N + $_{N}$C$_{2}$ inequality constraints, where $_{N}$C$_{2}$ is shorthand for the standard combinatorial formula. Calling the first Nd variables x$_{ik}$ with i = 1 . . . N and k = 1 . . . d and the last variable ω , the problem may be posed as follows: 

<img src='content_image/339662.jpg'>

The constraints in (A1) are split into three categories; the first category, containing N 

