# Densest local sphere-packing diversity: General concepts and application to two dimensions 

Adam B. Hopkins and Frank H. Stillinger 

Department of Chemistry, Princeton University, Princeton, New Jersey 08544 

Salvatore Torquato 

Department of Chemistry, Princeton Institute for the Science and Technology of Materials, Program in Applied and Computational Mathematics, Princeton Center for Theoretical Science, Princeton University, Princeton, New Jersey 08544 School of Natural Sciences, Institute for Advanced Study, Princeton, New Jersey 08544 

## Abstract 

The densest local packings of N identical nonoverlapping spheres within a radius R$_{min}$ ( N ) of a fixed central sphere of the same size are obtained using a nonlinear programming method operating in conjunction with a stochastic search of configuration space. Knowledge of R$_{min}$ ( N ) in d -dimensional Euclidean space$_{R}$ d allows for the construction both of a realizability condition for pair correlation functions of sphere packings and an upper bound on the maximal density of infinite sphere packings in$_{R}$ $^{d}$. In this paper, we focus on the two-dimensional circular disk problem. We find and present the putative densest packings and corresponding R$_{min}$ ( N ) for selected values of N up to N = 348 and use this knowledge to construct such a realizability condition and upper bound. We additionally analyze the properties and characteristics of the maximally dense packings, finding significant variability in their symmetries and contact networks, and that the vast majority differ substantially from the triangular lattice even for large N . Our work has implications for packaging problems, nucleation theory, and surface physics. 

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