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

TL;DR

This paper investigates the densest local arrangements of identical disks in two dimensions, using computational methods to find optimal packings, analyze their properties, and derive bounds relevant to sphere packing problems.

Contribution

It introduces a nonlinear programming and stochastic search approach to identify densest local packings in 2D and derives realizability conditions and bounds for sphere packings.

Findings

Identified densest local packings up to N=348 disks.

Discovered significant variability in symmetry and contact networks.

Most packings differ from the triangular lattice even at large N.

Abstract

The densest local packings of N identical nonoverlapping spheres within a radius Rmin(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 Rmin(N) in d-dimensional Euclidean space 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 this paper, we focus on the two-dimensional circular disk problem. We find and present the putative densest packings and corresponding Rmin(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.