The Densest Local Packings of Identical Spheres in Three Dimensions

We have used a novel algorithm combining nonlinear programming methods with a random search of configuration space to find the densest local packings of spheres in three-dimensional Euclidean space. Our results reveal a wealth of information about packings of spheres, including counterintuitive results concerning the physics of dilute mixtures of spherical solute particles in a solvent composed of same-size spheres and about the presence of unjammed spheres (rattlers) in the densest local structures.   Read more

The Densest Local Packings of Identical Disks in Two Dimensions

  N=15, point group D_5h

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.  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 a realizability condition for the pair correlation functions of sphere packings and an upper bound on the maximal density of infinite sphere packings.  Read more

The Densest Local Packings of Spheres in Any Dimension and the Golden Ratio

The optimal spherical code problem involves the placement of the centers of N nonoverlapping spheres of unit diameter onto the surface of a sphere of radius R such that R is minimized. We prove that in any dimension, all solutions between unity and the golden ratio to the optimal spherical code problem for N spheres are also solutions to the corresponding DLP problem.   Read more