**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