Spherical codes, maximal local packing density, and the golden ratio

The densest local packing (DLP) problem in d-dimensional Euclidean space R(d) involves the placement of N nonoverlapping spheres of unit diameter near an additional fixed unit-diameter sphere such that the greatest distance from the center of the fixed sphere to the centers of any of the N surrounding spheres is minimized. Solutions to the DLP problem are relevant to the realizability of pair correlation functions for packings of nonoverlapping spheres and might prove useful in improving upon the best known upper bounds on the maximum packing fraction of sphere packings in dimensions greater than 3. The optimal spherical code problem in R(d) 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. It is proved that in any dimension, all solutions between unity and the golden ratio tau to the optimal spherical code problem for N spheres are also solutions to the corresponding DLP problem. It follows that for any packing of nonoverlapping spheres of unit diameter, a spherical region of radius less than or equal to tau centered on an arbitrary sphere center cannot enclose a number of sphere centers greater than 1 more than the number that than can be placed on the region's surface. (C) 2010 American Institute of Physics. [doi:10.1063/1.3372627]