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

TL;DR

This paper explores the relationship between spherical codes and local sphere packing densities, revealing that solutions within the golden ratio are common to both problems and establishing bounds on sphere arrangements.

Contribution

It proves that solutions to the optimal spherical code problem within the golden ratio are also solutions to the densest local packing problem in any dimension.

Findings

Solutions between 1 and the golden ratio are shared by both problems.

A spherical region of radius up to the golden ratio cannot contain more than one additional sphere center than on its surface.

The results provide bounds relevant to sphere packing density and realizability of pair correlation functions.

Abstract

The densest local packing (DLP) problem in d-dimensional Euclidean space Rd 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 three. The optimal spherical code problem in Rd 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 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 the golden ratio centered on an arbitrary sphere center cannot enclose a number of sphere centers greater than one more than the number that can be placed on the region's surface.