Deriving Finite Sphere Packings

TL;DR

This paper introduces an analytical method to derive all minimally rigid sphere packings in three dimensions for up to 10 spheres, revealing structural patterns and oscillations in the number of maximal contact packings.

Contribution

The paper presents a novel analytical approach to enumerate all sphere packings satisfying minimal rigidity constraints for small n, extending to preliminary results for larger n.

Findings

All minimally rigid packings for n <= 9 have 3n-6 contacts.

Non-rigid packings satisfying minimal rigidity constraints appear for n >= 9.

The number of ground state packings oscillates with n.

Abstract

Sphere packing problems have a rich history in both mathematics and physics; yet, relatively few analytical analyses of sphere packings exist, and answers to seemingly simple questions are unknown. Here, we present an analytical method for deriving all packings of n spheres in R^3 satisfying minimal rigidity constraints (>= 3 contacts per sphere and >= 3n-6 total contacts). We derive such packings for n <= 10, and provide a preliminary set of maximal contact packings for 10 < n <= 20. The resultant set of packings has some striking features, among them: (i) all minimally rigid packings for n <= 9 have 3n-6 contacts, (ii) non-rigid packings satisfying minimal rigidity constraints arise for n >= 9, (iii) the number of ground states (i.e. packings with the maximal number of contacts) oscillates with respect to n, (iv) for 10 <= n <= 20 there are only a small number of packings with the maximal number of contacts, and for 10 <= n < 13 these are all commensurate with the HCP lattice. The general method presented here may have applications to other related problems in mathematics, such as the Erdos repeated distance problem and Euclidean distance matrix completion problems.