Deriving Finite Sphere Packings

Abstract

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 maximum 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 exactly 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 maximum number of contacts) oscillates with respect to n, (iv) for 10 ≤ n ≤ 20 there are only a small number of packings with the maximum 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 Erdös repeated distance problem and Euclidean distance matrix completion problems.

Colloquium : Toward living matter with colloidal particles

“…One recent set of results (Arkus et al, 2011;Holmes-Cerfon, 2016) is the complete enumeration of all structures that can form from N colloidal spheres, for N ≤ 14 particles ( Figure 1A). Even for this small N , a rich landscape of structures emerges, from which many possible "reactions," or dynamical pathways between states, can be examined.…”

“…Arkus and coworkers (Arkus et al, 2009(Arkus et al, , 2011) developed a method to enumerate all clusters of N particles that have at least 3N − 6 contacts. The method has two steps.…”

Selected Open Problems in Discrete Geometry and Optimization

“…The main results are: C.9/ D 21; C.10/ D 25 [38] and C.11/ D 29 [46]. However, the status of the mathematical rigour of the approaches of [38] as well as [46] remains to be seen. For C.n/ in general, when n is an arbitrary positive integer, we have the following estimates proved in [40] and [41].…”

“…In connection with this problem we call the reader's attention to the very recent and highly elegant article of Hayes [45]. It gives an overview of the computational methods presented in the papers [38] and [46] that are based on exhaustive enumeration and elementary geometry. The main results are: C.9/ D 21; C.10/ D 25 [38] and C.11/ D 29 [46].…”

“…It gives an overview of the computational methods presented in the papers [38] and [46] that are based on exhaustive enumeration and elementary geometry. The main results are: C.9/ D 21; C.10/ D 25 [38] and C.11/ D 29 [46]. However, the status of the mathematical rigour of the approaches of [38] as well as [46] remains to be seen.…”