Enumerating rigid sphere packings

TL;DR

This paper systematically enumerates rigid packings of identical spheres up to 14 or 19 particles, revealing unexpected geometrical features and expanding understanding of possible configurations in physical and biological systems.

Contribution

It provides the first comprehensive enumeration of rigid sphere clusters for small sizes, highlighting the prevalence of hypostatic clusters and unusual geometries.

Findings

Enumerated all rigid clusters for n≤14 and maximum contact clusters for n≤19.

Discovered the high occurrence of hypostatic clusters with fewer than 3n-6 contacts.

Identified unusual geometries that could inform material design and physical understanding.

Abstract

Packing problems, which ask how to arrange a collection of objects in space to meet certain criteria, are important in a great many physical and biological systems, where geometrical arrangements at small scales control behaviour at larger ones. In many systems there is no single, optimal packing that dominates, but rather one must understand the entire set of possible packings. As a step in this direction we enumerate rigid clusters of identical hard spheres for $n\leq 14$, and clusters with the maximum number of contacts for $n\leq 19$. A rigid cluster is one that cannot be continuously deformed while maintaining all contacts. This is a nonlinear notion that arises naturally because such clusters are the metastable states when the spheres interact with a short-range potential, as is the case in many nano- or micro-scale systems. We expect these lists are nearly complete, except for a small number of highly singular clusters (linearly floppy but nonlinearly rigid.) The data contains some major geometrical surprises, such as the prevalence of hypostatic clusters: those with less than the $3n-6$ contacts generically necessary for rigidity. We discuss these and several other unusual clusters, whose geometries may shed insight into physical mechanisms, pose mathematical and computational problems, or bring inspiration for designing new materials.