Ball Packings with Periodic Constraints

TL;DR

This paper investigates the properties of periodic ball packings in Euclidean space, establishing conditions for strict jamming, characterizing periodic unjamming on sublattices, and providing examples of jammed packings with specific densities.

Contribution

It extends known rigidity conditions to strict jamming, characterizes periodic jamming on sublattices, and presents new examples of jammed packings with particular densities.

Findings

A packing is consistently strictly jammed iff it is strictly jammed with respect to its period and consistently periodically jammed.

Finitely many strictly jammed packings of m unit balls exist.

An example of a low-density consistently periodically jammed packing is provided.

Abstract

We call a periodic ball packing in d-dimensional Euclidean space periodically (strictly) jammed with respect to a period lattice if there are no nontrivial motions of the balls that preserve the period (that maintain some period with smaller or equal volume). In particular, we call a packing consistently periodically (strictly) jammed if it is periodically (strictly) jammed on every one of its periods. After extending a well-known bar framework and stress condition to strict jamming, we prove that a packing with period Lambda is consistently strictly jammed if and only if it is strictly jammed with respect to Lambda and consistently periodically jammed. We next extend a result about rigid unit mode spectra in crystallography to characterize periodic jamming on sublattices. After that, we prove that there are finitely many strictly jammed packings of m unit balls and other similar results. An interesting example shows that the size of the first sublattice on which a packing is first periodically unjammed is not bounded. Finally, we find an example of a consistently periodically jammed packing of low density \delta = \frac{4 \pi}{6 \sqrt{3} + 11} + \epsilon ~ 0.59, where \epsilon is an arbitrarily small positive number. Throughout the paper, the statements for the closely related notions of periodic infinitesimal rigidity and affine infinitesimal rigidity for tensegrity frameworks are also given.