Ball Packings with Periodic Constraints

Abstract

We call a periodic ball packing in $$d$$d-dimensional Euclidean space periodically (resp. strictly) jammed with respect to a period lattice $$\varLambda $$? if there are no nontrivial motions of the balls that preserve $$\varLambda $$? (resp. that maintain some period with smaller or equal volume). In particular, we call a packing consistently periodically jammed (resp. consistently strictly jammed) if it is periodically (resp. 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 $$\varLambda $$? is consistently strictly jammed if and only if it is strictly jammed with respect to $$\varLambda $$? 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 $$m$$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 $$\delta = \frac{4 \pi }{6 \sqrt{3} + 11} + \varepsilon \approx 0.59$$?=4?63+11+??0.59, where $$\varepsilon $$? 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.