Ball Packings with Periodic Constraints

• Published in: Discrete & computational geometry 2014-12, Vol.52 (4), p.754-779
• Main Authors: Connelly, Robert, Shen, Jeffrey D., Smith, Alexander D.
• Format: Article
• Language: English
• Subjects: Combinatorics; Computational Mathematics and Numerical Analysis; Density; Geometry; Jamming; Lattices; Materials science; Mathematics; Mathematics and Statistics; Preserves; Rigidity; Spectra; Tensegrity; Texts
• ISSN: 0179-5376; 1432-0444
• DOI: 10.1007/s00454-014-9636-z

We call a periodic ball packing in d -dimensional Euclidean space periodically (resp. strictly ) jammed with respect to a period lattice Λ if there are no nontrivial motions of the balls that preserve Λ (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 Λ is consistently strictly jammed if and only if it is strictly jammed with respect to Λ 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 δ = 4 π 6 3 + 11 + ε ≈ 0.59 , where ε 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.