Ball Packings with Periodic Constraints
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454_2014_Article_9636.pdf
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Author(s)
Connelly, Robert
Shen, Jeffrey D.
Smith, Alexander D.
Date Issued
September 2014
Journal
Discrete & Computational Geometry
Publisher
Springer US
Citation
Connelly, Robert, Jeffrey D. Shen, and Alexander D. Smith. “Ball Packings with Periodic Constraints.” Discrete & Computational Geometry 52.4 (2014): 754–779.
Version
Author's final manuscript
Abstract
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=4π/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.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use.
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DOI of Published Version
https://doi.org/10.1007/s00454-014-9636-z
