Ball Packings with Periodic Constraints
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...
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| Format: | Article |
| Language: | English |
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Springer US
2016
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| Online Access: | http://hdl.handle.net/1721.1/104662 https://orcid.org/0000-0001-8835-3588 |
| _version_ | 1811072074100244480 |
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| author | Connelly, Robert Shen, Jeffrey D. Smith, Alexander D. |
| author2 | Massachusetts Institute of Technology. Department of Mathematics |
| author_facet | Massachusetts Institute of Technology. Department of Mathematics Connelly, Robert Shen, Jeffrey D. Smith, Alexander D. |
| author_sort | Connelly, Robert |
| collection | MIT |
| description | 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. |
| first_indexed | 2024-09-23T09:00:27Z |
| format | Article |
| id | mit-1721.1/104662 |
| institution | Massachusetts Institute of Technology |
| language | English |
| last_indexed | 2024-09-23T09:00:27Z |
| publishDate | 2016 |
| publisher | Springer US |
| record_format | dspace |
| spelling | mit-1721.1/1046622022-09-30T12:48:44Z Ball Packings with Periodic Constraints Connelly, Robert Shen, Jeffrey D. Smith, Alexander D. Massachusetts Institute of Technology. Department of Mathematics Shen, Jeffrey D. 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. National Science Foundation (U.S.) (Cornell University. Research Experience for Undergraduates. Grant DMS-1156350) 2016-10-06T20:57:27Z 2016-10-06T20:57:27Z 2014-09 2014-08 2016-08-18T15:41:12Z Article http://purl.org/eprint/type/JournalArticle 0179-5376 1432-0444 http://hdl.handle.net/1721.1/104662 Connelly, Robert, Jeffrey D. Shen, and Alexander D. Smith. “Ball Packings with Periodic Constraints.” Discrete & Computational Geometry 52.4 (2014): 754–779. https://orcid.org/0000-0001-8835-3588 en http://dx.doi.org/10.1007/s00454-014-9636-z Discrete & Computational Geometry 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. Springer Science+Business Media New York application/pdf Springer US Springer US |
| spellingShingle | Connelly, Robert Shen, Jeffrey D. Smith, Alexander D. Ball Packings with Periodic Constraints |
| title | Ball Packings with Periodic Constraints |
| title_full | Ball Packings with Periodic Constraints |
| title_fullStr | Ball Packings with Periodic Constraints |
| title_full_unstemmed | Ball Packings with Periodic Constraints |
| title_short | Ball Packings with Periodic Constraints |
| title_sort | ball packings with periodic constraints |
| url | http://hdl.handle.net/1721.1/104662 https://orcid.org/0000-0001-8835-3588 |
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