Rigidity of spherical codes
A packing of spherical caps on the surface of a sphere (that is, a spherical code) is called rigid or jammed if it is isolated within the space of packings. In other words, aside from applying a global isometry, the packing cannot be deformed. In this paper, we systematically study the rigidity o...
| Main Authors: | , , , |
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| Format: | Article |
| Language: | en_US |
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Mathematical Sciences Publishers
2012
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| Online Access: | http://hdl.handle.net/1721.1/71122 |
| _version_ | 1826208077426196480 |
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| author | Cohn, Henry Jiao, Yang Kumar, Abhinav Torquato, Salvatore |
| author2 | Massachusetts Institute of Technology. Department of Mathematics |
| author_facet | Massachusetts Institute of Technology. Department of Mathematics Cohn, Henry Jiao, Yang Kumar, Abhinav Torquato, Salvatore |
| author_sort | Cohn, Henry |
| collection | MIT |
| description | A packing of spherical caps on the surface of a sphere (that is, a spherical code) is
called rigid or jammed if it is isolated within the space of packings. In other words,
aside from applying a global isometry, the packing cannot be deformed. In this
paper, we systematically study the rigidity of spherical codes, particularly
kissing configurations. One surprise is that the kissing configuration of the
Coxeter–Todd lattice is not jammed, despite being locally jammed (each
individual cap is held in place if its neighbors are fixed); in this respect, the
Coxeter–Todd lattice is analogous to the face-centered cubic lattice in three
dimensions. By contrast, we find that many other packings have jammed
kissing configurations, including the Barnes–Wall lattice and all of the best
kissing configurations known in four through twelve dimensions. Jamming
seems to become much less common for large kissing configurations in higher
dimensions, and in particular it fails for the best kissing configurations known in
25 through 31 dimensions. Motivated by this phenomenon, we find new kissing configurations in
these dimensions, which improve on the records set in 1982 by the laminated
lattices. |
| first_indexed | 2024-09-23T14:00:06Z |
| format | Article |
| id | mit-1721.1/71122 |
| institution | Massachusetts Institute of Technology |
| language | en_US |
| last_indexed | 2024-09-23T14:00:06Z |
| publishDate | 2012 |
| publisher | Mathematical Sciences Publishers |
| record_format | dspace |
| spelling | mit-1721.1/711222022-10-01T18:28:51Z Rigidity of spherical codes Cohn, Henry Jiao, Yang Kumar, Abhinav Torquato, Salvatore Massachusetts Institute of Technology. Department of Mathematics Kumar, Abhinav Kumar, Abhinav A packing of spherical caps on the surface of a sphere (that is, a spherical code) is called rigid or jammed if it is isolated within the space of packings. In other words, aside from applying a global isometry, the packing cannot be deformed. In this paper, we systematically study the rigidity of spherical codes, particularly kissing configurations. One surprise is that the kissing configuration of the Coxeter–Todd lattice is not jammed, despite being locally jammed (each individual cap is held in place if its neighbors are fixed); in this respect, the Coxeter–Todd lattice is analogous to the face-centered cubic lattice in three dimensions. By contrast, we find that many other packings have jammed kissing configurations, including the Barnes–Wall lattice and all of the best kissing configurations known in four through twelve dimensions. Jamming seems to become much less common for large kissing configurations in higher dimensions, and in particular it fails for the best kissing configurations known in 25 through 31 dimensions. Motivated by this phenomenon, we find new kissing configurations in these dimensions, which improve on the records set in 1982 by the laminated lattices. 2012-06-08T14:50:22Z 2012-06-08T14:50:22Z 2011-11 2011-05 Article http://purl.org/eprint/type/JournalArticle 1465-3060 1364-0380 http://hdl.handle.net/1721.1/71122 Cohn, Henry et al. “Rigidity of spherical codes.” Geometry & Topology 15.4 (2011): 2235-2273. © Copyright 2011 Mathematical Sciences Publishers en_US http://dx.doi.org/10.2140/gt.2011.15.2235 Geometry and Topology 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. application/pdf Mathematical Sciences Publishers Kumar via Michael Noga |
| spellingShingle | Cohn, Henry Jiao, Yang Kumar, Abhinav Torquato, Salvatore Rigidity of spherical codes |
| title | Rigidity of spherical codes |
| title_full | Rigidity of spherical codes |
| title_fullStr | Rigidity of spherical codes |
| title_full_unstemmed | Rigidity of spherical codes |
| title_short | Rigidity of spherical codes |
| title_sort | rigidity of spherical codes |
| url | http://hdl.handle.net/1721.1/71122 |
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