Deriving Finite Sphere Packings
| Title: | Deriving Finite Sphere Packings |
| Author: |
Arkus, Natalie; Manoharan, Vinothan N.; Brenner, Michael P.
Note: Order does not necessarily reflect citation order of authors. |
| Citation: | Arkus, Natalie, Vinothan N. Monoharan, and Michael P. Brenner. 2011. Deriving Finite Sphere Packings. SIAM Journal on Discrete Mathematics 25(4): 1860-1901. |
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| Abstract: | Sphere packing problems have a rich history in both mathematics and physics; yet, relatively few analytical analyses of sphere packings exist, and answers to seemingly simple questions are unknown. Here, we present an analytical method for deriving all packings of n spheres in R3 satisfying minimal rigidity constraints (≥ 3 contacts per sphere and ≥ 3n − 6 total contacts). We derive such packings for n ≤ 10 and provide a preliminary set of maximum contact packings for 10 < n ≤ 20. The resultant set of packings has some striking features; among them are the following: (i) all minimally rigid packings for n ≤ 9 have exactly 3n−6 contacts; (ii) nonrigid packings satisfying minimal rigidity constraints arise for n ≥ 9; (iii) the number of ground states (i.e., packings with the maximum number of contacts) oscillates with respect to n; (iv) for 10 ≤ n ≤ 20 there are only a small number of packings with the maximum number of contacts, and for 10 ≤ n < 13 these are all commensurate with the hexagonal close-packed lattice. The general method presented here may have applications to other related problems in mathematics, such as the Erdos repeated distance problem and Euclidean distance matrix completion problems. |
| Published Version: | doi:10.1137/100784424 |
| Other Sources: | http://arxiv.org/abs/1011.5412 |
| Terms of Use: | This article is made available under the terms and conditions applicable to Open Access Policy Articles, as set forth at http://nrs.harvard.edu/urn-3:HUL.InstRepos:dash.current.terms-of-use#OAP |
| Citable link to this page: | http://nrs.harvard.edu/urn-3:HUL.InstRepos:6098792 |
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