NDLI: Deriving Finite Sphere Packings

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Brenner, Michael P. Arkus, Natalie Manoharan, Vinothan N.
Copyright Year	2011
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 $\mathbb{R}^3$ satisfying minimal rigidity constraints ($\geq 3$ contacts per sphere and $\geq 3n-6$ total contacts). We derive such packings for $n \leq 10$ and provide a preliminary set of maximum contact packings for $10 < n \leq 20$. The resultant set of packings has some striking features; among them are the following: (i) all minimally rigid packings for $n \leq 9$ have exactly $3n-6$ contacts; (ii) nonrigid packings satisfying minimal rigidity constraints arise for $n \geq 9$; (iii) the number of ground states (i.e., packings with the maximum number of contacts) oscillates with respect to n; (iv) for $10 \leq n \leq 20$ there are only a small number of packings with the maximum number of contacts, and for $10 \leq n < 13$ 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 Erds repeated distance problem and Euclidean distance matrix completion problems.
Starting Page	1860
Ending Page	1901
Page Count	42
File Format	PDF
ISSN	08954801
DOI	10.1137/100784424
e-ISSN	10957146
Journal	SIAM Journal on Discrete Mathematics (SJDMEC)
Issue Number	4
Volume Number	25
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2011-12-20
Access Restriction	Subscribed
Subject Keyword	sphere distance equations colloid packing
Content Type	Text
Resource Type	Article
Subject	Mathematics

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