Deriving Finite Sphere Packings


Arkus, N. ; Manoharan, V. N. ; Brenner, M. P. Deriving Finite Sphere Packings . SIAM Journal on Discrete Mathematics 2011, 25, 1860.


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 Erdös repeated distance problem and Euclidean distance matrix completion problems.


See also "The Science of Sticky Spheres" by Brian Hayes (American Scientist 2012).

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Last updated on 03/26/2016