Deriving Finite Sphere Packings

Authors:
Natalie Arkus;Vinothan N. Manoharan;Michael P. Brenner
Affiliations:
narkus@post.harvard.edu and vnm@seas.harvard.edu and brenner@seas.harvard.edu;-;-
Venue:
SIAM Journal on Discrete Mathematics
Year:
2011

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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 $\mathbb{R}^3$ 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