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 ErdÃ¶s repeated distance problem and Euclidean distance matrix completion problems.

%B SIAM Journal on Discrete Mathematics %V 25 %P 1860 %G eng %U http://dx.doi.org/10.1137/100784424 %N 4