On Arrangements of Six, Seven, and Eight Spheres: Maximal Bonding of Monatomic Ionic Compounds

Let $C(n)$ be the solution to the contact number problem, i.e., the maximum number of touching pairs among any packing of $n$ congruent spheres in ${\mathbb{R}}^3$. We prove the long conjectured values of $C(6)=12,C(7)=15$, and $C(8)=18$. The proof strategy generalizes under an extensive case analysis to $C(9)=21,C(10)=25,C(11)=29,C(12)=33$, and $C(13)=36$. These results have great importance for condensed matter physics, materials science, crystallography, organic and physical chemistry of interfaces.