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

TL;DR

This paper proves the long-standing conjectured maximum contact numbers for packings of 6, 7, and 8 spheres in three-dimensional space, and extends the proof to larger numbers, impacting various scientific fields.

Contribution

It establishes the exact maximum contact numbers for small sphere packings and generalizes the proof to larger packings through extensive case analysis.

Findings

C(6)=12, C(7)=15, C(8)=18 proven

Extended to C(9)=21, C(10)=25, C(11)=29, C(12)=33, C(13)=36

Results are significant for physics, chemistry, and materials science

Abstract

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.