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

2016/03/27 by Samuel Reid, Reid, Samuel
Chemical Engineering · Chemistry · Materials Science · #52C17 #Crystallography and molecular interactions #FOS: Mathematics #Ionic liquids properties and applications #Material Dynamics and Properties #Metric Geometry (math.MG) #Pickering emulsions and particle stabilization #Quasicrystal Structures and Properties

paper · pdf · doi:10.48550/arxiv.1603.08201

openalex publication_date 2016/03/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let C(n) be the solution to the contact number problem, i.e., the maximum\nnumber of touching pairs among any packing of n congruent spheres in\n\ℝ³. We prove the long conjectured values of C(6)=12, C(7)=15, and\nC(8)=18. The proof strategy generalizes under an extensive case analysis to\nC(9)=21, C(10) = 25, C(11) = 29, C(12) = 33, and C(13) = 36. These results\nhave great importance for condensed matter physics, materials science,\ncrystallography, organic and physical chemistry of interfaces.\n

Citations

• Sphere Packing with Limited Overlap
• Upper Bounds for Non-Congruent Sphere Packings
• On Contact Numbers of Finite Lattice Sphere Packings of 20-27 Balls
• The hydration number of Na+ in liquid water

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