FCC and HCP Maximal Sphere Packing: π/(3√2) ≈ 0.7405 — E8 Intelligence Research

FINDING: FCC and HCP are the two maximal-density sphere packings, both achieving packing fraction π/(3√2) ≈ 0.7405; the "π√2/6" in the query is a mis-transcription of this same constant (π√2/6 = π/(3√2) = 0.74048). | MATH: - SC: PF = (4/3)πr³/(2r)³ = π/6 ≈ 0.5236 - BCC: PF = 2·(4/3)πr³/(4r/√3)³ = π√3/8 ≈ 0.6802 - FCC: PF = 4·(4/3)πr³/(2√2·r)³ = π/(3√2) = π√2/6 ≈ 0.7405 - HCP: identical PF = π/(3√2), with ideal c/a = √(8/3) ≈ 1.633 (matches the 1.63 cited in the German video) - Kepler conjecture (Hales 1998/2014): π/(3√2) is the global maximum for any sphere packing in ℝ³. - HCP ideal axial ratio: c/a = √(8/3) = 2√(2/3) = 1.63299… - FCC/HCP coordination number Z = 12; kissing number in 3D = 12. - FCC = cubic close packing (ABCABC); HCP = hexagonal close packing (ABAB). Both are stackings of the same 2D triangular lattice (density π/(2√3) ≈ 0.9069). - Elastic constants differ up to ~20% between FCC and HCP hard-sphere crystals despite identical free energy, pressure, and bulk modulus (P Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com