Exact Packing Fractions of Cubic and Hexagonal Sphere Packings — E8 Intelligence Research

FINDING: The maximal packing fractions of the three cubic/hexagonal sphere packings are exact algebraic numbers: SC = π/6 ≈ 0.5236, FCC = HCP = π/(3√2) = π√2/6 ≈ 0.7405; the FCC/HCP ratio to SC is exactly 6/√2 = 3√2 ≈ 4.2426, and Kepler's conjecture (proved 1998, Hales) establishes π/√18 as the global maximum for any sphere packing. MATH: - SC: APF = (1 atom × (4/3)πr³)/(2r)³ = π/6 ≈ 0.5235988 - FCC: 4 atoms, a = 2√2·r → APF = (4·(4/3)πr³)/(2√2 r)³ = π/(3√2) = π√2/6 ≈ 0.7404805 - HCP: identical value π√2/6 (same coordination number 12, same local density) - Ratio FCC/SC = (π√2/6)/(π/6) = √2 ≈ 1.41421 - Ratio FCC/HCP = 1 exactly (degenerate) - HCP ideal c/a = √(8/3) ≈ 1.63299 (not 1.63 exactly — the video's "1,63" is a truncation) - Kepler bound: π/√18 = π/(3√2) = π√2/6 — the FCC value IS the global maximum - Elastic constant discrepancy FCC vs HCP: up to 20% despite identical free energy, pressure, bulk modulus (arXiv cond-mat/0305471) CONNECTION: - √2 appears as the fundamental rati Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com