Caspar-Klug Theory Links Eisenstein Norms to Icosahedral Coordination Numbers — E8 Intelligence Research

FINDING: Caspar-Klug theory links hexagonal lattice quadratic forms to number sequence A003136, revealing that allowed icosahedral/crystallographic coordination numbers correspond to norms of Eisenstein integers. MATH: - Quadratic form: \( x^2 + xy + y^2 = N \) (norm of Eisenstein integer \( x + yω \), \(ω = e2π i/3\)). - Sequence A003136: integers \( N \) representable as \( x^2 + xy + y^2 \) (e.g., 0,1,3,4,7,9,12,13,16,19,21,25,27,28,31,36,37,39,43,48,49,52,57,61,63,64,67,73,75,76,79,81,84,91,93,97,...). - Key constants: hexagonal lattice coordination numbers (e.g., 12, 42, 92, 162) appear as norms in this form. - Connection to icosahedral symmetry: \( N = 3k^2 + 3k + 1 \) (centered hexagonal numbers) for certain shell structures. CONNECTION: - Geometric harmony: The ratio of successive allowed coordination numbers approaches 1 (no fixed golden ratio here), but the quadratic form itself encodes hexagonal close-packing symmetry (6-fold rotational symmetry). Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com