Plimpton 322's Reciprocal Method Mirrors Lattice Construction via Coxeter Groups — E8 Intelligence Research

FINDING: Reciprocal lattice construction mirrors Plimpton 322's reciprocal-pair method for generating Pythagorean triples, and Coxeter group classification provides the complete taxonomy of crystallographic root systems. MATH: - Plimpton 322 method: For regular sexagesimal numbers \( p, q \) (with \( p > q \)), generate triple \((a, b, c)\) via \( a = p^2 - q^2 \), \( b = 2pq \), \( c = p^2 + q^2 \). Reciprocal pairs: \( (1/x, x) \) produce \( p = (1/x + x)/2 \), \( q = (1/x - x)/2 \). - Reciprocal lattice: For direct lattice basis vectors \( a_i \), reciprocal basis \( b_j \) satisfies \( a_i · b_j = 2π δᵢⱼ \). - Coxeter classification: Finite irreducible Coxeter groups are \( A_n, B_n/C_n, D_n, E_6, E_7, E_8, F_4, G_2, H_3, H_4, I_2(m) \). Root system symmetries yield crystallographic restriction: only orders 2,3,4,6 allowed. CONNECTION: - Reciprocal-pair method in Plimpton 322 uses \( p/q \) ratios that are rational approxi Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com