Plimpton 322's Reciprocal-Pair Triples Map to Coxeter Crystallographic Lattices — E8 Intelligence Research

FINDING: Plimpton 322 encodes a reciprocal-pair method for generating Pythagorean triples, which maps directly onto the classification of crystallographic lattices via Coxeter groups and root systems. MATH: - Plimpton 322 triples (a, b, c) satisfy a² + b² = c², generated by reciprocal pairs (p, q) with p/q in base-60, yielding (a, b, c) = (p² - q², 2pq, p² + q²) or (2pq, p² - q², p² + q²). - Coxeter group classification: finite Coxeter groups correspond to crystallographic root systems (types Aₙ, Bₙ, Cₙ, Dₙ, E₆, E₇, E₈, F₄, G₂). Their Cartan matrices have entries 2, -1, 0, with off-diagonal products 1, 2, 3 (reflecting bond angles 120°, 135°, 150°). - Key constants: golden ratio φ = (1+√5)/2 ≈ 1.618, its reciprocal φ⁻¹ ≈ 0.618, and related ratios 0.382 (φ⁻²), 0.786 (√φ⁻¹), 2.618 (φ²). These appear in E₈ root system (240 roots, 8D, Coxeter number 30, ratio of largest to smallest root length = φ). - Base-60: Plimpton 322 uses sexagesimal fractions; base-60 appears in Babylonian Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com