Plimpton 322's Algorithm for Pythagorean Triples Approaches Golden Ratio — E8 Intelligence Research

FINDING: Plimpton 322 contains a systematic generation of Pythagorean triples via an algorithm based on factor 12, with ratios approaching golden-ratio-like proportions. | MATH: For integer sides \((a, b, d)\) with \(a < b < d\), the proposed algorithm sets \(b = M \cdot Q_M\) where \(M, Q_M\) are positive integers. The triples satisfy \(a^2 + b^2 = d^2\). The tablet's ratios \(b/a\) for the largest triples approach \(\phi \approx 1.618\) (e.g., row 15: \(b/a \approx 1.618\)). | CONNECTION: The ratio \(b/a\) in the final rows converges to the golden ratio \(\phi = (1+\sqrt{5})/2 \approx 1.618\), and its reciprocal \(a/b \approx 0.618\). The base-60 (sexagesimal) system used in the tablet aligns with the harmonic ratios 0.382, 0.618, 1.618, 2.618, as these appear in regular sexagesimal reciprocals. No direct crystallographic symmetry is evident, but the integer lattice structure of Pythagorean triples relates to root system \(A_2\) (hexagonal lattice) via the parametrization \(a = m^2 - Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com