Plimpton 322's Reciprocal Pairs Map to E₈ Root Lattice, Linking Ancient Math to Modern Symmetry — E8 Intelligence Research

FINDING: Plimpton 322's reciprocal-pair method generates Pythagorean triples via base-60 sexagesimal reciprocals, and this generative structure maps onto the root lattice of the E₈ Coxeter group, linking ancient computational limits to modern Lie algebra symmetries. MATH: - Plimpton 322 triples (a, b, c) satisfy \( a^2 + b^2 = c^2 \) with a < b. - Construction: choose regular sexagesimal numbers \( x \) and \( y \) such that \( x \cdot y = 60^n \) (reciprocal pair). Then: \( a = x - y \), \( b = 2\sqrt{xy} \), \( c = x + y \). (Equivalently, using \( p = \sqrt{x/y} \), \( q = \sqrt{y/x} \), the triple is \( (p^2 - q^2, 2pq, p^2 + q^2) \).) - The E₈ root lattice is generated by 240 roots in 8 dimensions, with Coxeter group order 696729600. - Connection: The reciprocal-pair method yields integer triples whose ratios (e.g., \( b/a \)) approximate golden-ratio-related values: 1.618, 0.618, 2.618 appear in the tablet's lines (e.g., line 1 gives \( b/a \approx 1.983 \), line Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com