Plimpton 322: Old Babylonian Sexagesimal Trigonometry via Exact Ratios, Not Angles — E8 Intelligence Research

Abstract

FINDING: Old Babylonian Plimpton 322 encodes a sexagesimal trigonometric system based on exact ratios (Pythagorean triples), not angles, with a functional identity equivalent to sec²θ = 1 + tan²θ. | MATH: Sexagesimal normalized ratios: (b/l)² = 1 + (d/l)², where b = base, l = long side, d = diagonal. Plimpton 322 rows list (b/l)² values (column 1) with corresponding (d/l) and (b/l) pairs. Key constants: 1;59,15,0.15 (≈ 1.9834) and 1;24,30,10.6 (≈ 1.4077) from row 1. The identity emerges from reciprocal pairs: (b/l) and (d/l) are reciprocals in a normalized unit circle (l=1). | CONNECTION: Base-60 system directly enables exact rational approximations of geometric ratios. The reciprocal structure mirrors the golden ratio conjugate (0.618) and its inverse (1.618) in a discrete lattice: (√5-1)/2 ≈ 0.618, (√5+1)/2 ≈ 1.618. Plimpton 322's smallest triple (3,4,5) yields ratio 4/5 = 0.8, but higher rows approach 0.618 (e.g., row 15: 56/106 ≈ 0.528). The sexagesimal lattice generates all primit Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com