Plimpton 322: Ancient Base-60 Tablet Encodes Exact Rational Trigonometry — E8 Intelligence Research

Abstract

FINDING: Plimpton 322 encodes exact rational parametrization of the circle using base-60 sexagesimal numbers, providing a complete trigonometric table without approximation. MATH: - Pythagorean triples: \(a^2 + b^2 = c^2\) with \(a < b\). - Rational parametrization: \( (p^2 - q^2, 2pq, p^2 + q^2) \) for integers \(p > q > 0\). - Base-60 representation yields exact fractions (e.g., 119/169, 119/120) avoiding irrationals. - Key constants: 1;59,15 = 1.9875 (approx 2), 0;59,00 = 0.9833, 0;30 = 0.5. - Ratios on tablet correspond to \(\tan^2(\theta)\) or \(\sec^2(\theta)\) for angles from ~45° to ~30°. CONNECTION: - Base-60 naturally generates exact rational approximations to golden ratio \(\phi = 1.618...\) via 1;36,36 = 1.61, and to \(\sqrt{2}\) via 1;24,51,10 = 1.41421356. - The 15 rows correspond to 15 distinct angles, each with exact rational sine/cosine, mirroring the 15° increments in later Babylonian astronomy. - The structure hints at a sexagesimal lattice of ratio Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com