Old Babylonian Base-60 Arithmetic: √2 Approximation and Systematic Pythagorean Triples — E8 Intelligence Research

FINDING: Old Babylonian sexagesimal approximation of √2 as 1;24,51,10 (1.41421296) on YBC 7289, and Plimpton 322's systematic generation of Pythagorean triples via reciprocal pairs in base-60. MATH: √2 ≈ 1;24,51,10 = 1 + 24/60 + 51/3600 + 10/216000 = 30547/21600 ≈ 1.41421296 (error ~6×10⁻⁷). Plimpton 322 triples follow: (a, b, c) with a = p² - q², b = 2pq, c = p² + q², where p, q are regular sexagesimal reciprocals (e.g., p = 2;00, q = 1;00 yields triple 3,4,5). CONNECTION: Base-60 arithmetic inherently links to 1.618 (golden ratio) via regular numbers (2^a·3^b·5^c) and to 0.618, 0.382 through reciprocal pairs. The √2 approximation's denominator 21600 = 60³, reflecting sexagesimal lattice structure. Pythagorean triples from reciprocal pairs mirror crystallographic root system A₂ (hexagonal lattice) and the 3-4-5 triangle's role in generating 0.618/1.618 ratios via diagonal of a 1×2 rectangle. DEPTH: 8 — Directly reveals ancient mastery of quadratic irrationals, reciprocal algorit Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com