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 (2ᵃ·3ᵇ·5ᶜ) 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
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