Babylonian Geometric Proof of √2 Irrationality Predates Greek Methods — E8 Intelligence Research

FINDING: Babylonian geometric proof of √2 irrationality via recursive diagonal algorithm, predating Greek methods. MATH: √2 = 1;24,51,10 in base-60 (1.41421296...). Recursive algorithm: start with side s, diagonal d ≈ s·1;25; iterate d' = d + 2s, s' = d + s (Pell-type recurrence). Proof by infinite descent: if √2 = p/q, then p² = 2q² → contradiction via geometric scaling of a unit square's diagonal. CONNECTION: Ratio 1.414 (≈ √2) links to silver ratio δ_s = 1 + √2 ≈ 2.414, and to base-60 sexagesimal fractions (1;24,51,10). The recursive algorithm generates convergents 1/1, 3/2, 7/5, 17/12, 41/29, … whose ratios approach √2; these appear in Babylonian tablet YBC 7289. Geometric harmony: the diagonal of a unit square (√2) is the edge of a square of area 2, linking to the doubling of area (Plato's Meno). No direct 0.382/0.618/0.786/1.618/2.618 ratios, but the Pell recurrence is isomorphic to the silver mean (1+√2) and its reciprocal (√2-1 ≈ 0.414). DEPTH: 8 — Profound because it sho Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com