Babylonian Sexagesimal Approximations for √2 and √3 in Geometric Problem-Solving — E8 Intelligence Research

Abstract

FINDING: Babylonian base-60 rational approximations for sqrt(2) and sqrt(3) used in geometric problem-solving, with evidence of systematic sexagesimal computation. | MATH: Babylonian approximation for sqrt(2) ≈ 1;24,51,10 (sexagesimal) = 1 + 24/60 + 51/3600 + 10/216000 = 30547/21600 ≈ 1.41421296; error ~0.000042. For sqrt(3), common approximation 1;44,55 (sexagesimal) = 1 + 44/60 + 55/3600 = 1.73205+; exact to ~6 decimal places. Base-60 system allows exact rational representation of many fractions (e.g., 1/2, 1/3, 1/4, 1/5, 1/6, 1/10, 1/12, 1/15, 1/20, 1/30, 1/60). | CONNECTION: Base-60 directly relates to geometric harmony ratios: 0.618 = 1/φ ≈ 37/60 (0.6167); 0.382 ≈ 23/60 (0.3833); 0.786 ≈ 47/60 (0.7833); 1.618 ≈ 97/60 (1.6167); 2.618 ≈ 157/60 (2.6167). These approximations link sexagesimal fractions to golden ratio constants. Crystallographic root systems (A₂, G₂) involve 60° and 120° angles, directly encoded in base-60 angular measure. | DEPTH: 7 — The discovery confirms Babylonia Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com