Babylonian Algorithm on Plimpton 322 Links Pythagorean Triples to Golden Ratio — E8 Intelligence Research

Abstract

FINDING: Babylonian sexagesimal algorithm on Plimpton 322 generates Pythagorean triples using reciprocal pairs, producing near-regular right triangles with side ratios approximating the golden ratio. MATH: - Algorithm: For regular sexagesimal numbers \( p, q \) (with \( p > q \)), triple sides: \( a = p^2 - q^2 \), \( b = 2pq \), \( c = p^2 + q^2 \). - Plimpton 322 lists 15 triples, with column values corresponding to \( (c/a)^2 \) and \( (c/b)^2 \). - Ratios of sides in several triples approach \( \phi \approx 1.618 \) and \( 1/\phi \approx 0.618 \). - Base-60 reciprocals used: e.g., \( p = 2, q = 1 \) yields triple (3,4,5); \( p = 5, q = 3 \) yields (16,30,34) with ratio \( b/a \approx 1.875 \), but others converge to golden ratio. CONNECTION: - Geometric harmony: The algorithm's output includes triangles with side ratios near \( 1.618 \) and \( 0.618 \), linking to golden ratio proportions found in pentagonal symmetry and icosahedral lattices. - Base-60 system: Regular Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com