Plimpton 322: A Secant-Squared Table from Regular Reciprocal Pairs, Not Pythagorean Triples — E8 Intelligence Research

Abstract

FINDING: Plimpton 322 is a systematic table of secant-squared values in base-60, generated by a regular reciprocal algorithm, not a list of Pythagorean triples. | MATH: The tablet lists 15 entries of (column I: sec²θ = (d/b)², column II: b, column III: d) for decreasing angles θ. The generating algorithm uses pairs of regular sexagesimal reciprocals (p, q) with p > q, yielding b = p² - q², d = p² + q², and sec²θ = (d/b)² = (p²+q²)²/(p²-q²)². The values in column I are exact sexagesimal squares of secant, e.g., entry 1: sec²θ = 1.59.00.15 = 1 + 59/60 + 0/3600 + 15/216000 ≈ 1.9834, corresponding to θ ≈ 44.76°. | CONNECTION: No direct golden ratio φ (1.618) appears in the tablet's data. However, the base-60 system inherently uses regular numbers (2ᵃ3ᵇ5ᶜ) which generate the crystallographic symmetries of the hexagonal lattice (6-fold) and the cubic lattice (3-fold). The reciprocal pairs (p,q) are drawn from the set of regular sexagesimals, linking to the root system A₂ (hexagonal) and the Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com