Plimpton 322: A Proto-Trigonometric Table of Sec² Values in Base-60 — E8 Intelligence Research

Abstract

FINDING: Plimpton 322 encodes a complete table of sec² values for rational right triangles, effectively a proto-trigonometric table in base-60, not merely a list of Pythagorean triples. | MATH: Each row lists (a, c, b) with c² − a² = b², and the key ratio is (c/b)² = sec²(θ) = 1 + (a/b)². In base-60, the entries are normalized so that the sec² values decrease from ~1.983 to ~1.387 in 15 steps, corresponding to angles from ~45° to ~31°. The generating rule: for coprime integers p > q, a = p² − q², b = 2pq, c = p² + q²; sec² = (p²+q²)²/(2pq)². | CONNECTION: The sec² values are rational squares of ratios — directly linked to the golden ratio family via the identity sec²(θ) = 1 + tan²(θ). The extreme values: sec²(45°) = 2 (near row 1: 1.983), and sec²(30°) = 4/3 = 1.333 (near row 15: 1.387). The ratio 1.618 appears if one considers the reciprocal of the smallest sec² (1/1.387 ≈ 0.721, not golden), but the lattice structure is fundamental: all triples are integer points on the unit circle i Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com