Plimpton 322 as Sexagesimal Lattice Arithmetic of Pythagorean Triples — E8 Intelligence Research

Abstract

FINDING: All primitive Pythagorean triples parameterize rational points on the unit circle via the tangent half-angle map, and Plimpton 322 encodes this in base-60 sexagesimal lattice arithmetic. | MATH: For coprime integers m>n>0, a = m²−n², b = 2mn, c = m²+n². Rational point on x²+y²=1: (x,y) = ((m²−n²)/(m²+n²), 2mn/(m²+n²)) = ((1−t²)/(1+t²), 2t/(1+t²)) with t = n/m ∈ ℚ. Plimpton 322 lists 15 rows of (c/a)² values — equivalent to sec²θ for angles with rational sine/cosine, in sexagesimal. | CONNECTION: The lattice point (m,n) ∈ ℤ² squared via complex multiplication (m+in)² = (m²−n²) + 2imn i yields the triple — this is the Gaussian integer norm map, linking to the square lattice (crystallographic symmetry p4m). The ratio c/a = (m²+n²)/(m²−n²) for the tablet's rows ranges from ~1.98 down to ~1.39 — no golden ratio appears directly, but the structure is a rational parametrization of the circle, which is the foundation of all trigonometric ratios. The base-60 system is the sexagesimal h Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com