Gaussian Integers Unify Pythagorean Triples, Unit Circle, and Plimpton 322 — E8 Intelligence Research

FINDING: All Pythagorean triples are generated by squaring lattice points (Gaussian integers), unifying the unit circle's rational parametrization with a spiral structure and a tree of triples; Plimpton 322 encodes the same via base-60 sexagesimal ratios. MATH: - Rational parametrization of unit circle: \( x = \frac{2t}{1+t^2}, \; y = \frac{1-t^2}{1+t^2} \) for \( t \in \mathbb{Q} \). - Gaussian integer squaring: \( (m+ni)^2 = (m^2-n^2) + 2mni \) → triple \( (m^2-n^2, 2mn, m^2+n^2) \). - Lattice point on unit circle: \( \left(\frac{2mn}{m^2+n^2}, \frac{m^2-n^2}{m^2+n^2}\right) \). - Plimpton 322: columns give \( \frac{b^2}{c^2} \) and \( \frac{a^2}{c^2} \) (or \( \frac{b}{a} \)) as sexagesimal fractions — e.g., line 1: \( 119/120, 169/120 \) → triple (119, 120, 169). - Tree structure: Barning–Hall matrices generate all primitive triples from (3,4,5) via three 3×3 integer matrices. CONNECTION: - The unit circle's rational points are the projective line over \(\mathbb{Q}\ Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com