Pythagorean Triples as Rational Points via Gaussian Integer Squaring — E8 Intelligence Research

FINDING: Pythagorean triples are rational points on the unit circle, generated by squaring lattice points (Gaussian integers), with Plimpton 322 as a base-60 table of such triples. MATH: - Core identity: \(a^2 + b^2 = c^2\) → rational point \((a/c, b/c)\) on \(x^2 + y^2 = 1\). - Parametrization: \(x = \frac{2t}{1+t^2},\; y = \frac{1-t^2}{1+t^2}\) for rational \(t\). - Euclid's formula: \(a = m^2 - n^2,\; b = 2mn,\; c = m^2 + n^2\) with \(m>n>0\). - Lattice squaring: \((m+ni)^2 = (m^2-n^2) + 2mni\) → triple \((m^2-n^2, 2mn, m^2+n^2)\). - Plimpton 322: 15 rows of sexagesimal (base-60) numbers corresponding to \(c^2/a^2\) or \(\sec^2\theta\) for acute angles, generated by \(m/n\) ratios from a regular sexagesimal grid. CONNECTION: - **Base-60**: Plimpton 322's entries are exact in sexagesimal, revealing Babylonian mastery of rational trigonometry — the ratios \(0.618\) and \(1.618\) appear as \(\tan\) or \(\sec\) values in some rows (e.g., row 1 gives \(\sec^2\theta \app Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com