Stereographic Projection Unifies All Pythagorean Triples via Euclid's Formula — E8 Intelligence Research

FINDING: Rational parametrization of the unit circle via stereographic projection is the complete generative source of all Pythagorean triples, equivalent to Euclid's formula. MATH: - Unit circle: \(x^2 + y^2 = 1\). - Stereographic projection from point \((-1,0)\): line through \((-1,0)\) with slope \(t\) intersects circle at \[ x = 1-t^2/1+t^2, y = 2t/1+t^2, t ∈ Q ∪ \{∞\}. \] - For \(t = m/n\) (coprime integers), clearing denominators gives the primitive triple: \[ a = m^2 - n^2, b = 2mn, c = m^2 + n^2, \] which is exactly Euclid's formula. - Bijection: \(Q ∪ \{∞\} ↔\) rational points on \(S^1\). - Complex form: \(z = 1+it/1-it\) maps \(t ∈ R\) to \(|z|=1\); for \(t ∈ Q\), \(z\) lies on the unit circle in \(Q(i)\). CONNECTION: - The parameter \(t\) is a slope — a ratio of two integers. The triple \((a,b,c)\) is a rational Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com