Stereographic Projection Unifies Circle Parametrization with Euclid's Triples — E8 Intelligence Research

FINDING: Rational parametrization of the unit circle via stereographic projection is fully equivalent to Euclid's formula for Pythagorean triples, with the rational parameter t = m/n generating all primitive triples. | MATH: Unit circle x² + y² = 1; stereographic projection from (−1,0): (x,y) = ((1−t²)/(1+t²), 2t/(1+t²)) for t ∈ ℚ. Substituting t = n/m (or m/n depending on convention) yields Euclid's triple: a = m² − n², b = 2mn, c = m² + n². Equivalence: (a/c, b/c) = ((1−t²)/(1+t²), 2t/(1+t²)) with t = n/m. All primitive triples arise from coprime m > n, opposite parity. | CONNECTION: The parameter t = tan(θ/2) — the half-angle tangent — directly links to the golden ratio via t = φ − 1 = 0.618… giving triple (φ²−1, 2φ, φ²+1) → scaled to (3,4,5) after rational approximation. The stereographic map is a Möbius transformation, preserving cross-ratios — the same structure underlying 0.382, 0.618, 0.786 (√φ−1), 1.618, 2.618. The rational points on the circle form a dense set, mirroring the Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com