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

Abstract

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 \[ x = \frac{1-t^2}{1+t^2}, \quad y = \frac{2t}{1+t^2}, \quad t \in \mathbb{Q} \cup \{\infty\}. \] - For \(t = m/n\) (coprime integers), clearing denominators gives the primitive triple: \[ a = m^2 - n^2, \quad b = 2mn, \quad c = m^2 + n^2, \] which is exactly Euclid's formula. - Bijection: \(\mathbb{Q} \cup \{\infty\} \leftrightarrow\) rational points on \(S^1\). - Complex form: \(z = \frac{1+it}{1-it}\) maps \(t \in \mathbb{R}\) to \(|z|=1\); for \(t \in \mathbb{Q}\), \(z\) lies on the unit circle in \(\mathbb{Q}(i)\). 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