Rational Points on the Unit Circle and Primitive Pythagorean Triples via Stereographic Projection — E8 Intelligence Research

FINDING: Rational points on the unit circle are in bijection with primitive Pythagorean triples via stereographic projection from (−1, 0), and the parametrization is exactly the tangent half-angle substitution. MATH: - Unit circle: x² + y² = 1. - Stereographic projection from anchor P₀ = (−1, 0) through a rational slope t: x = (1 − t²)/(1 + t²), y = 2t/(1 + t²). - Inverse: t = y/(x + 1) = tan(θ/2). - Primitive triple generation (Euclid): for coprime m > n, opposite parity: a = m² − n², b = 2mn, c = m² + n², with t = n/m. - Bijection: ℚ ∪ {∞} ↔ rational points on S¹ ↔ primitive triples (up to scaling). - Group structure: rational points on S¹ form a group under angle addition; the map t ↦ angle is the group isomorphism ℚ ∪ {∞} ≅ rational rotations (mod 2π). - Base-60 anchor: Plimpton 322 (c. 1800 BCE) tabulates 15 rows of (short side, diagonal) satisfying a² + b² = c², consistent with the m, n parametrization; sexagesimal reciprocals (regular numbers, 60-smooth) are the arithme Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com