Spiral and Tree Structure in Pythagorean Triples via Stereographic Projection — E8 Intelligence Research

Abstract

FINDING: All primitive Pythagorean triples parameterize rational points on the unit circle via a stereographic projection from a lattice point, revealing a hidden spiral and tree structure. MATH: - Core identity: \(a^2 + b^2 = c^2\) ⇔ \((a/c)^2 + (b/c)^2 = 1\) → rational point \((x,y)\) on unit circle. - Euclid's formula: \(a = m^2 - n^2,\; b = 2mn,\; c = m^2 + n^2\) with \(m>n>0\), coprime, opposite parity. - Rational parametrization: \(x = \frac{2t}{1+t^2},\; y = \frac{1-t^2}{1+t^2}\) for \(t \in \mathbb{Q}\) (stereographic projection from \((-1,0)\)). - Lattice squaring: \((m+in)^2 = (m^2 - n^2) + i(2mn)\) → complex square maps lattice \(\mathbb{Z}[i]\) to Pythagorean triples. - Spiral/tree: Berggren's tree — three linear transformations generate all primitive triples from \((3,4,5)\). CONNECTION: - The unit circle is the compact form of the real line via stereographic projection — a Möbius transformation. - The lattice \(\mathbb{Z}[i]\) (Gaussian integers) is a 2D Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com