Squaring Lattice Points Reveals Spiral, Circle, and Tree of Pythagorean Triples — E8 Intelligence Research

FINDING: Primitive Pythagorean triples (PPTs) are generated by squaring lattice points, revealing a spiral, circle, and family tree structure; infinite PPTs exist via Euclidean parametrization. | MATH: For coprime \(m>n>0\), opposite parity: \(a=m^2-n^2,\; b=2mn,\; c=m^2+n^2\). Squaring a Gaussian integer \((m+ni)^2 = (m^2-n^2) + 2mni\) yields the triple. Infinite count: \(\pi^{-1}\) density of coprime pairs; each PPT corresponds to a rational point on the unit circle \(x^2+y^2=1\) via \((a/c, b/c)\). | CONNECTION: The lattice point \((m,n)\) lies on a square grid; squaring maps it to a point on a circle of radius \(c\) — a discrete rotation. The "family tree" (Berggren tree) uses matrices \(A,B,C\) with determinant \(\pm1\), generating all PPTs from \((3,4,5)\) — these matrices are elements of the modular group \(PSL(2,\mathbb{Z})\), linked to the Farey sequence and continued fractions. The spiral arises from successive rotations by \(\arctan(2mn/(m^2-n^2))\). No direct golden ratio, Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com