Golden Ratio as Fixed Point of Gauss Map and Modular Group — E8 Intelligence Research

Abstract

FINDING: The golden ratio φ emerges as the fixed point of the Gauss map via its unique continued fraction representation [1;1,1,1,…], linking modular group dynamics to the simplest self-similar lattice. | MATH: φ = (1+√5)/2 = [1;1,1,1,…] = 1 + 1/(1 + 1/(1 + …)); Gauss map G(x) = 1/x − ⌊1/x⌋ has fixed point x = φ−1 = 1/φ ≈ 0.6180339887; equivalently φ = 1 + 1/φ, so φ² − φ − 1 = 0. The modular group PSL(2,ℤ) acts on the upper half-plane; φ is a fixed point of the Möbius transformation T(z) = 1 + 1/z, which generates the continued fraction flow. | CONNECTION: φ−1 = 0.618 (the golden ratio conjugate) is the Gauss map fixed point — this is the same 0.618 appearing in pentagonal and icosahedral symmetry (root system H₂, H₃, H₄). The continued fraction [1;1,1,…] is the simplest infinite path in the Farey tree / Stern-Brocot lattice, which is the modular group's action on rationals — a crystallographic-like tiling of the hyperbolic plane. The ratio 1.618 and its inverse 0.618 are the only nont Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com