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

FINDING: The Gauss map's fixed point in the modular group PSL(2,Z) is the golden ratio, which generates the fundamental domain's hyperbolic tiling and encodes the Farey/Stern-Brocot tree. | MATH: The Gauss map \( T(x) = \{1/x\} \) (fractional part) has fixed point \( x = {√5-1}{2} = φ - 1 = 0.6180339... \) (since \( T(φ-1) = φ-1 \)). The modular group \( PSL(2,Z) \) acts on the upper half-plane \( H \) via \( z ↦ az+b/cz+d \), \( ad-bc=1 \). Its fundamental domain \( F = \{ z ∈ H : |z| ≥ 1, |(z)| ≤ 1/2 \} \) has cusp at \( i∞ \), elliptic points at \( i \) (order 2) and \( ω = eiπ/3 \) (order 3). The golden ratio appears as the fixed point of the hyperbolic element \( {pmatrix} 1 & 1 \\ 1 & 2 {pmatrix} \) (or its inverse), whose eigenvalue is \( φ^2 = 2.618... \). The continued fraction of \( φ-1 \) is \( [0;1,1,1,...] \), the unique purely periodic continued fraction Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com