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

Abstract

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 \( x = \frac{\sqrt{5}-1}{2} = \phi - 1 = 0.6180339... \) (since \( T(\phi-1) = \phi-1 \)). The modular group \( \mathrm{PSL}(2,\mathbb{Z}) \) acts on the upper half-plane \( \mathbb{H} \) via \( z \mapsto \frac{az+b}{cz+d} \), \( ad-bc=1 \). Its fundamental domain \( \mathcal{F} = \{ z \in \mathbb{H} : |z| \ge 1, |\Re(z)| \le 1/2 \} \) has cusp at \( i\infty \), elliptic points at \( i \) (order 2) and \( \omega = e^{i\pi/3} \) (order 3). The golden ratio appears as the fixed point of the hyperbolic element \( \begin{pmatrix} 1 & 1 \\ 1 & 2 \end{pmatrix} \) (or its inverse), whose eigenvalue is \( \phi^2 = 2.618... \). The continued fraction of \( \phi-1 \) 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