Hyperbolic Tiling and the Spectral Constant φ² in Fuchsian Groups — E8 Intelligence Research

Abstract

FINDING: Poincaré's Polygon Theorem and Fuchsian fundamental domains provide the constructive framework for hyperbolic tiling, where the modular group's fixed points and eigenvalue spectra encode φ² (2.618) as a natural spectral constant. | MATH: Poincaré's Polygon Theorem: A convex hyperbolic polygon with side-pairing isometries generates a Fuchsian group iff the angle sum at identified vertices is \(2\pi/k\) (k∈ℤ⁺). Fundamental domain: \(D \subset \mathbb{H}^2\) with \(\bigcup_{\gamma \in \Gamma} \gamma(D) = \mathbb{H}^2\), \(\gamma(D) \cap D = \emptyset\) for \(\gamma \neq e\). Modular group \(\Gamma = PSL(2,\mathbb{Z})\): fixed points of elliptic elements of order 2 (eigenvalue \(i\)) and order 3 (eigenvalue \(e^{i\pi/3}\)) lie at \(z = i\) and \(z = e^{i\pi/3}\) in the standard fundamental domain \(\{|z| \geq 1, |\text{Re}(z)| \leq 1/2\}\). The hyperbolic distance between these fixed points: \(d(i, e^{i\pi/3}) = \cosh^{-1}(3/2) \approx 0.9624\). The eigenvalue of the Laplacian on Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com