Golden Ratio, Root Systems, and Non-Periodic Order in Substitution Tilings — E8 Intelligence Research

Abstract

FINDING: Fibonacci substitution tilings and cyclotomic aperiodic substitution tilings (CAST) reveal deep links between golden-ratio scaling, root-system geometry, and non-periodic order. | MATH: Fibonacci substitution rule: \(0 \to 01,\; 1 \to 0\) (or \(a \to ab,\; b \to a\)); inflation multiplier \(\lambda = \phi = (1+\sqrt{5})/2 \approx 1.618\); substitution matrix \(M = \begin{pmatrix}1&1\\1&0\end{pmatrix}\) with eigenvalues \(\phi\) and \(-1/\phi\). CAST: vertices in \(\mathbb{Z}[\zeta_{2n}]\) (2n-th cyclotomic field); inflation multipliers are algebraic integers with modulus \(>1\); substitution matrices have Perron–Frobenius eigenvalue equal to the inflation factor. | CONNECTION: Fibonacci tiling's Fourier spectrum is supported on \(\mathbb{Z}[\phi]\) — a rank-2 module over \(\mathbb{Z}\), with golden-ratio scaling \(1/\phi = \phi-1 \approx 0.618\). CAST generalizes this to cyclotomic fields, which contain the symmetry groups of regular n-gons (dihedral \(D_n\)) — directly linkin Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com