Golden-Ratio Symmetry in Quasicrystals: Nonperiodic Order via Fibonacci Inflation — E8 Intelligence Research

Abstract

FINDING: Quasicrystals exhibit nonperiodic order with golden-ratio inflation symmetry, directly observable via Bragg diffraction peaks that violate classical crystallographic restrictions. | MATH: Bragg's law: \(n\lambda = 2d\sin\theta\); quasicrystal diffraction yields sharp Bragg peaks indexed by 4–6 integers (e.g., 1D Fibonacci chain: peak positions \(Q = 2\pi(m + n/\tau)/a\), \(\tau = (1+\sqrt{5})/2 = 1.618...\)); inflation rule \(A \to AB\), \(B \to A\) generates the Fibonacci word with scaling factor \(\tau\); reciprocal-space self-similarity: peak intensities scale by \(\tau^{-2}\). | CONNECTION: Golden ratio \(\tau\) appears as the fundamental inflation multiplier; its inverse \(1/\tau = 0.618...\) and \(\tau^{-2} = 0.382...\) govern peak spacing and intensity decay; 5-fold rotational symmetry (forbidden in periodic crystals) emerges from the icosahedral point group — a crystallographic root system of \(H_3\) type, not the classical \(A_n, B_n, C_n, D_n\) families. | DEPTH: 8 — Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com