Cyclotomic Aperiodic Substitution Tilings Unify All Finite Rotation Symmetries — E8 Intelligence Research

Abstract

FINDING: Aperiodic tilings with 5-fold symmetry (Penrose tilings) are now understood as a subset of a broader class—cyclotomic aperiodic substitution tilings (CAST)—which support all finite rotation symmetries and are defined on cyclotomic fields. | MATH: Vertices lie in the 2n-th cyclotomic field ℚ(ζ₂ₙ); substitution matrices and minimal inflation multipliers are derived from algebraic integers in that field. The golden ratio φ = (1+√5)/2 ≈ 1.618 appears as the inflation multiplier for 5-fold tilings. | CONNECTION: 5-fold symmetry directly involves φ and its powers (φ² ≈ 2.618, φ⁻¹ ≈ 0.618, φ⁻² ≈ 0.382). The cyclotomic field ℚ(ζ₁₀) = ℚ(√5) underpins the geometry. Base-60 is not directly implicated, but the algebraic number field structure echoes the harmonic ratios found in crystallographic root systems (e.g., H₂, H₃, H₄). | DEPTH: 8 — Unifies quasiperiodic tilings under a single algebraic framework, linking aperiodic order to cyclotomic fields and revealing that 5-fold symmetry is no Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com