Cyclotomic Aperiodic Substitution Tilings Unify Finite-Rotational Nonperiodic Systems — E8 Intelligence Research

FINDING: Cyclotomic Aperiodic Substitution Tilings (CAST) unify a broad class of non-periodic tilings with finite rotational symmetry, supported on 2n-th cyclotomic fields, with explicit substitution matrices and minimal inflation multipliers. | MATH: Vertices lie in ℤ[ζ₂ₙ] where ζ₂ₙ = e^(2πi/2n). Inflation multiplier λ is an algebraic integer in the cyclotomic field; minimal λ often relates to the norm of (1−ζ₂ₙ) or its real parts. For n=5 (decagonal), λ = τ² = φ² = 2.618… (where φ = (1+√5)/2 = 1.618…). For n=3 (hexagonal), λ = 2 (or √3 in some variants). Substitution matrix M has Perron–Frobenius eigenvalue = λ² (area scaling). | CONNECTION: Directly encodes 5-fold (n=5) and 3-fold (n=3) crystallographic-forbidden symmetries via algebraic integers. The ratio 1/φ = 0.618… and 1/φ² = 0.382… appear as edge-length ratios in the substitution rules. The minimal inflation multiplier for n=5 is φ² = 2.618, and its reciprocal 0.382 is the scaling factor of the smaller tile. Base-60 connection Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com