Cyclotomic Aperiodic Substitution Tilings Unify Non-Repeating Patterns — E8 Intelligence Research

FINDING: Cyclotomic Aperiodic Substitution Tilings (CAST) unify a broad class of non-repeating tilings whose vertices lie in the 2n-th cyclotomic field, with minimal inflation multipliers tied to algebraic integers. | MATH: Vertices in ℚ(ζ₂ₙ), ζ₂ₙ = e^{2πi/2n}; substitution matrix eigenvalues (Perron–Frobenius) give inflation multipliers λ; minimal λ are algebraic integers in ℤ[ζ₂ₙ]. For Penrose (n=5), λ = φ² = (1+√5)/2 ≈ 2.618; for Ammann–Beenker (n=8), λ = 1+√2 ≈ 2.414; for Hat/Einstein tile (n=6? — actually n=3? — the hat uses √3, so λ = 1+√3 ≈ 2.732 or related). | CONNECTION: Directly encodes golden ratio φ = 1.618, its square 2.618, and silver ratio 1+√2; also 0.618 = φ−1 appears as inverse inflation. Cyclotomic fields ℚ(ζ₂ₙ) contain real subfields ℚ(ζ₂ₙ+ζ₂ₙ⁻¹) = ℚ(cos(π/n)), whose units include 2cos(π/n) — e.g., n=5 gives φ, n=8 gives √2+1, n=12 gives 2+√3. These are exactly the minimal inflation multipliers. Crystallographic restriction: only n = 1,2,3,4,6 allow periodic tilings Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com