CAST Tilings: Cyclotomic Fields Link Aperiodic Order to Algebraic Integers — E8 Intelligence Research

FINDING: CAST tilings are substitution tilings with vertices in the 2n-th cyclotomic field, and their substitution matrices have eigenvalues that are algebraic integers in that field, linking aperiodic order to cyclotomic number theory. MATH: The 2n-th cyclotomic field is ℚ(ζ₂ₙ) where ζ₂ₙ = eπi/n. Substitution matrix eigenvalues are algebraic integers in this field. For n=5 (decagonal tilings), the minimal inflation multiplier is τ² = φ² = (3+√5)/2 ≈ 2.618, where φ = (1+√5)/2 ≈ 1.618. The characteristic polynomial of the substitution matrix often factors over ℚ(ζ₂ₙ), yielding eigenvalues like 1, τ, τ², and their Galois conjugates (e.g., 1-τ ≈ -0.618, 1/τ ≈ 0.618). For n=3 (hexagonal), eigenvalues involve √3; for n=4 (octagonal), eigenvalues involve √2. CONNECTION: The eigenvalues directly yield the golden ratio φ and its powers (1.618, 2.618) and reciprocals (0.618, 0.382) for n=5, which are the key ratios in Penrose tilings and quasicrystal diffraction. The 2n-th cycloto Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com