Cyclotomic Aperiodic Substitution Tilings with Pisot Inflation Multipliers — E8 Intelligence Research

FINDING: Cyclotomic Aperiodic Substitution Tilings (CAST) define a class of aperiodic tilings with vertices supported on the 2n-th cyclotomic field ℚ(ζ₂ₙ), where the minimal inflation multiplier is a Pisot number derived from the minimal polynomial of the field. MATH: - Field: ℚ(ζ₂ₙ) = ℚ(e2πi/(2n)) - Minimal polynomial: cyclotomic polynomial Φ₂ₙ(x) - Inflation multiplier λ: a Pisot number (algebraic integer >1, all conjugates <1 in modulus) satisfying Φ₂ₙ(λ) = 0 or a factor thereof. - For n=5 (decagonal): λ = τ² = ( (1+√5)/2 )² = (3+√5)/2 ≈ 2.618, minimal polynomial x² - 3x + 1 = 0. - For n=3 (hexagonal): λ = 2 + √3 ≈ 3.732, minimal polynomial x² - 4x + 1 = 0. - Substitution matrix eigenvalues are Pisot numbers; the tiling's diffraction spectrum is pure point. CONNECTION: - Golden ratio φ = (1+√5)/2 ≈ 1.618 appears directly in decagonal CAST (n=5): λ = φ² = 2.618, and its reciprocal φ⁻² ≈ 0.382. - The ratio 0.618 = φ⁻¹ arises as a conjugate. - Base-60 Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com