Penrose Tilings Generalised to Cyclotomic Fields via CAST Inflation — E8 Intelligence Research

FINDING: Penrose tilings are aperiodic due to substitution rules with inflation multiplier φ², and a new class (CAST) generalises this to cyclotomic fields. | MATH: Inflation multiplier = φ² = ( (1+√5)/2 )² = (3+√5)/2 ≈ 2.618. Substitution matrix eigenvalues are algebraic integers in ℚ(√5). CAST tilings have vertices in ℚ(ζ₂ₙ), the 2n-th cyclotomic field, with minimal inflation multipliers often units in that field. | CONNECTION: φ² = 2.618 is the golden ratio squared, directly linking to pentagonal symmetry (5-fold, forbidden in periodic crystals). The cyclotomic field ℚ(ζ₁₀) = ℚ(√5, i) underlies Penrose tilings; CAST extends to other n, e.g., n=5 gives decagonal (10-fold) quasicrystal symmetry. Ratios 0.382 (φ⁻²), 0.618 (φ⁻¹), 1.618 (φ) appear in tile side lengths and inflation scaling. | DEPTH: 8 — Unifies aperiodic order with algebraic number theory, revealing that aperiodicity arises from irrational scaling in cyclotomic fields, a deep link between geometry and number theory Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com