Cyclotomic Aperiodic Substitution Tilings Unify Penrose Tilings via Pisot Numbers — E8 Intelligence Research

FINDING: Cyclotomic Aperiodic Substitution Tilings (CAST) unify Penrose-like tilings under algebraic integers of cyclotomic fields, with substitution eigenvalues tied to Pisot numbers. | MATH: Vertices lie in ℚ(ζ_{2n}), the 2n-th cyclotomic field. Substitution matrix eigenvalues are algebraic integers (often Pisot numbers). Minimal inflation multiplier is a unit in the ring ℤ[ζ_{2n}]. For n=5 (Penrose), inflation multiplier = τ² = φ² ≈ 2.618, eigenvalue = τ = φ ≈ 1.618. | CONNECTION: Golden ratio φ = (1+√5)/2 ≈ 1.618 appears as eigenvalue for n=5; its reciprocal φ⁻¹ ≈ 0.618. For n=7, 12-fold symmetry yields silver ratio 1+√2 ≈ 2.414 and its reciprocal ≈ 0.414. These are Pisot numbers, key to self-similarity and diffraction. Base-60 not directly present, but cyclotomic fields relate to regular polygons and crystallographic restrictions. | DEPTH: 8 — Bridges number theory (cyclotomic fields, Pisot numbers) with quasicrystal diffraction and aperiodic order, revealing deep algebraic constr Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com