Cyclotomic Aperiodic Substitution Tilings Unify Finite Rotational Symmetries — E8 Intelligence Research

FINDING: Cyclotomic Aperiodic Substitution Tilings (CAST) define a class of substitution tilings whose vertices lie in the 2n-th cyclotomic field, unifying many known aperiodic tilings with finite rotational symmetry. | MATH: Vertices in \\(\\mathbb{Z}[\\zeta_{2n}]\\), \\(\\zeta_{2n}=e^{i\\pi/n}\\). Substitution matrix \\(M\\) has Perron–Frobenius eigenvalue \\(\\lambda\\) (inflation multiplier) which is a unit in \\(\\mathbb{Z}[\\zeta_{2n}]\\). Vertex density \\(\\rho = \\lim_{k\\to\\infty} N_k / \\text{Area}_k\\) is given by the Perron eigenvector components normalized by the area of the inflated tile. Minimal inflation multipliers are algebraic integers of degree \\(\\varphi(2n)/2\\) (real subfield). | CONNECTION: For \\(n=3\\) (hexagonal), \\(\\zeta_6 = e^{i\\pi/3}\\), the real subfield \\(\\mathbb{Z}[\\zeta_6+\\zeta_6^{-1}] = \\mathbb{Z}[\\sqrt{3}]\\) — inflation multipliers include \\(1+\\sqrt{3} \\approx 2.732\\) and \\(2+\\sqrt{3} \\approx 3.732\\). For \\(n=5\\) (decagonal), real subfield \\(\\mathbb{Z}[\\sqrt{5}]\\) yields golde Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com