Pisot Substitutions Bridge Aperiodic Order and Hyperbolic Geometry via Cyclotomic Tilings — E8 Intelligence Research

FINDING: Hyperbolic tilings via Pisot substitutions (Jacobi-Perron) connect aperiodic order to non-Euclidean geometry; CAST tilings formalize cyclotomic substitution systems with algebraic inflation multipliers. | MATH: Pisot units (e.g., 1.618, 2.618) as inflation multipliers; Jacobi-Perron algorithm generates simultaneous Diophantine approximations for algebraic numbers; CAST vertices in ℚ(ζ₂ₙ) (2n-th cyclotomic field); substitution matrix eigenvalues = Pisot numbers; minimal inflation multipliers are algebraic integers >1 with all conjugates <1. | CONNECTION: Pisot numbers 1.618 (φ), 2.618 (φ²), 0.618 (1/φ) are the exact inflation ratios for self-similar tilings; cyclotomic fields encode 2n-fold rotational symmetry (crystallographic restriction: n=1,2,3,4,6 in Euclidean plane, but hyperbolic allows n≥5); hyperbolic tilings admit regular polygons with any n, linking to root systems Aₙ, Dₙ, E₆,₇,₈ via Coxeter groups. | DEPTH: 8 — This unifies three deep structures: (1) Pisot numbers a Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com