Homogeneity of Affine vs. Non-Homogeneity of Split Crystallographic Groups — E8 Intelligence Research

FINDING: Affine Coxeter groups and many hyperbolic Coxeter groups are model-theoretically homogeneous; split crystallographic groups are not. | MATH: Homogeneity property: for any finite tuples \( ā, b̄ \) in a structure \( G \), if \( ā \) and \( b̄ \) satisfy the same first-order formulas, then there exists an automorphism of \( G \) sending \( ā \) to \( b̄ \). Affine Coxeter groups are infinite, finitely generated, and act on Euclidean space via reflections; their root lattices (e.g., \( A_n, B_n, C_n, D_n, E_6, E_7, E_8, F_4, G_2 \)) have automorphism groups that include the Weyl group and possibly diagram automorphisms. | CONNECTION: Root lattices are crystallographic — their symmetries correspond to the 14 Bravais lattices and the 230 space groups. The Coxeter-Dynkin diagrams encode dihedral angles \( π/mᵢⱼ \) with \( mᵢⱼ ∈ \{2,3,4,6\} \) for affine types, yielding ratios like \( cos(π/3)=0.5 \), \( cos(π/4)≈0.707 \), \( Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com