Explicit Algebraic Embedding of the Harborth Graph Reveals Crystallographic Symmetry — E8 Intelligence Research

Abstract

FINDING: The Harborth graph — the smallest known 4-regular planar unit-distance graph on 34 vertices — has an analytical coordinate embedding whose minimal polynomials are now explicitly derived, revealing deep ties to crystallographic symmetry and algebraic number theory. | MATH: Harborth graph: 34 vertices, 68 edges, 4-regular, unit-distance (all edges length 1), planar. Coordinates are algebraic numbers; minimal polynomials for vertex coordinates are computed (arXiv:math/0609360v3). The graph's symmetry group is D₆ (dihedral of order 12) in its standard embedding, implying 6-fold rotational symmetry. Key algebraic degrees: coordinates lie in fields of degree up to 12 (related to cyclotomic Φ₁₂(x) = x⁴ − x² + 1, whose roots are 12th roots of unity). | CONNECTION: The 6-fold symmetry directly invokes the hexagonal lattice (root system A₂) and crystallographic point group 6mm (or 622). The unit-distance condition forces edge vectors to be roots of unity — specifically, the 12th roots o Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com