SU(5) grand unified theory, its polytopes and 5-fold symmetric aperiodic tiling

Abstract

We associate the lepton–quark families with the vertices of the 4D polytopes 5-cell [Formula: see text] and the rectified 5-cell [Formula: see text] derived from the [Formula: see text] Coxeter–Dynkin diagram. The off-diagonal gauge bosons are associated with the root polytope [Formula: see text] whose facets are tetrahedra and the triangular prisms. The edge-vertex relations are interpreted as the [Formula: see text] charge conservation. The Dynkin diagram symmetry of the [Formula: see text] diagram can be interpreted as a kind of particle-antiparticle symmetry. The Voronoi cell of the root lattice consists of the union of the polytopes [Formula: see text] whose facets are 20 rhombohedra. We construct the Delone (Delaunay) cells of the root lattice as the alternating 5-cell and the rectified 5-cell, a kind of dual to the Voronoi cell. The vertices of the Delone cells closest to the origin consist of the root vectors representing the gauge bosons. The faces of the rhombohedra project onto the Coxeter plane as thick and thin rhombs leading to Penrose-like tiling of the plane which can be used for the description of the 5-fold symmetric quasicrystallography. The model can be extended to [Formula: see text] and even to [Formula: see text] by noting the Coxeter–Dynkin diagram embedding [Formula: see text]. Another embedding can be made through the relation [Formula: see text] for more popular [Formula: see text]. Appendix A includes the quaternionic representations of the Coxeter–Weyl groups [Formula: see text] which can be obtained directly from [Formula: see text] by projection. This leads to relations of the [Formula: see text] polytopes with the quasicrystallography in 4D and [Formula: see text] polytopes. Appendix B discusses the branching of the polytopes in terms of the irreducible representations of the Coxeter–Weyl group [Formula: see text].