SU(5) Grand Unified Theory, its Polytopes and 5-fold Symmetric Aperiodic Tiling

TL;DR

This paper explores the geometric and algebraic structures underlying SU(5) Grand Unified Theory, linking polytopes, lattice theory, and quasicrystal tilings to particle symmetries and gauge bosons.

Contribution

It introduces a novel geometric model connecting SU(5) GUT with polytopes, lattice structures, and Penrose-like tilings, providing new insights into particle symmetries and quasicrystallography.

Findings

Vertices of polytopes represent lepton-quark families.

Rhombohedral facets lead to Penrose-like tilings.

Model extends to SO(10) and SO(11) theories.

Abstract

We associate the lepton-quark families with the vertices of the 4D polytopes 5-cell and the rectified 5-cell derived from the SU(5) Coxeter-Dynkin diagram. The off-diagonal gauge bosons are associated with the root poytope (1000)A4 whose facets are tetrahedra and the triangular prisms. The edge-vertex relations are interpreted as the SU(5) charge conservation. The Dynkin diagram symmetry of the SU(5) 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 (1000)A4 + (0100)A4 + (0010)A4 + (0001)A4 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 consists 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 SO(10) and even to SO(11) by noting the Coxeter-Dynkin diagram embedding in A4 in D5 in B5. Another embedding can be made through the relation A4 in D5 in E6 for more popular GUT's.