### Abstract

We associate the lepton–quark families with the vertices of the 4D polytopes 5-cell (Formula presented.) and the rectified 5-cell (Formula presented.) derived from the (Formula presented.) Coxeter–Dynkin diagram. The off-diagonal gauge bosons are associated with the root polytope (Formula presented.) whose facets are tetrahedra and the triangular prisms. The edge-vertex relations are interpreted as the (Formula presented.) charge conservation. The Dynkin diagram symmetry of the (Formula presented.) 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 presented.) 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 presented.) and even to (Formula presented.) by noting the Coxeter–Dynkin diagram embedding (Formula presented.). Another embedding can be made through the relation (Formula presented.) for more popular (Formula presented.). Appendix A includes the quaternionic representations of the Coxeter–Weyl groups (Formula presented.) which can be obtained directly from (Formula presented.) by projection. This leads to relations of the (Formula presented.) polytopes with the quasicrystallography in 4D and (Formula presented.) polytopes. Appendix B discusses the branching of the polytopes in terms of the irreducible representations of the Coxeter–Weyl group (Formula presented.).

Original language | English |
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Journal | International Journal of Geometric Methods in Modern Physics |

DOIs | |

Publication status | Accepted/In press - Dec 21 2017 |

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### Keywords

- (Formula presented.) GUT
- Coxeter groups
- Delone cells
- polytope
- Voronoi cell

### ASJC Scopus subject areas

- Physics and Astronomy (miscellaneous)

### Cite this

*International Journal of Geometric Methods in Modern Physics*. https://doi.org/10.1142/S0219887818500561