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

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].

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Explicit construction of the Voronoi and Delaunay cells ofW(An) andW(Dn) lattices and their facets

Acta Crystallographica Section A Foundations and Advances ◽

10.1107/s2053273318007842 ◽

2018 ◽

Vol 74 (5) ◽

pp. 499-511 ◽

Cited By ~ 2

Author(s):

Mehmet Koca ◽

Nazife Ozdes Koca ◽

Abeer Al-Siyabi ◽

Ramazan Koc

Keyword(s):

Voronoi Cell ◽

Explicit Construction ◽

Weyl Groups ◽

Root Lattice ◽

Coxeter Number ◽

Aperiodic Tiling ◽

Higher Dimensional ◽

The Face ◽

Coxeter Plane ◽

Root Polytope

Voronoi and Delaunay (Delone) cells of the root and weight lattices of the Coxeter–Weyl groupsW(An) andW(Dn) are constructed. The face-centred cubic (f.c.c.) and body-centred cubic (b.c.c.) lattices are obtained in this context. Basic definitions are introduced such as parallelotope, fundamental simplex, contact polytope, root polytope, Voronoi cell, Delone cell,n-simplex,n-octahedron (cross polytope),n-cube andn-hemicube and their volumes are calculated. The Voronoi cell of the root lattice is constructed as the dual of the root polytope which turns out to be the union of Delone cells. It is shown that the Delone cells centred at the origin of the root latticeAnare the polytopes of the fundamental weights ω1, ω2,…, ωnand the Delone cells of the root latticeDnare the polytopes obtained from the weights ω1, ωn−1and ωn. A simple mechanism explains the tessellation of the root lattice by Delone cells. It is proved that the (n−1)-facet of the Voronoi cell of the root latticeAnis an (n−1)-dimensional rhombohedron and similarly the (n−1)-facet of the Voronoi cell of the root latticeDnis a dipyramid with a base of an (n−2)-cube. The volume of the Voronoi cell is calculatedviaits (n−1)-facet which in turn can be obtained from the fundamental simplex. Tessellations of the root lattice with the Voronoi and Delone cells are explained by giving examples from lower dimensions. Similar considerations are also worked out for the weight latticesAn* andDn*. It is pointed out that the projection of the higher-dimensional root and weight lattices on the Coxeter plane leads to theh-fold aperiodic tiling, wherehis the Coxeter number of the Coxeter–Weyl group. Tiles of the Coxeter plane can be obtained by projection of the two-dimensional faces of the Voronoi or Delone cells. Examples are given such as the Penrose-like fivefold symmetric tessellation by theA4root lattice and the eightfold symmetric tessellation by theD5root lattice.

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‘Grand unified theory of maths’ nets Abel Prize

Nature ◽

10.1038/d41586-018-03423-x ◽

2018 ◽

Author(s):

Davide Castelvecchi

Keyword(s):

Unified Theory ◽

Grand Unified Theory

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The Grand Unified Theory Methods of Contract Interpretation and Result Oriented Surplus Maximization

SSRN Electronic Journal ◽

10.2139/ssrn.2493484 ◽

2014 ◽

Author(s):

Kyle Chen

Keyword(s):

Unified Theory ◽

Grand Unified Theory

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Deep in the Bones Lie Memories and Hopes: A Grand Unified Theory

Studia Liturgica ◽

10.1177/0039320720981017 ◽

2021 ◽

Vol 51 (1) ◽

pp. 22-30

Author(s):

Daniel P. McCarthy

Keyword(s):

Unified Theory ◽

Grand Unified Theory ◽

Hagia Sophia ◽

The Sacrifice

Christ’s bones are missing at the Holy Sepulchre; St Peter’s bones remain in his basilica; Hagia Sophia was not built on bones. The absence, presence, or lack of bones effects different emphases on memory (anamnesis) and fulfillment (eschatology). In Jerusalem we witness our future glory (eschatology) already revealed in our history (anamnesis); in Rome we recall (anamnesis) the sacrifice of martyrs whose bones remain until the general resurrection (eschatology), even while we venerate the saints in light; at Hagia Sophia liturgy itself, rather than bones, provides the context for remembering the whole Christ in the power of the Spirit. Celebrating liturgy over the bones of martyrs in Rome, while venerating their sacrifice, may have accentuated the sacrificial character of the eucharistic liturgy in the Christian west, whereas in the Christian east the eschatological glory already revealed in our history and in liturgy may have shaped the eschatological character of liturgy.

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