Quaternionic Representation of Snub 24-Cell and Its Dual Polytope
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Uniform Polychora
BRIDGES Mathematical Connections in Art, Music, and Science Uniform Polychora Jonathan Bowers 11448 Lori Ln Tyler, TX 75709 E-mail: [email protected] Abstract Like polyhedra, polychora are beautiful aesthetic structures - with one difference - polychora are four dimensional. Although they are beyond human comprehension to visualize, one can look at various projections or cross sections which are three dimensional and usually very intricate, these make outstanding pieces of art both in model form or in computer graphics. Polygons and polyhedra have been known since ancient times, but little study has gone into the next dimension - until recently. Definitions A polychoron is basically a four dimensional "polyhedron" in the same since that a polyhedron is a three dimensional "polygon". To be more precise - a polychoron is a 4-dimensional "solid" bounded by cells with the following criteria: 1) each cell is adjacent to only one other cell for each face, 2) no subset of cells fits criteria 1, 3) no two adjacent cells are corealmic. If criteria 1 fails, then the figure is degenerate. The word "polychoron" was invented by George Olshevsky with the following construction: poly = many and choron = rooms or cells. A polytope (polyhedron, polychoron, etc.) is uniform if it is vertex transitive and it's facets are uniform (a uniform polygon is a regular polygon). Degenerate figures can also be uniform under the same conditions. A vertex figure is the figure representing the shape and "solid" angle of the vertices, ex: the vertex figure of a cube is a triangle with edge length of the square root of 2. -
Maximal Subgroups of the Coxeter Group W(H4) and Quaternions
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Linear Algebra and its Applications 412 (2006) 441–452 www.elsevier.com/locate/laa Maximal subgroups of the Coxeter group W(H4) and quaternions Mehmet Koca a, Ramazan Koç b,∗, Muataz Al-Barwani a, Shadia Al-Farsi a aDepartment of Physics, College of Science, Sultan Qaboos University, P.O. Box 36, Al-Khod 123, Muscat, Oman bDepartment of Physics, Faculty of Engineering, Gaziantep University, 27310 Gaziantep, Turkey Received 9 October 2004; accepted 17 July 2005 Available online 12 September 2005 Submitted by H. Schneider Abstract The largest finite subgroup of O(4) is the non-crystallographic Coxeter group W(H4) of order 14,400. Its derived subgroup is the largest finite subgroup W(H4)/Z2 of SO(4) of order 7200. Moreover, up to conjugacy, it has five non-normal maximal subgroups of orders 144, two 240, 400 and 576. Two groups [W(H2) × W(H2)]Z4 and W(H3) × Z2 possess non- crystallographic structures with orders 400 and 240 respectively. The groups of orders 144, 240 and 576 are the extensions of the Weyl groups of the root systems of SU(3) × SU(3), SU(5) and SO(8) respectively. We represent the maximal subgroups of W(H4) with sets of quaternion pairs acting on the quaternionic root systems. © 2005 Elsevier Inc. All rights reserved. AMS classification: 20G20; 20F05; 20E34 Keywords: Structure of groups; Quaternions; Coxeter groups; Subgroup structure ∗ Corresponding author. Tel.: +90 342 360 1200; fax: +90 342 360 1013. -
On Regular Polytopes Is [7]
ON REGULAR POLYTOPES Luis J. Boya Departamento de F´ısica Te´orica Universidad de Zaragoza, E-50009 Zaragoza, SPAIN [email protected] Cristian Rivera Departamento de F´ısica Te´orica Universidad de Zaragoza, E-50009 Zaragoza, SPAIN cristian [email protected] MSC 05B45, 11R52, 51M20, 52B11, 52B15, 57S25 Keywords: Polytopes, Higher Dimensions, Orthogonal groups Abstract Regular polytopes, the generalization of the five Platonic solids in 3 space dimensions, exist in arbitrary dimension n 1; now in dim. 2, 3 and 4 there are extra polytopes, while in general≥− dimensions only the hyper-tetrahedron, the hyper-cube and its dual hyper-octahedron ex- ist. We attribute these peculiarites and exceptions to special properties of the orthogonal groups in these dimensions: the SO(2) = U(1) group being (abelian and) divisible, is related to the existence of arbitrarily- sided plane regular polygons, and the splitting of the Lie algebra of the O(4) group will be seen responsible for the Schl¨afli special polytopes in 4-dim., two of which percolate down to three. In spite of dim. 8 being also special (Cartan’s triality), we argue why there are no extra polytopes, while it has other consequences: in particular the existence of the three division algebras over the reals R: complex C, quater- nions H and octonions O is seen also as another feature of the special arXiv:1210.0601v1 [math-ph] 1 Oct 2012 properties of corresponding orthogonal groups, and of the spheres of dimension 0,1,3 and 7. 1 Introduction Regular Polytopes are the higher dimensional generalization of the (regu- lar) polygons in the plane and the (five) Platonic solids in space. -
Four-Dimensional Regular Polytopes
faculty of science mathematics and applied and engineering mathematics Four-dimensional regular polytopes Bachelor’s Project Mathematics November 2020 Student: S.H.E. Kamps First supervisor: Prof.dr. J. Top Second assessor: P. Kiliçer, PhD Abstract Since Ancient times, Mathematicians have been interested in the study of convex, regular poly- hedra and their beautiful symmetries. These five polyhedra are also known as the Platonic Solids. In the 19th century, the four-dimensional analogues of the Platonic solids were described mathe- matically, adding one regular polytope to the collection with no analogue regular polyhedron. This thesis describes the six convex, regular polytopes in four-dimensional Euclidean space. The focus lies on deriving information about their cells, faces, edges and vertices. Besides that, the symmetry groups of the polytopes are touched upon. To better understand the notions of regularity and sym- metry in four dimensions, our journey begins in three-dimensional space. In this light, the thesis also works out the details of a proof of prof. dr. J. Top, showing there exist exactly five convex, regular polyhedra in three-dimensional space. Keywords: Regular convex 4-polytopes, Platonic solids, symmetry groups Acknowledgements I would like to thank prof. dr. J. Top for supervising this thesis online and adapting to the cir- cumstances of Covid-19. I also want to thank him for his patience, and all his useful comments in and outside my LATEX-file. Also many thanks to my second supervisor, dr. P. Kılıçer. Furthermore, I would like to thank Jeanne for all her hospitality and kindness to welcome me in her home during the process of writing this thesis. -
Platonic Solids Generate Their Four-Dimensional Analogues
This is a repository copy of Platonic solids generate their four-dimensional analogues. White Rose Research Online URL for this paper: https://eprints.whiterose.ac.uk/85590/ Version: Accepted Version Article: Dechant, Pierre-Philippe orcid.org/0000-0002-4694-4010 (2013) Platonic solids generate their four-dimensional analogues. Acta Crystallographica Section A : Foundations of Crystallography. pp. 592-602. ISSN 1600-5724 https://doi.org/10.1107/S0108767313021442 Reuse Items deposited in White Rose Research Online are protected by copyright, with all rights reserved unless indicated otherwise. They may be downloaded and/or printed for private study, or other acts as permitted by national copyright laws. The publisher or other rights holders may allow further reproduction and re-use of the full text version. This is indicated by the licence information on the White Rose Research Online record for the item. Takedown If you consider content in White Rose Research Online to be in breach of UK law, please notify us by emailing [email protected] including the URL of the record and the reason for the withdrawal request. [email protected] https://eprints.whiterose.ac.uk/ 1 Platonic solids generate their four-dimensional analogues PIERRE-PHILIPPE DECHANT a,b,c* aInstitute for Particle Physics Phenomenology, Ogden Centre for Fundamental Physics, Department of Physics, University of Durham, South Road, Durham, DH1 3LE, United Kingdom, bPhysics Department, Arizona State University, Tempe, AZ 85287-1604, United States, and cMathematics Department, University of York, Heslington, York, YO10 5GG, United Kingdom. E-mail: [email protected] Polytopes; Platonic Solids; 4-dimensional geometry; Clifford algebras; Spinors; Coxeter groups; Root systems; Quaternions; Representations; Symmetries; Trinities; McKay correspondence Abstract In this paper, we show how regular convex 4-polytopes – the analogues of the Platonic solids in four dimensions – can be constructed from three-dimensional considerations concerning the Platonic solids alone. -
Remarks on the Cohomology of Finite Fundamental Groups of 3–Manifolds
Geometry & Topology Monographs 14 (2008) 519–556 519 arXiv version: fonts, pagination and layout may vary from GTM published version Remarks on the cohomology of finite fundamental groups of 3–manifolds SATOSHI TOMODA PETER ZVENGROWSKI Computations based on explicit 4–periodic resolutions are given for the cohomology of the finite groups G known to act freely on S3 , as well as the cohomology rings of the associated 3–manifolds (spherical space forms) M = S3=G. Chain approximations to the diagonal are constructed, and explicit contracting homotopies also constructed for the cases G is a generalized quaternion group, the binary tetrahedral group, or the binary octahedral group. Some applications are briefly discussed. 57M05, 57M60; 20J06 1 Introduction The structure of the cohomology rings of 3–manifolds is an area to which Heiner Zieschang devoted much work and energy, especially from 1993 onwards. This could be considered as part of a larger area of his interest, the degrees of maps between oriented 3– manifolds, especially the existence of degree one maps, which in turn have applications in unexpected areas such as relativity theory (cf Shastri, Williams and Zvengrowski [41] and Shastri and Zvengrowski [42]). References [1,6,7, 18, 19, 20, 21, 22, 23] in this paper, all involving work of Zieschang, his students Aaslepp, Drawe, Sczesny, and various colleagues, attest to his enthusiasm for these topics and the remarkable energy he expended studying them. Much of this work involved Seifert manifolds, in particular, references [1, 6, 7, 18, 20, 23]. Of these, [6, 7, 23] (together with [8, 9]) successfully completed the programme of computing the ring structure H∗(M) for any orientable Seifert manifold M with 1 2 3 3 G := π1(M) infinite. -
Half-Bps M2-Brane Orbifolds 3
HALF-BPS M2-BRANE ORBIFOLDS PAUL DE MEDEIROS AND JOSÉ FIGUEROA-O’FARRILL Abstract. Smooth Freund–Rubin backgrounds of eleven-dimensional supergravity of the form AdS X7 and preserving at least half of the supersymmetry have been recently clas- 4 × sified. Requiring that amount of supersymmetry forces X to be a spherical space form, whence isometric to the quotient of the round 7-sphere by a freely-acting finite subgroup of SO(8). The classification is given in terms of ADE subgroups of the quaternions embed- dedin SO(8) as the graph of an automorphism. In this paper we extend this classification by dropping the requirement that the background be smooth, so that X is now allowed to be an orbifold of the round 7-sphere. We find that if the background preserves more than half of the supersymmetry, then it is automatically smooth in accordance with the homo- geneity conjecture, but that there are many half-BPS orbifolds, most of them new. The classification is now given in terms of pairs of ADE subgroups of quaternions fibred over the same finite group. We classify such subgroups and then describe the resulting orbi- folds in terms of iterated quotients. In most cases the resulting orbifold can be described as a sequence of cyclic quotients. Contents List of Tables 2 1. Introduction 3 How to use this paper 5 2. Spherical orbifolds 7 2.1. Spin orbifolds 8 2.2. Statement of the problem 9 3. Finite subgroups of the quaternions 10 4. Goursat’s Lemma 12 arXiv:1007.4761v3 [hep-th] 25 Aug 2010 4.1. -
From the Icosahedron to E8
From the Icosahedron to E8 John C. Baez Department of Mathematics University of California Riverside, California 92521, USA and Centre for Quantum Technologies National University of Singapore Singapore 117543 [email protected] December 22, 2017 Abstract The regular icosahedron is connected to many exceptional objects in mathematics. Here we describe two constructions of the E8 lattice from the icosahedron. One uses a subring of the quaternions called the \icosians", while the other uses du Val's work on the resolution of Kleinian singularities. Together they link the golden ratio, the quaternions, the quintic equation, the 600- cell, and the Poincar´ehomology 3-sphere. We leave it as a challenge to the reader to find the connection between these two constructions. In mathematics, every sufficiently beautiful object is connected to all others. Many exciting adven- tures, of various levels of difficulty, can be had by following these connections. Take, for example, the icosahedron|that is, the regular icosahedron, one of the five Platonic solids. Starting from this it is just a hop, skip and a jump to the E8 lattice, a wonderful pattern of points in 8 dimensions! As we explore this connection we shall see that it also ties together many other remarkable entities: the golden ratio, the quaternions, the quintic equation, a highly symmetrical 4-dimensional shape called the 600-cell, and a manifold called the Poincar´ehomology 3-sphere. Indeed, the main problem with these adventures is knowing where to stop. The story we shall tell is just a snippet of a longer one involving the McKay correspondence and quiver representations. -
The Classification and the Conjugacy Classes of the Finite Subgroups of the Sphere Braid Groups Daciberg Gonçalves, John Guaschi
The classification and the conjugacy classes of the finite subgroups of the sphere braid groups Daciberg Gonçalves, John Guaschi To cite this version: Daciberg Gonçalves, John Guaschi. The classification and the conjugacy classes of the finite subgroups of the sphere braid groups. Algebraic and Geometric Topology, Mathematical Sciences Publishers, 2008, 8 (2), pp.757-785. 10.2140/agt.2008.8.757. hal-00191422 HAL Id: hal-00191422 https://hal.archives-ouvertes.fr/hal-00191422 Submitted on 26 Nov 2007 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. The classification and the conjugacy classes of the finite subgroups of the sphere braid groups DACIBERG LIMA GONC¸ALVES Departamento de Matem´atica - IME-USP, Caixa Postal 66281 - Ag. Cidade de S˜ao Paulo, CEP: 05311-970 - S˜ao Paulo - SP - Brazil. e-mail: [email protected] JOHN GUASCHI Laboratoire de Math´ematiques Nicolas Oresme UMR CNRS 6139, Universit´ede Caen BP 5186, 14032 Caen Cedex, France. e-mail: [email protected] 21st November 2007 Abstract Let n ¥ 3. We classify the finite groups which are realised as subgroups of the sphere braid 2 Õ group Bn ÔS . -
Wythoffian Skeletal Polyhedra
Wythoffian Skeletal Polyhedra by Abigail Williams B.S. in Mathematics, Bates College M.S. in Mathematics, Northeastern University A dissertation submitted to The Faculty of the College of Science of Northeastern University in partial fulfillment of the requirements for the degree of Doctor of Philosophy April 14, 2015 Dissertation directed by Egon Schulte Professor of Mathematics Dedication I would like to dedicate this dissertation to my Meme. She has always been my loudest cheerleader and has supported me in all that I have done. Thank you, Meme. ii Abstract of Dissertation Wythoff's construction can be used to generate new polyhedra from the symmetry groups of the regular polyhedra. In this dissertation we examine all polyhedra that can be generated through this construction from the 48 regular polyhedra. We also examine when the construction produces uniform polyhedra and then discuss other methods for finding uniform polyhedra. iii Acknowledgements I would like to start by thanking Professor Schulte for all of the guidance he has provided me over the last few years. He has given me interesting articles to read, provided invaluable commentary on this thesis, had many helpful and insightful discussions with me about my work, and invited me to wonderful conferences. I truly cannot thank him enough for all of his help. I am also very thankful to my committee members for their time and attention. Additionally, I want to thank my family and friends who, for years, have supported me and pretended to care everytime I start talking about math. Finally, I want to thank my husband, Keith. -
The Double Cover of the Icosahedral Symmetry Group and Quark Mass
MADPHYS-10-1566 The Double Cover of the Icosahedral Symmetry Group and Quark Mass Textures Lisa L. Everett and Alexander J. Stuart Department of Physics, University of Wisconsin, Madison, WI, 53706, USA (Dated: October 25, 2018) We investigate the idea that the double cover of the rotational icosahedral symmetry group is the family symmetry group in the quark sector. The icosahedral ( 5) group was previously proposed as a viable family symmetry group for the leptons. To incorporateA the quarks, it is highly advantageous to extend the group to its double cover, as in the case of tetrahedral (A4) symmetry. We provide the basic group theoretical tools for flavor model-building based on the binary icosahedral group ′ and construct a model of the quark masses and mixings that yields many of the successful predictionsI of the well-known U(2) quark texture models. PACS numbers: 12.15Ff,12.60.Jv I. INTRODUCTION With the measurement of neutrino oscillations [1–6], an intriguing pattern of lepton mixing has emerged. The neu- trino oscillation data have revealed that two of the mixing angles of the Maki-Nakagawa-Sakata-Pontecorvo (MNSP) [8] mixing matrix are large and the third angle is bounded from above by the Cabibbo angle (for global fits, see [7]). This pattern, with its striking differences from the quark mixing angles of the Cabibbo-Kobayashi-Maskawa (CKM) matrix, has shifted the paradigm for addressing the Standard Model (SM) flavor puzzle. More precisely, while the quark sector previously indicated a flavor model-building framework based on the Froggatt- Nielsen mechanism [10] with continuous family symmetries (see also [9]), the large lepton mixing angles suggest discrete non-Abelian family symmetries. -
Topological Lensing in Spherical Spaces
Topological Lensing in Spherical Spaces Evelise Gausmann1, Roland Lehoucq2, Jean-Pierre Luminet1, Jean-Philippe Uzan3 and Jeffrey Weeks4 (1) D´epartement d’Astrophysique Relativiste et de Cosmologie, Observatoire de Paris – C.N.R.S. UMR 8629, F-92195 Meudon Cedex (France). (2) CE-Saclay, DSM/DAPNIA/Service d’Astrophysique, F-91191 Gif sur Yvette Cedex (France) (3) Laboratoire de Physique Th´eorique – C.N.R.S. UMR 8627, Bˆat. 210, Universit´eParis XI, F-91405 Orsay Cedex (France). (4) 15 Farmer St., Canton NY 13617-1120, USA. Abstract. This article gives the construction and complete classification of all three–dimensional spherical manifolds, and orders them by decreasing volume, in the context of multiconnected universe models with positive spatial curvature. It discusses which spherical topologies are likely to be detectable by crystallographic methods using three–dimensional catalogs of cosmic objects. The expected form of the pair separation histogram is predicted (including the location and height of the spikes) and is compared to computer simulations, showing that this method is stable with respect to observational uncertainties and is well suited for detecting spherical topologies. PACS numbers: 98.80.-q, 04.20.-q, 02.40.Pc 1. Introduction The search for the topology of our universe has focused mainly on candidate spacetimes with locally Euclidean or hyperbolic spatial sections. This was motivated on the one hand by the mathematical simplicity of three–dimensional Euclidean manifolds, and on the other hand by observational data which