Regular Polytopes

WE have described three families of regular polytopes: the simplexes an (viz., the triangle, tetrahedron, etc.), the cross polytopes βn (the square, octahedron, etc.), the measure polytopes γn (the square, cube, etc.); ...

Author: H. S. M. Coxeter

Publisher: Courier Corporation

ISBN: 9780486141589

Category: Mathematics

Page: 368

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Foremost book available on polytopes, incorporating ancient Greek and most modern work. Discusses polygons, polyhedrons, and multi-dimensional polytopes. Definitions of symbols. Includes 8 tables plus many diagrams and examples. 1963 edition.
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Abstract Regular Polytopes

In this chapter , we introduce the basic notation and concepts of the theory of abstract regular polytopes . Our notation will generally be patterned after the traditional theory , and will provide a convenient framework to study ...

Author: Peter McMullen

Publisher: Cambridge University Press

ISBN: 0521814960

Category: Mathematics

Page: 551

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Table of contents
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Geometric Regular Polytopes

As was pointed out in [99], strictly speaking we should talk about regular abstract polytopes. The term we use is influenced by euphony. 2. Observe that the concepts of 'complex polytope' introduced in [116] (see [30] for a ...

Author: Peter McMullen

Publisher: Cambridge University Press

ISBN: 9781108788311

Category: Mathematics

Page:

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Regular polytopes and their symmetry have a long history stretching back two and a half millennia, to the classical regular polygons and polyhedra. Much of modern research focuses on abstract regular polytopes, but significant recent developments have been made on the geometric side, including the exploration of new topics such as realizations and rigidity, which offer a different way of understanding the geometric and combinatorial symmetry of polytopes. This is the first comprehensive account of the modern geometric theory, and includes a wide range of applications, along with new techniques. While the author explores the subject in depth, his elementary approach to traditional areas such as finite reflexion groups makes this book suitable for beginning graduate students as well as more experienced researchers.
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The Classes of Higher Dimensional Polytopes in Chemical Physical and Biological Systems

ABSTRACT A direct construction of regular polytopes of dimension four was carried out by connecting threedimensional figures along whole flat faces included in the polytope. It was found that the images of a 24-cell cell known from ...

Author: Zhizhin, Gennadiy Vladimirovich

Publisher: IGI Global

ISBN: 9781799883760

Category: Mathematics

Page: 366

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The study of the geometry of structures that arise in a variety of specific natural systems, such as chemical, physical, biological, and geological, revealed the existence of a wide range of types of polytopes of the highest dimension that were unknown in classical geometry. At the same time, new properties of polytopes were discovered as well as the geometric patterns to which they obey. There is a need to classify these types of polytopes of the highest dimension by listing their properties and formulating the laws to which they obey. The Classes of Higher Dimensional Polytopes in Chemical, Physical, and Biological Systems explains the meaning of higher dimensions and systematically generalizes the results of geometric research in various fields of knowledge. This book is useful both for the fundamental development of geometry and for the development of branches of science related to human activities. It builds upon previous books published by the author on this topic. Covering areas such as heredity, geometry, and dimensions, this reference work is ideal for researchers, scholars, academicians, practitioners, industry professionals, instructors, and students.
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Polytopes

This generalizes also to many classes of abstract regular polytopes. The previous sections were aiming at the classification of all the finite polytopes among the universal locally toroidal regular polytopes {P1, P2}.

Author: Tibor Bisztriczky

Publisher: Springer Science & Business Media

ISBN: 9789401109246

Category: Mathematics

Page: 507

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The aim of this volume is to reinforce the interaction between the three main branches (abstract, convex and computational) of the theory of polytopes. The articles include contributions from many of the leading experts in the field, and their topics of concern are expositions of recent results and in-depth analyses of the development (past and future) of the subject. The subject matter of the book ranges from algorithms for assignment and transportation problems to the introduction of a geometric theory of polyhedra which need not be convex. With polytopes as the main topic of interest, there are articles on realizations, classifications, Eulerian posets, polyhedral subdivisions, generalized stress, the Brunn--Minkowski theory, asymptotic approximations and the computation of volumes and mixed volumes. For researchers in applied and computational convexity, convex geometry and discrete geometry at the graduate and postgraduate levels.
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Convex Polytopes

According to the inductive definition (see, for example, Coxeter [1], Fejes Tóth [3]) a d-polytope is regular provided all its facets and all its vertex figures are regular (d – 1)-polytopes. (Vertex figure is here understood in a more ...

Author: Branko Grünbaum

Publisher: Springer Science & Business Media

ISBN: 9781461300199

Category: Mathematics

Page: 471

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"The original edition [...] inspired a whole generation of grateful workers in polytope theory. Without it, it is doubtful whether many of the subsequent advances in the subject would have been made. The many seeds it sowed have since grown into healthy trees, with vigorous branches and luxuriant foliage. It is good to see it in print once again." --Peter McMullen, University College London
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Symmetries in Graphs Maps and Polytopes

Regular polytopes. Dover Publications, Inc., New York, third edition, 1973. 3. H. S. M. Coxeter, M. S. Longuet-Higgins, and J. C. P. Miller. Uniform polyhedra.Philos. Trans. Roy. Soc. London. Ser. A., 246:401–450 (6 plates), 1954. 4.

Author: Jozef Širáň

Publisher: Springer

ISBN: 9783319304519

Category: Mathematics

Page: 332

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This volume contains seventeen of the best papers delivered at the SIGMAP Workshop 2014, representing the most recent advances in the field of symmetries of discrete objects and structures, with a particular emphasis on connections between maps, Riemann surfaces and dessins d’enfant.Providing the global community of researchers in the field with the opportunity to gather, converse and present their newest findings and advances, the Symmetries In Graphs, Maps, and Polytopes Workshop 2014 was the fifth in a series of workshops. The initial workshop, organized by Steve Wilson in Flagstaff, Arizona, in 1998, was followed in 2002 and 2006 by two meetings held in Aveiro, Portugal, organized by Antonio Breda d’Azevedo, and a fourth workshop held in Oaxaca, Mexico, organized by Isabel Hubard in 2010.This book should appeal to both specialists and those seeking a broad overview of what is happening in the area of symmetries of discrete objects and structures.iv>
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Polytopes Combinations and Computation

Regular polytopes Without any doubt, the most classical polyhedra are the five Platonic Solids: the tetrahedron, the cube and the octahedron, the icosahedron and the dodecahedron. They are the only 3-dimensional regular polytopes, ...

Author: Gil Kalai

Publisher: Birkhäuser

ISBN: 9783034884389

Category: Mathematics

Page: 225

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Questions that arose from linear programming and combinatorial optimization have been a driving force for modern polytope theory, such as the diameter questions motivated by the desire to understand the complexity of the simplex algorithm, or the need to study facets for use in cutting plane procedures. In addition, algorithms now provide the means to computationally study polytopes, to compute their parameters such as flag vectors, graphs and volumes, and to construct examples of large complexity. The papers of this volume thus display a wide panorama of connections of polytope theory with other fields. Areas such as discrete and computational geometry, linear and combinatorial optimization, and scientific computing have contributed a combination of questions, ideas, results, algorithms and, finally, computer programs.
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Polytopes and Symmetry

A polytope P is said to be metrically regular or just regular if the action of г ( P ) on $ ( P ) is transitive . Equivalently , P is regular iff it is combinatorially regular and T ( P ) = Aut P. We expect the dimension of the symmetry ...

Author: Stewart A. Robertson

Publisher: Cambridge University Press

ISBN: 0521277396

Category: Mathematics

Page: 138

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This book describes a fresh approach to the classification of of convex plane polygons and of convex polyhedra according to their symmetry properties, based on ideas of topology and transformation group theory. Although there is considerable agreement with traditional treatments, a number of new concepts emerge that present classical ideas in a quite new way.
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Lectures on Polytopes

Here is one more aspect that makes the d-cubes and d-crosspolytopes remarkable: they are regular polytopes — polytopes with maximal symmetry. (We will not give a precise definition here.) There is an extensive and very beautiful theory ...

Author: Günter M. Ziegler

Publisher: Springer Science & Business Media

ISBN: 9780387943657

Category: Mathematics

Page: 370

View: 193

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Based on a graduate course at the Technische Universität, Berlin, these lectures present a wealth of material on the modern theory of convex polytopes. The straightforward exposition features many illustrations, and complete proofs for most theorems. With only linear algebra as a prerequisite, it takes the reader quickly from the basics to topics of recent research. The lectures introduce basic facts about polytopes, with an emphasis on methods that yield the results, discuss important examples and elegant constructions, and show the excitement of current work in the field. They will provide interesting and enjoyable reading for researchers as well as students.
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