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Author : Peter McMullen,Egon Schulte
Publisher : Cambridge University Press
Page : 580 pages
File Size : 53,7 Mb
Release : 2002-12-12
Category : Mathematics
ISBN : 0521814960
Table of contents
Author : Tibor Bisztriczky,Peter McMullen,Rolf Schneider,Asia Ivic Weiss
Publisher : Springer Science & Business Media
Page : 515 pages
File Size : 51,5 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9789401109246
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.
Author : Ilan Eldar
Publisher : Unknown
Page : 36 pages
File Size : 44,8 Mb
Release : 1971
Category : Polytopes
ISBN : STANFORD:36105046360215
Walkup and Klee studied the diameter of ordinary convex polytopes which is defined as the smallest integer k such that all pairs of vertices can be joined by a path of k or less neighboring vertices. The well known d-step (or Hirsch) conjecture for d dimensional polytopes with n facets states that the maximum diameter is n - d. Walkup and Klee showed the conjecture as correct for all n - d
Author : Peter McMullen
Publisher : Cambridge University Press
Page : 617 pages
File Size : 52,9 Mb
Release : 2020-02-20
Category : Mathematics
ISBN : 9781108788311
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.
Author : Csaba D. Toth,Joseph O'Rourke,Jacob E. Goodman
Publisher : CRC Press
Page : 1928 pages
File Size : 47,5 Mb
Release : 2017-11-22
Category : Computers
ISBN : 9781498711425
The Handbook of Discrete and Computational Geometry is intended as a reference book fully accessible to nonspecialists as well as specialists, covering all major aspects of both fields. The book offers the most important results and methods in discrete and computational geometry to those who use them in their work, both in the academic world—as researchers in mathematics and computer science—and in the professional world—as practitioners in fields as diverse as operations research, molecular biology, and robotics. Discrete geometry has contributed significantly to the growth of discrete mathematics in recent years. This has been fueled partly by the advent of powerful computers and by the recent explosion of activity in the relatively young field of computational geometry. This synthesis between discrete and computational geometry lies at the heart of this Handbook. A growing list of application fields includes combinatorial optimization, computer-aided design, computer graphics, crystallography, data analysis, error-correcting codes, geographic information systems, motion planning, operations research, pattern recognition, robotics, solid modeling, and tomography.
Author : Stewart A. Robertson
Publisher : Cambridge University Press
Page : 138 pages
File Size : 49,6 Mb
Release : 1984-01-26
Category : Mathematics
ISBN : 0521277396
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.
Author : Gabriel Cunningham,Mark Mixer,Egon Schulte
Publisher : American Mathematical Soc.
Page : 272 pages
File Size : 55,9 Mb
Release : 2021-04-06
Category : Education
ISBN : 9781470448974
The papers showcase the breadth of discrete geometry through many new methods and results in a variety of topics. Also included are survey articles on some important areas of active research. This volume is aimed at researchers in discrete and convex geometry and researchers who work with abstract polytopes or string C C-groups. It is also aimed at early career mathematicians, including graduate students and postdoctoral fellows, to give them a glimpse of the variety and beauty of these research areas. Topics covered in this volume include: the combinatorics, geometry, and symmetries of convex polytopes; tilings; discrete point sets; the combinatorics of Eulerian posets and interval posets; symmetries of surfaces and maps on surfaces; self-dual polytopes; string C C-groups; hypertopes; and graph coloring.
Author : Marston D. E. Conder,Antoine Deza,Asia Ivić Weiss
Publisher : Springer
Page : 333 pages
File Size : 46,8 Mb
Release : 2018-06-11
Category : Mathematics
ISBN : 9783319784342
This book consists of contributions from experts, presenting a fruitful interplay between different approaches to discrete geometry. Most of the chapters were collected at the conference “Geometry and Symmetry” in Veszprém, Hungary from 29 June to 3 July 2015. The conference was dedicated to Károly Bezdek and Egon Schulte on the occasion of their 60th birthdays, acknowledging their highly regarded contributions in these fields. While the classical problems of discrete geometry have a strong connection to geometric analysis, coding theory, symmetry groups, and number theory, their connection to combinatorics and optimization has become of particular importance. The last decades have seen a revival of interest in discrete geometric structures and their symmetry. The rapid development of abstract polytope theory has resulted in a rich theory featuring an attractive interplay of methods and tools from discrete geometry, group theory and geometry, combinatorial group theory, and hyperbolic geometry and topology. This book contains papers on new developments in these areas, including convex and abstract polytopes and their recent generalizations, tiling and packing, zonotopes, isoperimetric inequalities, and on the geometric and combinatorial aspects of linear optimization. The book is a valuable resource for researchers, both junior and senior, in the field of discrete geometry, combinatorics, or discrete optimization. Graduate students find state-of-the-art surveys and an open problem collection.
Author : Csaba D. Toth,Joseph O'Rourke,Jacob E. Goodman
Publisher : CRC Press
Page : 1557 pages
File Size : 54,9 Mb
Release : 2004-04-13
Category : Mathematics
ISBN : 9781420035315
While high-quality books and journals in this field continue to proliferate, none has yet come close to matching the Handbook of Discrete and Computational Geometry, which in its first edition, quickly became the definitive reference work in its field. But with the rapid growth of the discipline and the many advances made over the past seven years, it's time to bring this standard-setting reference up to date. Editors Jacob E. Goodman and Joseph O'Rourke reassembled their stellar panel of contributors, added manymore, and together thoroughly revised their work to make the most important results and methods, both classic and cutting-edge, accessible in one convenient volume. Now over more then 1500 pages, the Handbook of Discrete and Computational Geometry, Second Edition once again provides unparalleled, authoritative coverage of theory, methods, and applications. Highlights of the Second Edition: Thirteen new chapters: Five on applications and others on collision detection, nearest neighbors in high-dimensional spaces, curve and surface reconstruction, embeddings of finite metric spaces, polygonal linkages, the discrepancy method, and geometric graph theory Thorough revisions of all remaining chapters Extended coverage of computational geometry software, now comprising two chapters: one on the LEDA and CGAL libraries, the other on additional software Two indices: An Index of Defined Terms and an Index of Cited Authors Greatly expanded bibliographies
Author : Stanford University. Department of Operations Research. Operations Research House
Publisher : Unknown
Page : 28 pages
File Size : 43,7 Mb
Release : 1971
Category : Electronic
ISBN : STANFORD:36105046360207
Abstract polytopes include ordinary convex polytopes as a special case and are defined as systems satisfying certain combinatorial properties of ordinary polytopes. The Euler characteristic is the sum over i with alternating signs of the number of i-dimensional faces. For ordinary polytopes its value is +1. This relation, however, does not hold in general for abstract polytopes. Since the 3-dimensional abstract polytopes correspond 1-1 to triangulated 2-manifolds, the range of their Euler characteristic could be determined by applying known results of manifold theory. The paper investigates the range of the Euler characteristic of abstract polytopes in general. (Author).
Author : H. S. M. Coxeter
Publisher : Cambridge University Press
Page : 224 pages
File Size : 43,6 Mb
Release : 1991-04-26
Category : Mathematics
ISBN : 0521394902
The properties of regular solids exercise a fascination which often appeals strongly to the mathematically inclined, whether they are professionals, students or amateurs. In this classic book Professor Coxeter explores these properties in easy stages, introducing the reader to complex polyhedra (a beautiful generalization of regular solids derived from complex numbers) and unexpected relationships with concepts from various branches of mathematics: magic squares, frieze patterns, kaleidoscopes, Cayley diagrams, Clifford surfaces, crystallographic and non-crystallographic groups, kinematics, spherical trigonometry, and algebraic geometry. In the latter half of the book, these preliminary ideas are put together to describe a natural generalization of the Five Platonic Solids. This updated second edition contains a new chapter on Almost Regular Polytopes, with beautiful 'abstract art' drawings. New exercises and discussions have been added throughout the book, including an introduction to Hopf fibration and real representations for two complex polyhedra.
Author : Robert Connelly,Asia Ivić Weiss,Walter Whiteley
Publisher : Springer
Page : 374 pages
File Size : 42,5 Mb
Release : 2014-06-11
Category : Mathematics
ISBN : 9781493907816
This book contains recent contributions to the fields of rigidity and symmetry with two primary focuses: to present the mathematically rigorous treatment of rigidity of structures and to explore the interaction of geometry, algebra and combinatorics. Contributions present recent trends and advances in discrete geometry, particularly in the theory of polytopes. The rapid development of abstract polytope theory has resulted in a rich theory featuring an attractive interplay of methods and tools from discrete geometry, group theory, classical geometry, hyperbolic geometry and topology. Overall, the book shows how researchers from diverse backgrounds explore connections among the various discrete structures with symmetry as the unifying theme. The volume will be a valuable source as an introduction to the ideas of both combinatorial and geometric rigidity theory and its applications, incorporating the surprising impact of symmetry. It will appeal to students at both the advanced undergraduate and graduate levels, as well as post docs, structural engineers and chemists.
Author : Harold Scott Macdonald Coxeter,Chandler Davis,Erich W. Ellers
Publisher : American Mathematical Soc.
Page : 344 pages
File Size : 41,9 Mb
Release : 2024-06-03
Category : Mathematics
ISBN : 0821887602
This collection of essays on the legacy of mathematican Donald Coxeter is a mixture of surveys, updates, history, storytelling and personal memories covering both applied and abstract maths. Subjects include: polytopes, Coxeter groups, equivelar polyhedra, Ceva's theorum, and Coxeter and the artists.
Author : György Darvas
Publisher : Springer Nature
Page : 262 pages
File Size : 46,7 Mb
Release : 2022-01-01
Category : Mathematics
ISBN : 9783030880590
This volume is a collection of essays on complex symmetries. It is curated, emphasizing the analysis of the symmetries, not the various phenomena that display those symmetries themselves. With this, the volume provides insight to nonspecialist readers into how individual simple symmetries constitute complex symmetry. The authors and the topics cover many different disciplines in various sciences and arts. Simple symmetries, such as reflection, rotation, translation, similitude, and a few other simple manifestations of the phenomenon, are all around, and we are aware of them in our everyday lives. However, there are myriads of complex symmetries (composed of a bulk of simple symmetries) as well. For example, the well-known helix represents the combination of translational and rotational symmetry. Nature produces a great variety of such complex symmetries. So do the arts. The contributions in this volume analyse selected examples (not limited to geometric symmetries). These include physical symmetries, functional (meaning not morphological) symmetries, such as symmetries in the construction of the genetic code, symmetries in human perception (e.g., in geometry education as well as in constructing physical theories), symmetries in fractal structures and structural morphology, including quasicrystal and fullerene structures in stable bindings and their applications in crystallography and architectural design, as well as color symmetries in the arts. The volume is rounded of with beautiful illustrations and presents a fascinating panorama of this interdisciplinary topic.
Author : Stanford University. Department of Operations Research. Operations Research House,Ilan Adler,George Dantzig,Katta Murty
Publisher : Unknown
Page : 16 pages
File Size : 54,8 Mb
Release : 1970
Category : Polytopes
ISBN : STANFORD:36105046359977
Given a finite set E of n symbols a family S of subsets of E (called vertices) form an abstract polytope if (1) Each vertex is a subset of m symbols of E. (2) Every subset of m + 1 symbols of E contains either zero or two vertices (called adjacent). (3) Every pair of vertices V sup 0 and V sup * can be joined by a path V sup 0 = V sub 1 ..., V sub k = V sup * such that V sub i, V sub (i + 1) are adjacent and (V sub i) contained in (V sup 0) joined to (V sup *) i = 1 ..., k-1. It is shown that if two vertices of a given abstract polytope contain the same symbol (say x) then there exists a path such that every vertex along the path contains x. (Author).