Regular Complex Polytopes


Author: N.A
Publisher: CUP Archive
ISBN: N.A
Category:
Page: N.A
View: 7994
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Regular Complex Polytopes


Author: H. S. M. Coxeter
Publisher: Cambridge University Press
ISBN: 9780521394901
Category: Mathematics
Page: 224
View: 3006
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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.

Regular Polytopes


Author: H. S. M. Coxeter
Publisher: Courier Corporation
ISBN: 0486141586
Category: Mathematics
Page: 368
View: 6460
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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.

Handbook of Discrete and Computational Geometry, Third Edition


Author: Csaba D. Toth,Joseph O'Rourke,Jacob E. Goodman
Publisher: CRC Press
ISBN: 1351645919
Category: Computers
Page: 1928
View: 8769
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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 ?elds as diverse as operations research, molecular biology, and robotics. Discrete geometry has contributed signi?cantly 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 ?eld 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.

Handbook of Discrete and Computational Geometry, Third Edition


Author: Csaba D. Toth,Joseph O'Rourke,Jacob E. Goodman
Publisher: CRC Press
ISBN: 1498711421
Category: Computers
Page: 1928
View: 4768
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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 ?elds as diverse as operations research, molecular biology, and robotics. Discrete geometry has contributed signi?cantly 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 ?eld 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.

Regular Solids and Isolated Singularities


Author: Klaus Lamotke
Publisher: Springer-Verlag
ISBN: 3322917673
Category: Mathematics
Page: 224
View: 1462
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An Introduction to Finite Tight Frames


Author: Shayne F. D. Waldron
Publisher: Springer
ISBN: 0817648151
Category: Mathematics
Page: 587
View: 5849
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This textbook is an introduction to the theory and applications of finite tight frames, an area that has developed rapidly in the last decade. Stimulating much of this growth are the applications of finite frames to diverse fields such as signal processing, quantum information theory, multivariate orthogonal polynomials, and remote sensing. Featuring exercises and MATLAB examples in each chapter, the book is well suited as a textbook for a graduate course or seminar involving finite frames. The self-contained, user-friendly presentation also makes the work useful as a self-study resource or reference for graduate students, instructors, researchers, and practitioners in pure and applied mathematics, engineering, mathematical physics, and signal processing.

Abstract Regular Polytopes


Author: Peter McMullen,Egon Schulte
Publisher: Cambridge University Press
ISBN: 9780521814966
Category: Mathematics
Page: 551
View: 6054
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A modern, comprehensive review of abstract regular polytopes.

Regular polytopes


Author: Harold Scott Macdonald Coxeter
Publisher: N.A
ISBN: N.A
Category: Polytopes
Page: 321
View: 992
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Internationale Mathematische Nachrichten


Author: N.A
Publisher: N.A
ISBN: N.A
Category: Mathematics
Page: N.A
View: 4902
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Topological Crystallography

With a View Towards Discrete Geometric Analysis
Author: Toshikazu Sunada
Publisher: Springer Science & Business Media
ISBN: 4431541772
Category: Mathematics
Page: 229
View: 4728
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Geometry in ancient Greece is said to have originated in the curiosity of mathematicians about the shapes of crystals, with that curiosity culminating in the classification of regular convex polyhedra addressed in the final volume of Euclid’s Elements. Since then, geometry has taken its own path and the study of crystals has not been a central theme in mathematics, with the exception of Kepler’s work on snowflakes. Only in the nineteenth century did mathematics begin to play a role in crystallography as group theory came to be applied to the morphology of crystals. This monograph follows the Greek tradition in seeking beautiful shapes such as regular convex polyhedra. The primary aim is to convey to the reader how algebraic topology is effectively used to explore the rich world of crystal structures. Graph theory, homology theory, and the theory of covering maps are employed to introduce the notion of the topological crystal which retains, in the abstract, all the information on the connectivity of atoms in the crystal. For that reason the title Topological Crystallography has been chosen. Topological crystals can be described as “living in the logical world, not in space,” leading to the question of how to place or realize them “canonically” in space. Proposed here is the notion of standard realizations of topological crystals in space, including as typical examples the crystal structures of diamond and lonsdaleite. A mathematical view of the standard realizations is also provided by relating them to asymptotic behaviors of random walks and harmonic maps. Furthermore, it can be seen that a discrete analogue of algebraic geometry is linked to the standard realizations. Applications of the discussions in this volume include not only a systematic enumeration of crystal structures, an area of considerable scientific interest for many years, but also the architectural design of lightweight rigid structures. The reader therefore can see the agreement of theory and practice.

Extended Icosahedral Structures


Author: Marko V. Jaric
Publisher: Elsevier
ISBN: 0323162304
Category: Science
Page: 236
View: 685
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Extended Icosahedral Structures discusses the concepts about crystal structures with extended icosahedral symmetry. This book is organized into six chapters that focus on actual modeling of extended icosahedral crystal structures. This text first presents a tiling approach to the modeling of icosahedral quasiperiodic crystals. It then describes the models for icosahedral alloys based on random connections between icosahedral units, with particular emphasis on diffraction properties. Other chapters examine the glassy structures with only icosahedral orientational order and the extent of translational order in such structures relates to the nature of their growth. Lastly, the use of the polytope concept to rationalize and construct real structures with varying degrees of icosahedral order is addressed. This book is of great value to physicists, crystallographers, metallurgists, and beginners in the field of quasicrystals.

Simon Stevin


Author: Simon Stevin
Publisher: N.A
ISBN: N.A
Category: Mathematics
Page: N.A
View: 8657
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Polytopes and Symmetry


Author: Stewart A. Robertson,Robertson Stewart A.
Publisher: Cambridge University Press
ISBN: 9780521277396
Category: Mathematics
Page: 112
View: 3641
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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.

Kaleidoscopes

Selected Writings of H.S.M. Coxeter
Author: Harold Scott Macdonald Coxeter,F. Arthur Sherk,Peter McMullen,Canadian Mathematical Society,Anthony C. Thompson,Asia Ivic Weiss
Publisher: John Wiley & Sons
ISBN: 9780471010036
Category: Mathematics
Page: 439
View: 9319
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Coxeter was one of the leaders in geometry, algebra, topology and finite mathematics. This book is a collection of approximately 30 papers, organized by subject, published between 1931 and 1991. Covers such areas as polytopes, abstract groups and the evolution of the Coxeter Dynkin diagram. Features reproductions of original publications by Coxeter, some remarkably containing his own handwritten margin notes.

Leonardo : Journal of International Society for the Arts, Sciences and Technology


Author: N.A
Publisher: N.A
ISBN: N.A
Category: Art, Modern
Page: N.A
View: 2014
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International journal of contemporary visual artists.

Hamiltonian Submanifolds of Regular Polytopes


Author: Felix Effenberger
Publisher: Logos Verlag Berlin GmbH
ISBN: 3832527583
Category:
Page: 197
View: 1904
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This work is set in the field of combinatorial topology, sometimes also referred to as discrete geometric topology, a field of research in the intersection of topology, geometry, polytope theory and combinatorics. The main objects of interest in the field are simplicial complexes that carry some additional structure, forming combinatorial triangulations of the underlying PL manifolds. In particular, polyhedral manifolds as subcomplexes of the boundary complex of a convex regular polytope are investigated. Such a subcomplex is called k-Hamiltonian if it contains the full k-skeleton of the polytope. The notion of tightness of a PL-embedding of a triangulated manifold is closely related to its property of being a Hamiltonian subcomplex of some convex polytope. Tightness of a triangulated manifold is a topological condition, roughly meaning that any simplex-wise linear embedding of the triangulation into Euclidean space is ``as convex as possible''. It can thus be understood as a generalization of the concept of convexity. In even dimensions, there exist purely combinatorial conditions which imply the tightness of a triangulation. In this work, other sufficient and purely combinatorial conditions which can be applied to the odd-dimensional case as well are presented.

Tores et torsades


Author: Jean Charvolin,Jean-Francois Sadoc
Publisher: EDP Sciences
ISBN: 2759809242
Category: Science
Page: 172
View: 1661
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Cet ouvrage réunit dans un même cadre conceptuel les structures d'objets nanoscopiques aussi divers que les films toriques construits par des molécules amphiphiles ou des phospholipides et les phases ou fibres torsadées construites par des molécules de cristaux liquides, des polymères ou des macromolécules biologiques. Des objets non seulement divers mais aussi étonnamment complexes dans la mesure où les assemblages de leurs molécules y présentent localement de forts écarts à tout ordre régulier. Ces écarts à l'ordre, ou défauts, jouent un rôle essentiel dans le choix d'une forme d'organisation, le contrôle de sa taille ou de son inclusion dans une séquence hiérarchique. Dans la mesure où les mises en oeuvre des matériaux biologiques dans les organismes sont étroitement reliées à leurs structures et morphologies, l'étude de ces défauts prend là une importance particulière. Les auteurs développent une approche systématique et unifiée des défauts dans ces objets de la matière «molle» ou de la biologie en mettant en oeuvre le concept de frustration imaginé à l'origine pour décrire des systèmes de la matière « dure » présentant une grande variété d'écarts à l'ordre cristallin. Les outils géométriques et topologiques nécessaires à cette mise en oeuvre sont présentés dans le texte à l'aide de nombreuses illustrations faisant largement appel à l'intuition du lecteur, le formalisme rigoureux est néanmoins développé dans des appendices. Cette extension du concept de frustration de la matière « dure » vers la matière « molle » illustre remarquablement son universalité et suggère de nombreux développements. Cet ouvrage, synthèse unique sur un sujet très riche et très actuel, s'adresse aux chercheurs, enseignants et étudiants attentifs au rôle des défauts en matière condensée et biologie structurale.