Classical Tessellations and Three-Manifolds
This unusual book, richly illustrated with 29 colour illustrations and about 200 line drawings, explores the relationship between classical tessellations and three-manifolds. In his original and entertaining style, the author provides graduate students wi
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Jose Maria Montesinos
Classical Tessellations and Three-Manifolds With 225 Figures Including 29 Colour Illustrations
Springer-Verlag Berlin Heidelberg New York London Paris Tokyo
Jose Marfa Montesinos-Amilibia Facultad de Matematicas Universidad Complutense 28040 Madrid, Spain
Mathematics Subject Classification (1985): 57N10, 51 M20, 05845, 57M05, 57M12, 57M25, 51 F15, 51 M05, 51M10,20F38,52A25,52A45
ISBN-13:978-3-540-15291-0 DOl: 10.1007/978-3-642-61572-6
e-ISBN-13: 978-3-642-61572-6
Library of Congress Cataloging in Publication Data Montesinos, Jose, 1944- Classical tessellations and three-manifolds. (Universitext) Bibliography: p. 1. Tessellations (Mathematics) 2. Three-manifolds (Topology) I. Title. QA166.8.M66 1987 514'.223 87-20645 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. Duplication of this publication or parts thereof is only permitted under the provisions of the German Copyright Law of September 9, 1965, in its version of June 24, 1985, and a copyright fee must always be paid. Violations fall under the prosecution act of the German Copyright Law.
© Springer-Verlag Berlin Heidelberg 1987
2141/3140-543210
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Preface
"Mas has dicho, Sancho, de 10 que sabes (dixo Don Quixote), que hay algunos que se cansan en saber, y averiguar cosas que despues de sabidas, y averiguadas, no importa un ardite al entendimiento, ni a la memoria."
"You have said more than you know, Sancho", said Don Quixote, "for there are some who tire themselves out learning and proving things which, once learnt and proved, do not concern either 'the understanding 01' the memory a jot." Cervantes, Don Quixote, Part II, Chapter LXXV, Of the great Adventure of Montesinos' Cave in the heart of La Mancha, which the valorous Don Quixote brought to a happy ending.
This book explores a relationship between classical tessellations and three-manifolds. All of us are very familiar with the symmetrical ornamental motifs used in the decoration of walls and ceilings. Oriental palaces contain an abundance of these, and many examples taken from them will be found in the following pages. These are the so-called mosaics or symmetrical tessellations of the euclidean plane. Even though we can imagine or even create very many of them, in fact the rules governing them are quite restrictive, if our purpose is to understand the symmetric group of the tessellation, that is to say, the group consisting of the plane isometries which leave the tessellation invariant. From this point of view, it can be proved that there are precisely seventeen possible groups of symmetry for euclidean-plane tessellations, discounting those tessellations which can be considered as extensions of linear tessellations. On the surface of a sphere, i.e.
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