Non-Euclidean Geometries

NON-EUCLIDEAN GEOMETRIES Jinos Bolyai Memorial Volume Mathematics and Its Applications Managing Editor: M. HAZEWINKEL Centrefor Mathematics and Computer Science, Amsterdam, The Netherlands Volume 58 1 NON-EUCLIDEAN GEOMETRIES JBnos Bolyai Memorial Volume Edited by ANDRAS PREKOPA Rutgers Center for Operations Research, Piscataway, New Jersey, USA EMIL MOLNAR Budapest University of Technology and Economics, Hungary - Springer Library of Congress Control Number: 2 0 0 5 9 3 3 8 8 5 Printed on acid-free paper. O 2006 Springer Science+Business Media, Inc. All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, Inc., 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed in the United States of America. This plaquette depicting Jfinos Bolyai was made by Kinga SzCchenyi in commemoration of the 200th anniversary of Bolyai's birth. Contents Preface Andrds Prikopa, Emil Molndr Part I History The Revolution of JBnos Bolyai Andra's Prikopa Gauss and non-Euclidean geometry Jeremy Gray JBnos Bolyai's new face Elemir Kiss Part I1 Axiomatical and Logical Aspects Hyperbolic Geometry, Dimension-Free Walter Benx An Absolute Property of Four Mutually Tangent Circles H.S.M. Coxeter Remembering Donald Coxeter Asia Ivic Weiss. William Weiss Axiomatizations of hyperbolic and absolute geometries Victor Pambuccian Logical axiomatizations of space-time. Samples from the literature Hajnal Andrdka, Judit X. Madardsz and Istvdn Nimeti ... vul NON-EUCLIDEAN GEOMETRIES Part I11 Polyhedra, Volumes, Discrete Arrangements, Fractals Structures in Hyperbolic Space Robert Connelly The Symmetry of Optimally Dense Packings Charles Radin Flexible Octahedra in the Hyperbolic Space Hellmuth Stachel Fractal Geometry On Hyperbolic Manifolds Bernd 0. Stratmann A volume formula for generalised hyperbolic tetrahedra Akira Ushijima Part IV Tilings, Orbifolds and Manifolds, Visualization The Geometry of Hyperbolic Manifolds of Dimension at least 4 John G. Ratcliffe Real-Time Animation in Hyperbolic, Spherical, and Product Geometries Jeffrey R. Weeks On spontaneous surgery on knots and links A.D. Mednykh, V.S. Petrov Classification of tile-transitive 3-simplex tilings and their realizations E. Molndr - I. Prok - J. Szirmai Part V Differential Geometry Non-Euclidean Analysis Sigurdur Helgason Holonomy, geometry and topology of manifolds with Grassmann structure Neda Bokan, Paola Matzeu, Zoran Rakic' Hypersurfaces of type number 2 in the hyperbolic four-space OldiLich Kowalski, Masami Sekizawa How far does hyperbolic geometry generalize? Jdnos Szenthe Geometry of the point Finsler spaces Lajos Tamdssy Contents Part VI Physics Black hole perturbations Zoltcin P e e 6 An Idea Whose Time Has Returned Abraham A. Ungar PREFACE JBnos Bolyai is the greatest figure in the history of Hungarian mathematics. He solved the more than two thousand year old problem in connection with Euclid's fifth postulate and discovered non-Euclidean geometry. The glory of the discovery is shared by the Russian Nicolai Ivanovich Lobachevskii but it brought more pain and bitterness than joy to the innovators. Non-Euclidean geometry fundamentally changed our views about geometry and mathematics, in general. Some historians state that since the time of the ancient Greeks there has never been such a great revolution in mathematics than the one originating in the works of Bolyai and Lobachevskii. It became clear that geometry and reality may be different and geometry does not belong to natural sciences. The same is true for the other branches of mathematics. By the time the famous Greek mathematicians Thales and Pythagoras introduced deductive reasoning into mathematics and their followers, primarily Euclid, systematized mathematical knowledge and clarified which are the assertions that we accept without proof and which are the ones we need to prove, it was only a matter of time to learn that our mathematical way of thinking is based on abstract structures. Mathematics does not address reality in a direct manner but substitutes real life objects by abstract ones, determines their relations to each other and then solves the problems within their structures. These structures or axiomatic systems may or may not adequately describe reality. In successful cases they do and provide us with powerful tools for theoretical and practical problem solutions. For example, non-Euclidean geometrical structures, those created by Bolyai and Lobachevskii as well a,s more general ones, allowed for the development of modern physical theories in the twentieth century. By the end of the nineteenth century almost all branches of mathematics became collections of axiomatic systems and the deductive consequences of the statements within. The advent of computers made mathematics more powerful and contributed tremendously to its applicability. Interestingly, applications also began to use axiomatic systems. xii NON-EUCLIDEAN GEOMETRIES In fact, when we start to solve a practical problem first we create a mathematical model, a collection of notions, that represent real life objects, and their relations. Then we elaborate on it, derive its mathematical properties and solve the computational problems. To commemorate the 200th anniversary of the birth of the great scientist, the Hungarian Academy of Sciences, together with other institutions in Hungary and abroad, organized an international conference on hyperbolic geometry on July 6-12, 2002, in Budapest, at the headquarters of the Academy. Besides the Conference this volume is a tribute to the great scientist and his world-famous scientific achievements. ~ were commemorated in The looth, 150th and 1 7 5 ~anniversaries Kolozsv&r, Budapest and Budapest, respectively, but only lectures in the Hungarian language were presented. The 200th anniversary is special not only because of the round number of years that have elapsed since 1802 but because the Bolyai research reached a significant stage. When J h o s Bolyai died he left behind 14,000 pages of manuscript, out of which 3,000 contain his mathematical notes and the rest his utopian ideas about science and society. Some of these pages have been scrutinized earlier, mostly about 100 years ago, and important letters (e.g., . . . from nothing I have created a new, different world . . . ) and theories (e.g., foundations of the theory of complex numbers) have been discovered. However, most of the 14,000 pages remained unread until 1952. In that year Samu Benka, professor of history in Kolozsv6r (Cluj) began to arrange the manuscripts (which were put into chests after JBnos Bolyai's death on order of the commanding officer of the Marosv&&hely garrison). His work lasted sixteen years. During this time he also scrutinized the nonmathematical texts that had remained unread. Similarly, Elem& Kiss, professor of mathematics in Marosv&s&-hely(TBrgu Mureg) studied the mathematical texts during the 1990s. Both scholars have found interesting and important ideas in the manuscripts and presented them to the world. There are two places, where Bolyai manuscripts, documents and memorabilia are collected: the Teleki Library in Marosv5siirhely and the Library of the Hungarian Academy of Sciences in Budapest. Simultaneously with the conference an exhibition of the most important pieces of the latter was organized in the Gallery of the Academy. At the Conference there were 300 participants from 25 countries. We all were honoured that the most famous geometer of the time, the 95 years old Canadian professor H.S.M. Coxeter, came to Budapest to participate at the Conference and delivered the first plenary talk (the next year we learned the sad news that he had passed away). Preface xiii On the occasion of the anniversary Kinga Szkchenyi made a plaquette of JBnos Bolyai (accepting the relief of the mathematician on the faqade of the Palace of Culture of MarosvAsBrhely to be authentic). Copies of it were given to the main speakers and those who have done outstanding research in connection with J h o s Bolyai. The picture of the plaquette can be seen on page 20. Fig.1 of this volume. A special edition of the Appendix, sponsored by the Hungarian Academy of Sciences, printed in Latin, English and Hungarian, was published and given to all participants. The Hungarian National Bank issued a 3000 HUF face value silver coin designed by Gyorgy Kiss and the Hungarian Post issued special stamps for the anniversary. The latter could be purchased on site during the conference. We express our special thanks to the Hungarian Academy of Sciences for allowing the use of the main building of the Academy, together with its equipment. We are also grateful to the Manuscript Section of the Library of the H.A.S., especially to its head Marianne Rozsondai and researchers Bkla M&zi and Kiiroly Horiinyi, for the organization of the exhibition of the "Bolyai Collection" of the Academy. For the excellent organizational work in connection with the Conference our thanks should go to the Conference Organizing Group of the Computing and Automation Institute of the H.A.S., in particular to its head GusztBv Hencsey and his associate Viktor Richter, who were mainly in charge. Many thanks should go to the members of the Program and Organizing Committees as well as those who contributed to this volume, including Dr Attila Bolcskei and Ms Ildik6 Szab6, who made the collection of papers ready for print. Last but not least we express our thanks to Springer Publishers for the publication of this memorial volume. AndrBs Prkkopa member of the Hungarian Academy of Sciences chairman of the Jdnos Bolyai Conference editor Emil Molniir editor