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Configurations of Points and Lines Configurations of Points and Lines "RANKO'RüNBAUM 'RADUATE3TUDIES IN-ATHEMATICS 6OLUME !MERICAN-ATHEMATICAL3OCIETY Configurations of Points and Lines Configurations of Points and Lines Branko Grünbaum Graduate Studies in Mathematics Volume 103 American Mathematical Society Providence, Rhode Island Editorial Board David Cox (Chair) Steven G. Krantz Rafe Mazzeo Martin Scharlemann 2000 Mathematics Subject Classification. Primary 01A55, 01A60, 05–03, 05B30, 05C62, 51–03, 51A20, 51A45, 51E30, 52C30. For additional information and updates on this book, visit www.ams.org/bookpages/gsm-103 Library of Congress Cataloging-in-Publication Data Gr¨unbaum, Branko. Configurations of points and lines / Branko Gr¨unbaum. p. cm. — (Graduate studies in mathematics ; v. 103) Includes bibliographical references and index. ISBN 978-0-8218-4308-6 (alk. paper) 1. Configurations. I. Title. QA607.G875 2009 516.15—dc22 2009000303 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA. Requests can also be made by e-mail to [email protected]. c 2009 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10987654321 141312111009 Dedicated to the memory of my teachers: Milenko Vucˇkic´ (1911–1981) Stanko Bilinski (1909–1998) Abraham Halevi (Adolf) Fraenkel (1891–1965) Aryeh Dvoretzky (1916–2008) Contents Preface xi Chapter 1. Beginnings 1 §1.1. Introduction 1 §1.2. An informal history of configurations 8 §1.3. Basic concepts and definitions 14 §1.4. Tools for the study of configurations 26 §1.5. Symmetry 32 §1.6. Reduced Levi graphs 38 §1.7. Derived figures and other tools 47 Chapter 2. 3-Configurations 59 §2.0. Overview 59 §2.1. Existence of 3-configurations 60 §2.2. Enumeration of 3-configurations (Part 1) 69 §2.3. Enumeration of 3-configurations (Part 2) 81 §2.4. General constructions for combinatorial 3-configurations 89 §2.5. Steinitz’s theorem—the combinatorial part 92 §2.6. Steinitz’s theorem—the geometric part 102 §2.7. Astral 3-configurations with cyclic symmetry group 110 §2.8. Astral 3-configurations with dihedral symmetry group 121 §2.9. Multiastral 3-configurations 134 §2.10. Duality of astral 3-configurations 144 §2.11. Open problems (and a few exercises) 150 vii viii Contents Chapter 3. 4-Configurations 155 §3.0. Overview 155 §3.1. Combinatorial 4-configurations 156 §3.2. Existence of topological and geometric 4-configurations 161 §3.3. Constructions of geometric 4-configurations 169 §3.4. Existence of geometric 4-configurations 186 §3.5. Astral 4-configurations 190 §3.6. 2-Astral 4-configurations 203 §3.7. 3-Astral 4-configurations 212 §3.8. k-Astral configurations for k ≥ 4 223 §3.9. Open problems 231 Chapter 4. Other Configurations 233 §4.0. Overview 233 §4.1. 5-configurations 234 §4.2. k-configurations for k ≥ 6 239 §4.3. [3, 4]- and [4, 3]-configurations 242 §4.4. Unbalanced [q, k]-configurations with [q, k] =[3 , 4] 253 §4.5. Floral configurations 261 §4.6. Topological configurations 280 §4.7. Unconventional configurations 286 §4.8. Open problems 293 Chapter 5. Properties of Configurations 295 §5.0. Overview 295 §5.1. Connectivity of configurations 296 §5.2. Hamiltonian multilaterals 308 §5.3. Multilateral decompositions 322 §5.4. Multilateral-free configurations 327 §5.5. “Density” of trilaterals in configurations 342 §5.6. The dimension of a configuration 344 §5.7. Movable configurations 349 §5.8. Automorphisms and duality 359 §5.9. Open problems 374 Postscript 375 Appendix 377 Contents ix The Euclidean, projective, and extended Euclidean planes 377 References 385 Index 397 Preface It is easy to explain the concept of a configuration of points and lines to any ten-years-old youngster. Why then a book on this topic in a graduate series? There are several good reasons: • First and foremost, configurations are mathematically challenging even though easily accessible. • The study of configurations leans on many fields: classical geom- etry, combinatorics, topology, algebraic geometry, computing, and even analysis and number theory. • There is a visual appeal to many types of configurations. • There are opportunities for serious innovation that do not rely on long years of preliminary study. The truly remarkable aspect of configurations is the scarcity of results in a field that was explicitly started well over a century ago, and informally much earlier. One of the foremost aims of the present text is to make avail- able, essentially for the first time ever, a coherent account of the material. Historical aspects are presented in order to enable the reader to follow the advances (as well as the occasional retreats) of the understanding of configurations. As explained more fully in the text, an initial burst of en- thusiasm in the late nineteenth century produced several basic results. For almost a century, these were not matched by any comparably important new achievements. But near the end of the last century it turned out that the early results were incorrect, and this became part of the impetus for a reinvigorated study of configurations. xi xii Preface The recent realization that symmetries may play an important role in the investigations of configurations provided additional points of view on configurations. Together with the increased ability to actually draw configurations—made possible by advances in computer graphics—the stage was set for renewed efforts in correcting the ancient mistakes and to studying configurations that were never contemplated in the past. This text relies very heavily on the graphical presentation of config- urations. This is practically inevitable considering the topic and greatly simplifies the description of the many types of configurations dealt with. Most of the diagrams have been crafted using Mathematica,Geometers Sketchpad,andClarisDraw, often in combination. In many respects this is a “natural history” of configurations—the prop- erties and methods of generation depend to a large extent on the kind of configuration, and we present them in separate chapters and sections. We have avoided insisting on proofs of properties that are visually ob- vious to such an extent that formal proofs would needlessly lengthen the exposition and make it quite boring. We firmly believe that an appropriate diagram is as much of a valid argument as a pedantic verbal explanation, besides being more readily understandable. It is hoped that the reader will agree! The text is narrowly restricted to the topic of its title. There are many other kinds of configurations that might have been included. However, the nature of such configurations, for example, of points and planes, or of various higher-dimensional flats, is totally different from our topic. It is well possible that the early attempts to cover all possibilities led to very general definitions followed by very meager results. Two exceptions to the restricted character of the presentation concern combinatorial configurations and topological configurations. The former are essential to the theory of geometric configurations, and we present the topic with this aim in mind. We do not enlarge on the various more general as- pects of combinatorial designs and finite geometries—there are many excel- lent texts on these matters. Completely different is the situation regarding topological configurations. Very little is known about them, and the present text collects most of what is available. One other aspect not covered here is the detailed investigations of the hierarchies of some special configurations. It seems that at one time it was fashionable to start with a simple result, such as the theorem of Pascal, and generate a whole family of objects by permuting the starting elements, then considering all the intersections of the resulting lines and the lines generated by the obtained points, etc. This way one could secure a family of points or Preface xiii lines or whatever to be attached to one’s name. The interested reader may gain access to this literature through other means. For almost all the material covered, we provided as ample and detailed references as we were able to find. However, we did not give details con- cerning the programs that produced the various computer-generated enu- merations. The reason—besides lack of competence—is that the programs and the computers on which they run change too rapidly for any printed information to be of lasting value. The interested reader should contact the authors of these results to obtain the most up-to-date status. Results for which no reference is given are the author’s and appear here for the first time. Also, as noted in appropriate places, several colleagues have been kind enough to allow the inclusion of their unpublished results—I am greatly indebted to them for this courtesy. My gratitude goes to several other people and institutions. The Ameri- can Mathematical Society was extremely helpful at all stages of the prepa- ration of this text; in particular, allowing the illustrations to be in color has greatly increased the appeal of the book, as well as its instructional value.
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