Configurations of Points and Lines

Conﬁgurations of Points and Lines

"RANKO'RüNBAUM

'RADUATE3TUDIES IN-ATHEMATICS 6OLUME

!MERICAN-ATHEMATICAL3OCIETY http://dx.doi.org/10.1090/gsm/103 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 Grünbaum, Branko. Configurations of points and lines / Branko Grünbaum. 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 geometry, 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 available, 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 configurations. 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 properties 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 aspects 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 preparation 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. I greatly appreciate the attention of the editorial staff to detail and consis- tency, and the generous help they gave me through all stages of publication. The Department of Mathematics of the University of Washington supported my efforts in a variety of ways, both during the several times I gave graduate courses about configurations and in the preparation of the manuscript later on. The staff of the Mathematics Research Library at the university was very helpful in obtaining for me many of the old papers and books (and some new ones) and in guiding me through the labyrinths of the world of digital books and journals. Several stays at the Helen Riaboff Whiteley Center at the Friday Harbor Laboratories of the University of Washington provided the pleasant atmosphere and conditions conducive to work on this book. Special thanks go to my friends and coauthors of recent papers on configurations—L. W. Berman, M. Boben, J. Bokowski, T. Pisanski, and L. Schewe. Their insights and comments, as well as results, encouraged me greatly while adding to the joint enterprises. The students of the courses I gave on configurations have earned my gratitude for their interest in the topic, which inspired me to investigate many questions and write up material in lecture notes. xiv Preface

Last—but certainly not least— my thanks go to my wife Zdenka, not only for her patience and forbearance over the long haul of my study of conﬁgurations and the preparation of the manuscript of this book, but even more for her love and support during well over half a century. Branko Gr¨unbaum Seattle, October 23, 2008

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(3m+) construction, 177, 186 m#(b, c; d; µ), 123 (4m) construction, 172, 186 m#(b, c; d), 111, 144 (5/2m) construction, 186 m#(b1,b2,...,bh; b0), 136 (5/6m) construction, 174 m#(s1,t1; s2,t2; ...,sk,tk), 191 (5m) construction, 169, 186, 240 m#{s, t},200 (6m) construction, 171, 186 r-lateral, 308 (M1), 263, 272 #s-subfiguration, 21 (M2), 263, 265, 272 #s-superfiguration, 20 (M3), 263, 265, 272 1-connected, 301 (M4), 263, 268 2-connected, 85, 298 (DU-1) deleted unions construction, 180 3-configurations, 60 (DU-2) deleted unions constructions, 180 3-connected, 85 (k, g)-cage, 330 4-configurations, 156 5-configurations, 234 (nk), 1 (nk) configuration, 1 (pq,nk), 15 aggregate, 7 (t-A.m), 186 arranged configuration table, 99, 105 C3(n, a, b), 68 arrangement, 31, 163 C3(n, m), 67 astral, 34, 110, 203, 235, 240 E2,377 astral 3-configurations, 110 E3,378 astral 4-configurations, 190, 257 L(C), 28, 296 astral configurations, 335 LC(4), 231 automorphisms, 32, 296, 359 LC(k), 2, 235, 239, 299, 328 [3, 4]-configurations, 242 balanced, 16, 348 [4, 3]-configurations, 242 BB configuration, 126 [h1,h2]-orbital, 33 bipartite, 310 [q, k]-configuration, 15, 253, 318 bipartite graph, 93 BB(m; s, t), 150 bislim geometries, 23 c-connected, 30 bislim geometry, 15 h-astral, 34, 134 blocking set, 321 h-chiral, 135 blocks, 17 h-dihedral, 135 h-orbital, 34 cells, 31 k-astral, 191 characteristic path, 137, 193 k-configuration, 1, 15 chiral, 111, 284

397 398 Index

chiral configurations, 272 Fano configuration, 61, 105 circuit, 70 final polynomials, 63 circuit decomposition, 308 flag-transitive, 361 clade, 201 floral configurations, 261 cohort, 200, 207, 215 floret, 262 cohort symbol, 200 combinatorial configuration, 17, 242 generalized hexagons, 337 combinatorial type, 5, 18 generalized quadrangle, 330 combinatorially equivalent, 5, 18 geometric configuration, 16, 86, 102, 161, complementary graph, 30 243 complex plane, 66 geometric symmetries, 16 configuration, 1, 15 Georges configuration, 312 configuration graph, 28, 53 Georges graph, 311 configuration table, 17, 26 group of symmetries, 33 configurations of points and circles, 286 configurations of pseudolines, 21 Hamiltonian circuits, 156 connected, 30, 298 Hamiltonian multilateral, 25, 156, 308, 330, connected k-configuration, 100 374 connectivity, 296 constructions (DU-1) (DU-2), 186 incidence, 4 Cremona-Richmond, 329, 347 incidence systems, 324 Cremona-Richmond configuration, 2, 144 incidence matrix, 26 cubic curve, 245 incident, 15 cyclic 4-configurations, 231 independence graph, 308 cyclic astral configuration, 319 independent, 304 cyclic configuration, 65, 158, 234, 301, 348 independent family, 374 cyclic configuration C3(n), 60, 91 infinite [k]-configurations, 290 cyclic group cr,34 infinite k-configuration, 288 cyclic symmetry group, 110 inscribed/circumscribed family, 323 inscribed/circumscribed multilateral, 343, DD configurations, 122 374 deficiency graph, 28, 53, 70 inscribed/circumscribed pentalateral, 335 degrees of freedom, 76 irreducible, 89 density, 342 isomorphic, 5, 18 derived figures, 47 isomorphism, 359 Desargues configuration, 2, 87, 324, 345 diagonals of regular polygons, 210 König’s theorem, 93 diagrams, 16 dihedral astral, 351 Levi graph, 28, 296, 308, 330 dihedral group dr,34 Levi incidence matrix, 26, 67 dihedral symmetry group, 121 lines, 15 dimension, 24, 344 disconnected, 298 Martinetti graph, 28 disconnected configuration, 82, 131 Martinetti’s claim, 89 dual, 19 matching, 93 dual configuration, 348 Menger graph, 27 dualities, 32 mirrors, 34 duality, 19, 144, 242, 359 Möbius-Kantor configurations, 68, 105 movable configurations, 349, 374 edges, 31 multiastral, 34 EE configurations, 122 multiastral 3-configurations, 134 elements, 15 multilateral, 25, 98, 308 enumeration, 69 multilateral decomposition, 98, 100, 322 Euclidean, 377 multilateral path, 25, 100, 308 Euclidean plane E2,33 multilateral-free, 327 extended Euclidean plane E2+, 33, 235, 377, 382 oppositely selfpolar, 360 Index 399

orbit, 296 subfiguration, 21 orbit transitivity, 361 superfiguration, 19, 86, 102, 129 orbit type, 33 Sylvester’s problem, 7, 8, 242 orchard problem, 7, 8 symmetric Hamiltonian multilaterals, 319 orderly, 18 symmetry, 32, 359 orderly configuration table, 93, 105, 157 symmetry group, 33 orderly family, 324 symmetry type, 134 ordinary points, 61, 64, 167 systematic, 200, 214

Pappus configuration, 2, 4, 91 theorem of Pappus, 8 Pappus’s theorem, 85 topological configuration, 21, 61, 85, 161, parameters, 15 280 pentalateral-free, 337 trilateral, 71, 308, 342 points, 15 trilateral-free, 328 polar, 145, 199, 359 trivial, 200, 214 polarity, 33, 242 Tutte’s (3, 8)-cage, 330 polycyclic configurations, 35 polygon, 98 unsplittable, 305 prefiguration, 19, 170 projective, 377 vertices, 15, 31 projective plane P 2, 33, 379 voltage graphs, 40 projective transformation, 383 projectively h-astral, 135 weak realization, 102 protofloret, 263, 271 weakly flag-transitive, 365 pseudoline, 21, 61, 85, 163, 281 quadrilateral-free, 335 rank, 366 rational plane, 81, 231 realization, 18, 103 reciprocation, 33 reduced, 217 reduced Levi diagram, 126, 198 reduced Levi graph R(C), 38, 121 reducible, 89 regular, 343 remainder figures, 30, 48, 70 representation, 20, 102, 114 rigid, 350 RLG, 150 Roberts’s theorem, 8 selfdual, 19 selfduality, 19, 359 selfpolar*, 364 selfpolarity, 359 set-configurations, 17, 324 simplicial arrangements, 286 slim geometry, 15 span, 192 splittable, 305 sporadic, 201, 215 sporadic configurations, 205 Steiner triple system, 67, 254 Steinitz’s theorem, 92, 102, 231 subconfigurations, 163

Titles in This Series

103 Branko Grünbaum, Configurations of points and lines, 2009 102 Mark A. Pinsky, Introduction to Fourier analysis and wavelets, 2009 101 Ward Cheney and Will Light, A course in approximation theory, 2009 100 I. Martin Isaacs, Algebra: A graduate course, 2009 99 Gerald Teschl, Mathematical methods in quantum mechanics: With applications to Schrödinger operators, 2009 98 Alexander I. Bobenko and Yuri B. Suris, Discrete differential geometry: Integrable structure, 2008 97 David C. Ullrich, Complex made simple, 2008 96 N. V. Krylov, Lectures on elliptic and parabolic equations in Sobolev spaces, 2008 95 Leon A. Takhtajan, Quantum mechanics for mathematicians, 2008 94 James E. Humphreys, Representations of semisimple Lie algebras in the BGG category O,2008 93 Peter W. Michor, Topics in differential geometry, 2008 92 I. Martin Isaacs, Finite group theory, 2008 91 Louis Halle Rowen, Graduate algebra: Noncommutative view, 2008 90 Larry J. Gerstein, Basic quadratic forms, 2008 89 Anthony Bonato, A course on the web graph, 2008 88 Nathanial P. Brown and Narutaka Ozawa, C∗-algebras and finite-dimensional approximations, 2008 87 Srikanth B. Iyengar, Graham J. Leuschke, Anton Leykin, Claudia Miller, Ezra Miller, Anurag K. Singh, and Uli Walther, Twenty-four hours of local cohomology, 2007 86 Yulij Ilyashenko and Sergei Yakovenko, Lectures on analytic differential equations, 2007 85 John M. Alongi and Gail S. Nelson, Recurrence and topology, 2007 84 Charalambos D. Aliprantis and Rabee Tourky, Cones and duality, 2007 83 Wolfgang Ebeling, Functions of several complex variables and their singularities (translated by Philip G. Spain), 2007 82 Serge Alinhac and Patrick Gérard, Pseudo-differential operators and the Nash–Moser theorem (translated by Stephen S. Wilson), 2007 81 V. V. Prasolov, Elements of homology theory, 2007 80 Davar Khoshnevisan, Probability, 2007 79 William Stein, Modular forms, a computational approach (with an appendix by Paul E. Gunnells), 2007 78 Harry Dym, Linear algebra in action, 2007 77 Bennett Chow, Peng Lu, and Lei Ni, Hamilton’s Ricci flow, 2006 76 Michael E. Taylor, Measure theory and integration, 2006 75 Peter D. Miller, Applied asymptotic analysis, 2006 74 V. V. Prasolov, Elements of combinatorial and differential topology, 2006 73 Louis Halle Rowen, Graduate algebra: Commutative view, 2006 72 R. J. Williams, Introduction the the mathematics of finance, 2006 71 S. P. Novikov and I. A. Taimanov, Modern geometric structures and fields, 2006 70 Seán Dineen, Probability theory in finance, 2005 69 Sebastián Montiel and Antonio Ros, Curves and surfaces, 2005 68 Luis Caffarelli and Sandro Salsa, A geometric approach to free boundary problems, 2005 67 T.Y. Lam, Introduction to quadratic forms over fields, 2004 TITLES IN THIS SERIES

66 Yuli Eidelman, Vitali Milman, and Antonis Tsolomitis, Functional analysis, An introduction, 2004 65 S. Ramanan, Global calculus, 2004 64 A. A. Kirillov, Lectures on the orbit method, 2004 63 Steven Dale Cutkosky, Resolution of singularities, 2004 62 T. W. Körner, A companion to analysis: A second first and first second course in analysis, 2004 61 Thomas A. Ivey and J. M. Landsberg, Cartan for beginners: Differential geometry via moving frames and exterior differential systems, 2003 60 Alberto Candel and Lawrence Conlon, Foliations II, 2003 59 Steven H. Weintraub, Representation theory of finite groups: algebra and arithmetic, 2003 58 Cédric Villani, Topics in optimal transportation, 2003 57 Robert Plato, Concise numerical mathematics, 2003 56 E. B. Vinberg, A course in algebra, 2003 55 C. Herbert Clemens, A scrapbook of complex curve theory, second edition, 2003 54 Alexander Barvinok, A course in convexity, 2002 53 Henryk Iwaniec, Spectral methods of automorphic forms, 2002 52 Ilka Agricola and Thomas Friedrich, Global analysis: Differential forms in analysis, geometry and physics, 2002 51 Y. A. Abramovich and C. D. Aliprantis, Problems in operator theory, 2002 50 Y. A. Abramovich and C. D. Aliprantis, An invitation to operator theory, 2002 49 John R. Harper, Secondary cohomology operations, 2002 48 Y. Eliashberg and N. Mishachev, Introduction to the h-principle, 2002 47 A. Yu. Kitaev, A. H. Shen, and M. N. Vyalyi, Classical and quantum computation, 2002 46 Joseph L. Taylor, Several complex variables with connections to algebraic geometry and Lie groups, 2002 45 Inder K. Rana, An introduction to measure and integration, second edition, 2002 44 Jim Agler and John E. McCarthy, Pick interpolation and Hilbert function spaces, 2002 43 N. V. Krylov, Introduction to the theory of random processes, 2002 42 Jin Hong and Seok-Jin Kang, Introduction to quantum groups and crystal bases, 2002 41 Georgi V. Smirnov, Introduction to the theory of differential inclusions, 2002 40 Robert E. Greene and Steven G. Krantz, Function theory of one complex variable, third edition, 2006 39 Larry C. Grove, Classical groups and geometric algebra, 2002 38 Elton P. Hsu, Stochastic analysis on manifolds, 2002 37 Hershel M. Farkas and Irwin Kra, Theta constants, Riemann surfaces and the modular group, 2001 36 Martin Schechter, Principles of functional analysis, second edition, 2002 35 James F. Davis and Paul Kirk, Lecture notes in algebraic topology, 2001 34 Sigurdur Helgason, Differential geometry, Lie groups, and symmetric spaces, 2001 33 Dmitri Burago, Yuri Burago, and Sergei Ivanov, A course in metric geometry, 2001 32 Robert G. Bartle, A modern theory of integration, 2001

For a complete list of titles in this series, visit the AMS Bookstore at www.ams.org/bookstore/. This is the only book on the topic of geometric configurations of points and lines. It presents in detail the history of the topic, with its surges and declines since its beginning in 1876. It covers all the advances in the field since the revival of interest in geometric configurations some 20 years ago. The author’s contributions are central to this revival. In particular, he initiated the study of 4- configurations (that is, those that contain four points on each line, and four lines through each point); the results are fully described

in the text. The main novelty in the approach to all geometric Photo courtesy of Ina Mette. conﬁgurations is the concentration on their symmetries, which make it possible to deal with conﬁgurations of rather large sizes. The book brings the readers to the limits of present knowledge in a leisurely way, enabling them to enjoy the material as well as entice them to try their hand at expanding it.

For additional information and updates on this book, visit WWWAMSORGBOOKPAGESGSM

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