DOCSLIB.ORG

A Geometrical Picture Book

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
A Geometrical Picture Book Burkard Polster A Geometrical Picture Book With 405 Illustrations Springer Contents I Finite Geometries 1 1 Introduction via the Fano Plane 3 1.1 Geometries—Basic Facts and Conventions 3 1.2 Projective Planes 5 1.3 Affine Planes 7 1.4 Automorphisms 9 1.5 Polarities 11 1.6- Ovals and Hyperovals 12 1.7 Blocking Sets 13 1.8 Difference Sets and Singer Diagrams 13 1.9 Incidence Graphs 15 1.10 Spatial Models 15 2 Designs 19 2.1 The Smallest Nontrivial 2-Design 20 2.2 Hadamard Designs 21 2.2.1 The One-Point Extension of the Fano Plane 21 2.3 Steiner Triple Systems 23 2.3.1 Kirkman's Schoolgirl Problem 25 3 Configurations 27 3.1 Configurations with Three Points on a Line 28 xvi Contents 3.1.1 The Fano, Pappus, and Desargues Configurations . 29 3.1.2 The Configurations with Parameters (73) and (83) . 30 3.1.3 The Configurations with Parameters (93) 31 3.1.4 The Configurations with Parameters (IO3) 32 3.2 Configurations with Four Points on a Line 36 3.3 Tree-Planting Puzzles 37 4 Generalized Quadrangles 39 4.1 The Generalized Quadrangle of Order (2,2) 41 4.1.1 A Plane Model—The Doily 41 4.1.2 A Model on the Tetrahedron 43 4.1.3 A Model on the Icosahedron 48 4.1.4 A Model in Four-Space 50 4.2 The Petersen Graph 50 4.3 How to Construct the Models 53 4.4 The Generalized Quadrangle of Order (2,4) 58 4.5 The Generalized Quadrangle of Order (4,2) 60 4.6 Symmetric Designs and Generalized Quadrangles 65 4.7 Incidence Graph of a Generalized Quadrangle 65 5 The Smallest Three-Dimensional Projective Space 67 5.1 A Plane Model 69 5.1.1 Line Pencils 70 5.1.2 Subplanes 71 5.1.3 Spreads and Packings . 71 5.1.4 Reguli 75 5.1.5 Ovoids 76 5.1.6 A Labelling with Fano Planes 77 5.1.7 Fake Generalized Quadrangles . 80 5.1.8 The Hoffmann-Singleton Graph 81 5.2 Spatial Models 82 5.2.1 A Model on the Tetrahedron 82 5.2.2 Other Substructures 86 5.2.3 Spreads 88 5.2.4 A Model on the Icosahedron 90 5.3 Symmetric Designs Associated with Our Space 91 6 The Projective Plane of Order 3 93 6.1 More Models of the Affine Plane of Order 3 93 6.1.1 Two Triangular Models 94 6.1.2 Maximizing the Number of Straight Lines 96 6.2 Projective Extension 98 6.2.1 Blocking Sets 100 6.2.2 Desargues Configuration 101 Contents xvii 6.3 Singer Diagram and Incidence Graph 102 6.4 A Spatial Model on the Cube 103 6.5 Extending the Affine Plane to a 5-Design 104 7 The Projective Plane of Order 4 107 7.1 A Plane Model 108 7.2 Constructing the Plane Around a Unital 110 7.3 A partition into Three Fano Planes 112 7.4 A Spatial Model Around a Generalized Quadrangle 113 7.5 Another Partition into Fano Planes 115 7.6 Singer Diagram 116 8 The Projective Plane of Order 5 117 8.1 Beutelspacher's Model 117 8.2 A' Spatial Model on the Dodecahedron 120 8.3 The Desargues Configuration Revisited 121 9 Stargazing in Affine Planes up to Order 8 125 9.1 Star Diagrams of the Affine Planes of Orders 2 and 3 .... 126 9.2 Star Diagram of the Affine Plane of Order 4 128 9.3 Star Diagram of the Affine Plane of Order 5 129 9.4 Star Diagram of the Affine Plane of Order 7 130 9.4.1 The Pascal Configuration in a Conic 130 9.5 Star Diagram of the Affine Plane of Order 8 131 9.5.1 The Fano Configuration 132 9.5.2 An Oval That Is Not a Conic 133 10 Biplanes 135 10.1 The Biplane of Order 2 136 10.2 The Biplane of Order 3 140 10.3 The Three Biplanes of Order 4 143 10.3.1 A First Biplane of Order 4 143 10.3.2 A Second Biplane of Order 4 149 10.3.3 A Third Biplane of Order 4 151 10.4 Two Biplanes of Order 7 152 10.4.1 A First Biplane of Order 7 152 10.4.2 A Second Biplane of Order 7 152 10.5 A Biplane of Order 9 153 10.6 Blocking Sets 153 11 Semibiplanes 155 11.1 The Semibiplanes on Hypercubes 156 11.2 The Semibiplane of Order (12,5) on the Icosahedron .... 158 11.3 Folded Projective Planes 159 xviii Contents 12 The Smallest Benz Planes 161 12.1 The Smallest Inversive Planes 161 12.1.1 The Inversive Plane of Order 2 161 12.1.2 The Inversive Plane of Order 3 163 12.2 A Unifying Definition of Benz Planes 167 12.3 The Smallest Laguerre Planes 168 12.3.1 The Laguerre Plane of Order 2 168 12.3.2 The Laguerre Plane of Order 3 170 12.4 The Smallest Minkowski Planes 171 12.4.1 The Minkowski Plane of Order 2 171 12.4.2 The Minkowski Plane of Order 3 171 13 Generalized Polygons 173 13.1 The Generalized Hexagon of Order (1,2) 174 13.2 The Generalized Hexagon of Order (1,3) 175 13.3 The Two Generalized Hexagons of Order (2,2) 176 13.4 The Generalized Octagon of Order (1,2) 178 13.5 The Generalized 12-gon of Order (1,2) 179 13.6 Cages 180 14 Colour Pictures and Building the Models 185 15 Some Fun Games and Puzzles 197 15.1 The Game "Set"—Line Spotting in 4-D 197 15.2 Mill and Ticktacktoe on Geometries 199 15.3 Circular Walks on Geometries 200 15.4 Which Generalized Quadrangles Are Magical? 200 15.5 Question du Lapin 201 II Geometries on Surfaces 203 16 Introduction via Flat Affine Planes 205 16.1 Some More Basic Facts and Conventions 205 16.2 The Euclidean Plane—A Flat Affine Plane 206 16.2.1 Proper R2-Planes 206 16.2.2 Pencils, Parallel Classes, Generator-Only Pictures . 206 16.2.3 The Group Dimension 207 16.2.4 Topological Geometries 208 16.2.5 The Way Lines Intersect 208 16.2.6 Ovals and Maximal Arcs 209 16.3 Nonclassical R2-Planes 210 16.3.1 Moulton Planes 210 16.3.2 Shift Planes . 211 16.3.3 Arc Planes 213 Contents xix 16.3.4 Integrated Foliations 214 16.3.5 Gluing Constructions 214 16.4 Classification 217 16.5 Semibiplanes 218 17 Flat Circle Planes—An Overview 219 17.1 The "Axiom of Joining" and the "Map" 219 17.2 Nested Flat Circle Planes 221 17.3 Interpolation 222 18 Flat Projective Planes 223 18.1 Models of the Real Projective Plane 224 18.1.1 The Euclidean Plane Plus Its Line at Infinity .... 225 18.1.2 The Geometry of Great Circles 225 18.1.3 Mobius Strip—Antiperiodic 2-Unisolvent Set .... 226 18.1.4 A Disk Model 227 18.2 Recycled Nonclassical Projective Planes 228 18.3 Salzmann's Classification 229 19 Spherical Circle Planes 231 19.1 Intro via Ovoidal Spherical Circle Planes 231 19.2 The Miquel and Bundle Configurations 233 19.3 Nonclassical Flat Spherical Circle Planes 236 19.3.1 The Affine Part 236 19.3.2 Ewald's Affine Parts of Flat Mobius Planes 236 19.3.3 Steinke's Semiclassical Affine Parts 238 19.4 Classification 238 19.5 Subgeometries 238 19.5.1 Double Covers of Flat Projective Planes 239 19.5.2 Recycled Projective Planes 239 19.6 Lie Geometries Associated with Flat Mobius Planes .... 240 19.6.1 The Apollonius Problem 240 19.6.2 Semibiplanes 241 19.6.3 The Geometry of Oriented Lines and Circles .... 242 20 Cylindrical Circle Planes of Rank 3 245 20.1 Intro via Ovoidal Cylindrical Circle Planes 245 20.2 The Miquel and Bundle Configurations 247 20.3 Nonclassical Flat Cylindrical Circle Planes 249 20.3.1 Maurer's Construction 249 20.3.2 The Affine Part 250 20.3.3 The Artzy-Groh Construction 250 20.3.4 The Lowen-Pfiiller Construction 251 20.3.5 Steinke's Two Types of Semiclassical Planes 252 20.3.6 Integrated Flat Affine Planes 252 xx Contents 20.4 Classification 252 20.5 Subgeometries 252 20.5.1 Recycled Projective Planes 253 20.5.2 Semibiplanes 255 20.6 Lie Geometries Associated with Flat Laguerre Planes . 255 20.6.1 Biaffine Planes 255 20.6.2 More Semibiplanes 257 20.6.3 Generalized Quadrangles 258 21 Toroidal Circle Planes 259 21.1 Intro via the Classical Minkowski Plane 259 21.2 The Miquel, Bundle, and Rectangle Configurations ..... 261 21.3 Nonclassical Toroidal Circle Planes 262 21.3.1 The Affine Part 262 21.3.2 The Two Parts of a Toroidal Circle Plane 263 21.3.3 The Artzy-Groh Construction 264 21.3.4 Semiclassical Planes 264 21.3.5 Proper Toroidal Circle Planes 265 21.4 Classification 265 21.5 Subgeometries 266 21.5.1 Flat Projective Planes Minus Convex Disks 266 A Models on Regular Solids 267 B Mirror Technique Stereograms 269 B.I The Generalized Quadrangle of Order (4,2) 269 B.2 Two PCs with Parameters 103 272 B.3 The Smallest Projective Space 273 References 279 Index 287.
Load more

Recommended publications
  • Binomial Partial Steiner Triple Systems Containing Complete Graphs
    Graphs and Combinatorics DOI 10.1007/s00373-016-1681-3 ORIGINAL PAPER Binomial partial Steiner triple systems containing complete graphs Małgorzata Pra˙zmowska1 · Krzysztof Pra˙zmowski2 Received: 1 December 2014 / Revised: 25 January 2016 © The Author(s) 2016. This article is published with open access at Springerlink.com Abstract We propose a new approach to studies on partial Steiner triple systems con- sisting in determining complete graphs contained in them. We establish the structure which complete graphs yield in a minimal PSTS that contains them. As a by-product we introduce the notion of a binomial PSTS as a configuration with parameters of a minimal PSTS with a complete subgraph. A representation of binomial PSTS with at least a given number of its maximal complete subgraphs is given in terms of systems of perspectives. Finally, we prove that for each admissible integer there is a binomial PSTS with this number of maximal complete subgraphs. Keywords Binomial configuration · Generalized Desargues configuration · Complete graph Mathematics Subject Classification 05B30 · 05C51 · 05B40 1 Introduction In the paper we investigate the structure which (may) yield complete graphs contained in a (partial) Steiner triple system (in short: in a PSTS). Our problem is, in fact, a particular instance of a general question, investigated in the literature, which STS’s B Małgorzata Pra˙zmowska [email protected] Krzysztof Pra˙zmowski [email protected] 1 Faculty of Mathematics and Informatics, University of Białystok, ul. Ciołkowskiego 1M, 15-245 Białystok, Poland 2 Institute of Mathematics, University of Białystok, ul. Ciołkowskiego 1M, 15-245 Białystok, Poland 123 Graphs and Combinatorics (more generally: which PSTS’s) contain/do not contain a configuration of a prescribed type.
    [Show full text]
  • COMBINATORICS, Volume
    http://dx.doi.org/10.1090/pspum/019 PROCEEDINGS OF SYMPOSIA IN PURE MATHEMATICS Volume XIX COMBINATORICS AMERICAN MATHEMATICAL SOCIETY Providence, Rhode Island 1971 Proceedings of the Symposium in Pure Mathematics of the American Mathematical Society Held at the University of California Los Angeles, California March 21-22, 1968 Prepared by the American Mathematical Society under National Science Foundation Grant GP-8436 Edited by Theodore S. Motzkin AMS 1970 Subject Classifications Primary 05Axx, 05Bxx, 05Cxx, 10-XX, 15-XX, 50-XX Secondary 04A20, 05A05, 05A17, 05A20, 05B05, 05B15, 05B20, 05B25, 05B30, 05C15, 05C99, 06A05, 10A45, 10C05, 14-XX, 20Bxx, 20Fxx, 50A20, 55C05, 55J05, 94A20 International Standard Book Number 0-8218-1419-2 Library of Congress Catalog Number 74-153879 Copyright © 1971 by the American Mathematical Society Printed in the United States of America All rights reserved except those granted to the United States Government May not be produced in any form without permission of the publishers Leo Moser (1921-1970) was active and productive in various aspects of combin• atorics and of its applications to number theory. He was in close contact with those with whom he had common interests: we will remember his sparkling wit, the universality of his anecdotes, and his stimulating presence. This volume, much of whose content he had enjoyed and appreciated, and which contains the re• construction of a contribution by him, is dedicated to his memory. CONTENTS Preface vii Modular Forms on Noncongruence Subgroups BY A. O. L. ATKIN AND H. P. F. SWINNERTON-DYER 1 Selfconjugate Tetrahedra with Respect to the Hermitian Variety xl+xl + *l + ;cg = 0 in PG(3, 22) and a Representation of PG(3, 3) BY R.
    [Show full text]
  • Second Edition Volume I
    Second Edition Thomas Beth Universitat¨ Karlsruhe Dieter Jungnickel Universitat¨ Augsburg Hanfried Lenz Freie Universitat¨ Berlin Volume I PUBLISHED BY THE PRESS SYNDICATE OF THE UNIVERSITY OF CAMBRIDGE The Pitt Building, Trumpington Street, Cambridge, United Kingdom CAMBRIDGE UNIVERSITY PRESS The Edinburgh Building, Cambridge CB2 2RU, UK www.cup.cam.ac.uk 40 West 20th Street, New York, NY 10011-4211, USA www.cup.org 10 Stamford Road, Oakleigh, Melbourne 3166, Australia Ruiz de Alarc´on 13, 28014 Madrid, Spain First edition c Bibliographisches Institut, Zurich, 1985 c Cambridge University Press, 1993 Second edition c Cambridge University Press, 1999 This book is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 1999 Printed in the United Kingdom at the University Press, Cambridge Typeset in Times Roman 10/13pt. in LATEX2ε[TB] A catalogue record for this book is available from the British Library Library of Congress Cataloguing in Publication data Beth, Thomas, 1949– Design theory / Thomas Beth, Dieter Jungnickel, Hanfried Lenz. – 2nd ed. p. cm. Includes bibliographical references and index. ISBN 0 521 44432 2 (hardbound) 1. Combinatorial designs and configurations. I. Jungnickel, D. (Dieter), 1952– . II. Lenz, Hanfried. III. Title. QA166.25.B47 1999 5110.6 – dc21 98-29508 CIP ISBN 0 521 44432 2 hardback Contents I. Examples and basic definitions .................... 1 §1. Incidence structures and incidence matrices ............ 1 §2. Block designs and examples from affine and projective geometry ........................6 §3. t-designs, Steiner systems and configurations .........
    [Show full text]
  • Graph Theory Graph Theory (II)
    J.A. Bondy U.S.R. Murty Graph Theory (II) ABC J.A. Bondy, PhD U.S.R. Murty, PhD Universite´ Claude-Bernard Lyon 1 Mathematics Faculty Domaine de Gerland University of Waterloo 50 Avenue Tony Garnier 200 University Avenue West 69366 Lyon Cedex 07 Waterloo, Ontario, Canada France N2L 3G1 Editorial Board S. Axler K.A. Ribet Mathematics Department Mathematics Department San Francisco State University University of California, Berkeley San Francisco, CA 94132 Berkeley, CA 94720-3840 USA USA Graduate Texts in Mathematics series ISSN: 0072-5285 ISBN: 978-1-84628-969-9 e-ISBN: 978-1-84628-970-5 DOI: 10.1007/978-1-84628-970-5 Library of Congress Control Number: 2007940370 Mathematics Subject Classification (2000): 05C; 68R10 °c J.A. Bondy & U.S.R. Murty 2008 Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or trans- mitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms of licenses issued by the Copyright Licensing Agency. Enquiries concerning reproduction outside those terms should be sent to the publishers. The use of registered name, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant laws and regulations and therefore free for general use. The publisher makes no representation, express or implied, with regard to the accuracy of the information contained in this book and cannot accept any legal responsibility or liability for any errors or omissions that may be made.
    [Show full text]
  • The Pappus Configuration and the Self-Inscribed Octagon
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector MATHEMATICS THE PAPPUS CONFIGURATION AND THE SELF-INSCRIBED OCTAGON. I BY H. S. M. COXETER (Communic&ed at the meeting of April 24, 1976) 1. INTRODUCTION This paper deals with self-dual configurations 8s and 93 in a projective plane, that is, with sets of 8 or 9 points, and the same number of lines, so arranged that each line contains 3 of the points and each point lies on 3 of the lines. Combinatorially there is only one 83, but there are three kinds of 9s. However, we shall discuss only the kind to which the symbol (9s)i was assigned by Hilbert and Cohn-Vossen [24, pp. 91-98; 24a, pp. 103-1111. Both the unique 8s and this particular 9s occur as sub- configurations of the finite projective plane PG(2, 3), which is a 134. The 83 is derived by omitting a point 0, a non-incident line I, all the four points on E, and all the four lines through 0. Similarly, our 9s is derived from the 134 by omitting a point P, a line 1 through P, the remaining three points on I, and the remaining three lines through P. Thus the points of the 83 can be derived from the 9s by omitting any one of its points, but then the lines of the 83 are not precisely a subset of the lines of the 9s. Both configurations occur not only in PG(2, 3) but also in the complex plane, and the 93 occurs in the real plane.
    [Show full text]
  • Geometric Point-Circle Pentagonal Geometries from Moore Graphs
    Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 11 (2016) 215–229 Geometric point-circle pentagonal geometries from Moore graphs Klara Stokes ∗ School of Engineering Science, University of Skovde,¨ 54128 Skovde¨ Sweden Milagros Izquierdo Department of Mathematics, Linkoping¨ University, 58183 Linkoping¨ Sweden Received 27 December 2014, accepted 13 October 2015, published online 15 November 2015 Abstract We construct isometric point-circle configurations on surfaces from uniform maps. This gives one geometric realisation in terms of points and circles of the Desargues configura- tion in the real projective plane, and three distinct geometric realisations of the pentagonal geometry with seven points on each line and seven lines through each point on three distinct dianalytic surfaces of genus 57. We also give a geometric realisation of the latter pentago- nal geometry in terms of points and hyperspheres in 24 dimensional Euclidean space. From these, we also obtain geometric realisations in terms of points and circles (or hyperspheres) of pentagonal geometries with k circles (hyperspheres) through each point and k −1 points on each circle (hypersphere). Keywords: Uniform map, equivelar map, dessin d’enfants, configuration of points and circles Math. Subj. Class.: 05B30, 05B45, 14H57, 14N20, 30F10, 30F50, 51E26 1 Introduction A compact Klein surface S is a surface (possibly with boundary and non-orientable) en- dowed with a dianalytic structure, that is, the transition maps are holomorphic or antiholo- morphic (the conjugation z ! z is allowed). If the surface S admits analytic structure and is closed, then the surface is a Riemann surface.
    [Show full text]
  • Foundations of Projective Geometry
    Foundations of Projective Geometry Robin Hartshorne 1967 ii Preface These notes arose from a one-semester course in the foundations of projective geometry, given at Harvard in the fall term of 1966–1967. We have approached the subject simultaneously from two different directions. In the purely synthetic treatment, we start from axioms and build the abstract theory from there. For example, we have included the synthetic proof of the fundamental theorem for projectivities on a line, using Pappus’ Axiom. On the other hand we have the real projective plane as a model, and use methods of Euclidean geometry or analytic geometry to see what is true in that case. These two approaches are carried along independently, until the first is specialized by the introduction of more axioms, and the second is generalized by working over an arbitrary field or division ring, to the point where they coincide in Chapter 7, with the introduction of coordinates in an abstract projective plane. Throughout the course there is special emphasis on the various groups of transformations which arise in projective geometry. Thus the reader is intro- duced to group theory in a practical context. We do not assume any previous knowledge of algebra, but do recommend a reading assignment in abstract group theory, such as [4]. There is a small list of problems at the end of the notes, which should be taken in regular doses along with the text. There is also a small bibliography, mentioning various works referred to in the preparation of these notes. However, I am most indebted to Oscar Zariski, who taught me the same course eleven years ago.
    [Show full text]
  • AN INFINITE GRAPH of GIRTH 12 by ASIA Ivlfc WEISS
    TRANSACTIONS of the AMERICAN MATHEMATICAL SOCIETY Volume 283. Number 2, June 1984 AN INFINITE GRAPH OF GIRTH 12 BY ASIA IVlfc WEISS Abstract. From the regular hyperbolic honeycomb {3,6,3} we derive regular honeycombs with finite numbers of toroidal cells. Joining centers of faces of these honeycombs to the midpoints of its edges we obtain trivalent symmetrical graphs. We investigate the relation between these honeycombs, their groups and the graphs embedded in them. 1. Introduction. Coxeter and Whitrow [1950, p. 429] showed that the points whose four coordinates are proportional to the integral solutions of the Diophantine equation t2 — x2 —y2 — z2 — 1 are vertices of the hyperbolic honeycomb {4,4, 3}. Coxeter suggested that it would be useful to find coordinates for all regular hyperbolic honeycombs as integral solutions of Diophantine equations since as a by-product we would obtain a simple representation for generators of the symmetry groups of these honeycombs. Garner in his Ph.D. Thesis [1964] discovered coordi- nates for two more hyperbolic honeycombs {6,3,4} and {6,3,3}. In §3 we complete these investigations by giving coordinates for all cristallographic regular hyperbolic honeycombs as well as their truncations. This is then used in §5 to find the group whose Cayley diagram is an infinite trivalent symmetrical graph of girth 12. At the Conference on Graph Theory and Combinatorial Analysis held at the University of Waterloo, R. M. Foster [1966] presented a paper which included a table of 176 trivalent symmetrical graphs. Since then there has been a search to find more. A few more of these graphs have been found by Coxeter, Frucht, Harries, and by Foster himself.
    [Show full text]
  • Combinatorial Configurations, Quasiline Arrangements, And
    Combinatorial configurations, quasiline arrangements, and systems of curves on surfaces J¨urgenBokowski, Jurij Koviˇc,TomaˇzPisanski, and Arjana Zitnikˇ October 7, 2014 Abstract It is well known that not every combinatorial configuration admits a geo- metric realization with points and lines. Moreover, some of them do not even admit realizations with pseudoline arrangements, i.e., they are not topologi- cal. In this paper we provide a new topological representation by using and essentially generalizing the topological representation of oriented matroids in rank 3. These representations can also be interpreted as curve arrangements on surfaces. In particular, we generalize the notion of a pseudoline arrangement to the notion of a quasiline arrangement by relaxing the condition that two pseudo- lines meet exactly once and show that every combinatorial configuration can be realized as a quasiline arrangement in the real projective plane. We also generalize well-known tools from pseudoline arrangements such as sweeps or wiring diagrams. A quasiline arrangement with selected vertices belonging to the configuration can be viewed as a map on a closed surface. Such a map can be used to distinguish between two \distinct" realizations of a combinatorial configuration as a quasiline arrangement. Keywords: pseudoline arrangement, quasiline arrangement, projective plane, in- cidence structure, combinatorial configuration, topological configuration, geometric configuration, sweep, wiring diagram, allowable sequence of permutations, maps on a surface. Math. Subj. Class. (2010): 51E20, 52C30, 05B30. arXiv:1410.2350v1 [math.CO] 9 Oct 2014 1 Introduction For a long time researchers in configurations have neglected the difference between combinatorial and geometric configurations of points and lines. In the nineteenth century it was either tacitly assumed that these concepts coincide or the emphasis was on geometric existence of configurations and the two well-known combinatorial configurations (73) and (83) were automatically excluded from consideration.
    [Show full text]
  • Desargues, Doily, Dualities and Exceptional Isomorphisms
    AUSTRALASIAN JOURNAL OF COMBINATORICS Volume 57 (2013), Pages 251–270 Desargues, doily, dualities and exceptional isomorphisms Bernhild Stroppel Markus Stroppel LExMath — Lehrexportzentrum Mathematik Fakult¨at f¨ur Mathematik und Physik Universit¨at Stuttgart D-70550 Stuttgart Germany [email protected] F¨ur Herrn Professor Dr. G. Pickert aus Anlass seines 95. Geburtstags. Abstract We study various graphs associated with symmetric groups. This leads to a quick understanding of automorphisms, dualities and polarities both for the Desargues configuration and the smallest generalised quadrangle. The results are used to establish the existence of outer automorphisms of the symmetric group on six symbols, and the exceptional isomorphism between the alternating group on six symbols and the little projective group of the projective line over the field with nine elements. For the latter isomorphism, we present a new proof of uniqueness for the inversive plane of order 3. 1 Introduction The Desargues configuration is an incidence structure with ten points and ten lines; it plays an important role in the foundations of geometry (see 1.3 below). It turns out that it is also of great interest of its own; this is due to its inherent symmetry. We give a sleek description of this configuration first: F F 1.1 Definition. Let be a set of 5 elements. As point set, we take the set 2 of F F all subsets of size 2 in ,theset 3 of lines consists of all subsets of size 3, and D F , F , ⊂ incidence is inclusion: := 2 3 . It seems that this labelling dates back to [12, IV, §1].
    [Show full text]
  • Thickly Resolvable Designs by Amanda Malloch B.Sc., University
    Thickly Resolvable Designs by Amanda Malloch B.Sc., University of Victoria, 2007 M.Sc., University of Victoria, 2009 A Dissertation Submitted in Partial Fulfillment of the Requirements for the Degree of DOCTOR OF PHILOSOPHY in the Department of Mathematics and Statistics c Amanda Malloch, 2016 University of Victoria All rights reserved. This dissertation may not be reproduced in whole or in part, by photocopying or other means, without the permission of the author. ii Thickly Resolvable Designs by Amanda Malloch B.Sc., University of Victoria, 2007 M.Sc., University of Victoria, 2009 Supervisory Committee Dr. Peter Dukes, Supervisor (Department of Mathematics and Statistics) Dr. Kieka Mynhardt, Departmental Member (Department of Mathematics and Statistics) Dr. Frank Ruskey, Outside Member (Department of Computer Science) iii Supervisory Committee Dr. Peter Dukes, Supervisor (Department of Mathematics and Statistics) Dr. Kieka Mynhardt, Departmental Member (Department of Mathematics and Statistics) Dr. Frank Ruskey, Outside Member (Department of Computer Science) ABSTRACT In this dissertation, we consider a generalization of the historically significant problem posed in 1850 by Reverend Thomas Kirkman which asked whether it was possible for 15 schoolgirls to walk in lines of three to school for seven days so that no two of them appear in the same line on multiple days. This puzzle spawned the study of what we now call resolvable pairwise balanced designs, which balance pair coverage of points within blocks while also demanding that the blocks can be grouped in such a way that each group partitions the point-set. Our generalization aims to relax this condition slightly, so that each group of blocks balances point-wise coverage but each point occurs in each group σ times (instead of just once).
    [Show full text]
  • DESARGUES' THEOREM Two Triangles ABC and a B C Are Said To
    DESARGUES' THEOREM DONALD ROBERTSON Two triangles ABC and A0B0C0 are said to be in perspective axially when no two vertices are equal and when the three intersection points AC \A0C0, AB \A0B0 and BC\B0C0 are collinear. We say ABC and A0B0C0 are in perspective centrally when no two vertices are equal and when the intersection points A0A \ B0B, A0A \ C0C and B0B \ C0C are all equal. Girard Desargues, the father of projective geometry, proved the following theorem in the 17th century. It was first published in 1648 by Abraham Bosse. Theorem 1. Two triangles in the real projective plane are in perspective centrally if and only if they are in perspective axially. These properties of perspective only make sense when we know that any two lines intersect at exactly one point. Thus the natural setting for studying these properties is projective space, rather than the more familiar affine space. We restrict ourselves to projective planes because in higher dimensions Desargues' theorem follows from the fact that any two planes intersect in a line. 1. Affine and Projective Planes An affine plane consists of a set P of points and a set L of subsets of P satisfying the following three axioms. AP1 Any two distinct points lie on one and only one line. AP2 Given a line l and a point P not contained in l there is one and only one line which contains P and is disjoint from l. AP3 There are three points not all contained in a single line. Members of L are called lines.
    [Show full text]