A Geometrical Picture Book

Burkard Polster

With 405 Illustrations

Springer Contents

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