## A Geometrical Picture BookHow do you convey to your students, colleagues and friends some of the beauty of the kind of mathematics you are obsessed with? If you are a mathematician interested in finite or topological geometry and combinatorial designs, you could start by showing them some of the (400+) pictures in the "picture book". Pictures are what this book is all about; original pictures of everybody's favorite geometries such as configurations, projective planes and spaces, circle planes, generalized polygons, mathematical biplanes and other designs which capture much of the beauty, construction principles, particularities, substructures and interconnections of these geometries. The level of the text is suitable for advanced undergraduates and graduate students. Even if you are a mathematician who just wants some interesting reading you will enjoy the author's very original and comprehensive guided tour of small finite geometries and geometries on surfaces This guided tour includes lots of sterograms of the spatial models, games and puzzles and instructions on how to construct your own pictures and build some of the spatial models yourself. |

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### Contents

3 | |

Designs | 19 |

Configurations | 27 |

Generalized Quadrangles | 39 |

The Smallest ThreeDimensional Projective Space | 67 |

The Projective Plane of Order 3 93 | 92 |

The Projective Plane of Order 4 | 107 |

The Projective Plane of Order 5 | 117 |

Generalized Polygons 173 | 172 |

Colour Pictures and Building the Models | 185 |

Some Fun Games and Puzzles | 197 |

Introduction via Flat Affine Planes | 205 |

Flat Circle PlanesAn Overview | 219 |

Spherical Circle Planes 231 | 230 |

Cylindrical Circle Planes of Rank 3 | 245 |

Toroidal Circle Planes | 259 |

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### Common terms and phrases

2-design APC with parameters automorphisms Benz planes biplane of order Chapter classical projective planes complete graph construction correspond cube cylindrical circle plane derived geometry Desargues configuration dodecahedron edges Euclidean plane example Fano plane finite flat affine planes flat circle planes flat Laguerre plane flat Minkowski plane flat Möbius plane flat projective plane generator-only model generator-only picture grid group dimension hexagon homeomorphism Hussain graphs hyperoval icosahedron incidence graph interior points inversive plane isomorphic labels line pencils line set Minkowski planes Möbius strip Moulton planes nontrivial number of points ovoid Pappus configuration parallel classes partition Petersen graph plane of order plane of rank planes with group point set points and lines points are contained projective space quadrangle of order R*-plane real projective plane Section semibiplane sphere spherical circle plane star diagrams stereograms subplanes synthemes tetrahedron topological circles topological lines topological oval toroidal circle plane triangle unique vertex vertices