A Geometrical Picture Book

Overview

How 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 ...

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Overview

How 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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Product Details

  • ISBN-13: 9781461264262
  • Publisher: Springer New York
  • Publication date: 12/31/2012
  • Series: Universitext Series
  • Edition description: Softcover reprint of the original 1st ed. 1998
  • Edition number: 1
  • Pages: 292

Table of Contents

I Finite Geometries.- 1 Introduction via the Fano Plane.- 1.1 Geometries—Basic Facts and Conventions.- 1.2 Projective Planes.- 1.3 Affine Planes.- 1.4 Automorphisms.- 1.5 Polarities.- 1.6 Ovals and Hyperovals.- 1.7 Blocking Sets.- 1.8 Difference Sets and Singer Diagrams.- 1.9 Incidence Graphs.- 1.10 Spatial Models.- 2 Designs.- 2.1 The Smallest Nontrivial 2-Design.- 2.2 Hadamard Designs.- 2.2.1 The One-Point Extension of the Fano Plane.- 2.3 Steiner Triple Systems.- 2.3.1 Kirkman’s Schoolgirl Problem.- 3 Configurations.- 3.1 Configurations with Three Points on a Line.- 3.1.1 The Fano, Pappus, and Desargues Configurations.- 3.1.2 The Configurations with Parameters (73) and (83).- 3.1.3 The Configurations with Parameters (93).- 3.1.4 The Configurations with Parameters (1033).- 3.2 Configurations with Four Points on a Line.- 3.3 Tree-Planting Puzzles.- 4 Generalized Quadrangles.- 4.1 The Generalized Quadrangle of Order (2,2).- 4.1.1 A Plane Model—The Doily.- 4.1.2 A Model on the Tetrahedron.- 4.1.3 A Model on the Icosahedron.- 4.1.4 A Model in Four-Space.- 4.2 The Petersen Graph.- 4.3 How to Construct the Models.- 4.4 The Generalized Quadrangle of Order (2,4).- 4.5 The Generalized Quadrangle of Order (4,2).- 4.6 Symmetric Designs and Generalized Quadrangles.- 4.7 Incidence Graph of a Generalized Quadrangle.- 5 The Smallest Three-Dimensional Projective Space.- 5.1 A Plane Model.- 5.1.1 Line Pencils.- 5.1.2 Subplanes.- 5.1.3 Spreads and Packings.- 5.1.4 Reguli.- 5.1.5 Ovoids.- 5.1.6 A Labelling with Fano Planes.- 5.1.7 Fake Generalized Quadrangles.- 5.1.8 The Hoffmann-Singleton Graph.- 5.2 Spatial Models.- 5.2.1 A Model on the Tetrahedron.- 5.2.2 Other Substructures.- 5.2.3 Spreads.- 5.2.4 A Model on the Icosahedron.- 5.3 Symmetric Designs Associated with Our Space.- 6 The Projective Plane of Order 3.- 6.1 More Models of the Affine Plane of Order 3.- 6.1.1 Two Triangular Models.- 6.1.2 Maximizing the Number of Straight Lines.- 6.2 Projective Extension.- 6.2.1 Blocking Sets.- 6.2.2 Desargues Configuration.- 6.3 Singer Diagram and Incidence Graph.- 6.4 A Spatial Model on the Cube.- 6.5 Extending the Affine Plane to a 5-Design.- 7 The Projective Plane of Order 4.- 7.1 A Plane Model.- 7.2 Constructing the Plane Around a Unital.- 7.3 A partition into Three Fano Planes.- 7.4 A Spatial Model Around a Generalized Quadrangle.- 7.5 Another Partition into Fano Planes.- 7.6 Singer Diagram.- 8 The Projective Plane of Order 5.- 8.1 Beutelspacher’s Model.- 8.2 A Spatial Model on the Dodecahedron.- 8.3 The Desargues Configuration Revisited.- 9 Stargazing in Affine Planes up to Order 8.- 9.1 Star Diagrams of the Affine Planes of Orders 2 and 3.- 9.2 Star Diagram of the Affine Plane of Order 4.- 9.3 Star Diagram of the Affine Plane of Order 5.- 9.4 Star Diagram of the Affine Plane of Order 7.- 9.4.1 The Pascal Configuration in a Conic.- 9.5 Star Diagram of the Affine Plane of Order 8.- 9.5.1 The Fano Configuration.- 9.5.2 An Oval That Is Not a Conic.- 10 Biplanes.- 10.1 The Biplane of Order 2.- 10.2 The Biplane of Order 3.- 10.3 The Three Biplanes of Order 4.- 10.3.1 A First Biplane of Order 4.- 10.3.2 A Second Biplane of Order 4.- 10.3.3 A Third Biplane of Order 4.- 10.4 Two Biplanes of Order 7.- 10.4.1 A First Biplane of Order 7.- 10.4.2 A Second Biplane of Order 7.- 10.5 A Biplane of Order 9.- 10.6 Blocking Sets.- 11 Semibiplanes.- 11.1 The Semibiplanes on Hypercubes.- 11.2 The Semibiplane of Order (12,5) on the Icosahedron.- 11.3 Folded Projective Planes.- 12 The Smallest Benz Planes.- 12.1 The Smallest Inversive Planes.- 12.1.1 The Inversive Plane of Order 2.- 12.1.2 The Inversive Plane of Order 3.- 12.2 A Unifying Definition of Benz Planes.- 12.3 The Smallest Laguerre Planes.- 12.3.1 The Laguerre Plane of Order 2.- 12.3.2 The Laguerre Plane of Order 3.- 12.4 The Smallest Minkowski Planes.- 12.4.1 The Minkowski Plane of Order 2.- 12.4.2 The Minkowski Plane of Order 3.- 13 Generalized Polygons.- 13.1 The Generalized Hexagon of Order (1,2).- 13.2 The Generalized Hexagon of Order (1,3).- 13.3 The Two Generalized Hexagons of Order (2,2).- 13.4 The Generalized Octagon of Order (1,2).- 13.5 The Generalized 12-gon of Order (1,2).- 13.6 Cages.- 14 Colour Pictures and Building the Models.- 15 Some Fun Games and Puzzles.- 15.1 The Game “Set”—Line Spotting in 4-D.- 15.2 Mill and Ticktacktoe on Geometries.- 15.3 Circular Walks on Geometries.- 15.4 Which Generalized Quadrangles Are Magical?.- 15.5 Question du Lapin.- II Geometries on Surfaces.- 16 Introduction via Flat Affine Planes.- 16.1 Some More Basic Facts and Conventions.- 16.2 The Euclidean Plane—A Flat Affine Plane.- 16.2.1 Proper R2-Planes.- 16.2.2 Pencils, Parallel Classes, Generator-Only Pictures.- 16.2.3 The Group Dimension.- 16.2.4 Topological Geometries.- 16.2.5 The Way Lines Intersect.- 16.2.6 Ovals and Maximal Arcs.- 16.3 Nonclassical R2-Planes.- 16.3.1 Moulton Planes.- 16.3.2 Shift Planes.- 16.3.3 Arc Planes.- 16.3.4 Integrated Foliations.- 16.3.5 Gluing Constructions.- 16.4 Classification.- 16.5 Semibiplanes.- 17 Flat Circle Planes—An Overview.- 17.1 The “Axiom of Joining” and the “Map”.- 17.2 Nested Flat Circle Planes.- 17.3 Interpolation.- 18 Flat Projective Planes.- 18.1 Models of the Real Projective Plane.- 18.1.1 The Euclidean Plane Plus Its Line at Infinity.- 18.1.2 The Geometry of Great Circles.- 18.1.3 Möbius Strip—Antiperiodic 2-Unisolvent Set.- 18.1.4 A Disk Model.- 18.2 Recycled Nonclassical Projective Planes.- 18.3 Salzmann’s Classification.- 19 Spherical Circle Planes.- 19.1 Intro via Ovoidal Spherical Circle Planes.- 19.2 The Miquel and Bundle Configurations.- 19.3 Nonclassical Flat Spherical Circle Planes.- 19.3.1 The Affine Part.- 19.3.2 Ewald’s Affine Parts of Flat Möbius Planes.- 19.3.3 Steinke’s Semiclassical Affine Parts.- 19.4 Classification.- 19.5 Subgeometries.- 19.5.1 Double Covers of Flat Projective Planes.- 19.5.2 Recycled Projective Planes.- 19.6 Lie Geometries Associated with Flat Möbius Planes.- 19.6.1 The Apollonius Problem.- 19.6.2 Semibiplanes.- 19.6.3 The Geometry of Oriented Lines and Circles.- 20 Cylindrical Circle Planes of Rank 3.- 20.1 Intro via Ovoidal Cylindrical Circle Planes.- 20.2 The Miquel and Bundle Configurations.- 20.3 Nonclassical Flat Cylindrical Circle Planes.- 20.3.1 Mäurer’s Construction.- 20.3.2 The Affine Part.- 20.3.3 The Artzy-Groh Construction.- 20.3.4 The Löwen-Pfüller Construction.- 20.3.5 Steinke’s Two Types of Semiclassical Planes.- 20.3.6 Integrated Flat Affine Planes.- 20.4 Classification.- 20.5 Subgeometries.- 20.5.1 Recycled Projective Planes.- 20.5.2 Semibiplanes.- 20.6 Lie Geometries Associated with Flat Laguerre Planes.- 20.6.1 Biaffine Planes.- 20.6.2 More Semibiplanes.- 20.6.3 Generalized Quadrangles.- 21 Toroidal Circle Planes.- 21.1 Intro via the Classical Minkowski Plane.- 21.2 The Miquel, Bundle, and Rectangle Configurations.- 21.3 Nonclassical Toroidal Circle Planes.- 21.3.1 The Affine Part.- 21.3.2 The Two Parts of a Toroidal Circle Plane.- 21.3.3 The Artzy-Groh Construction.- 21.3.4 Semiclassical Planes.- 21.3.5 Proper Toroidal Circle Planes.- 21.4 Classification.- 21.5 Subgeometries.- 21.5.1 Flat Projective Planes Minus Convex Disks.- A Models on Regular Solids.- B Mirror Technique Stereograms.- B.1 The Generalized Quadrangle of Order (4,2).- B.3 The Smallest Projective Space.- References.

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