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 Moebius 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 Moebius 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 Moebius 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 Maurer's Construction.- 20.3.2 The Affine Part.- 20.3.3 The Artzy-Groh Construction.- 20.3.4 The Loewen-Pfuller 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.
