# Contents

 Preface xv Useful Supplements xvii My Teaching Background xviii Acknowledgments for the First Edition xviii Acknowledgments for This Edition xx How to Use This Book xxii How I Use this Book in a Course xxiii But, Do It Your Own Way xxiv Chapter Sequences xxv Message to the Reader xxvii Proof as Convincing Argument That Answers — Why? xxix Chapter 1 What is Straight? 1 Problem 1.1 When Do You Call a Line Straight? 2 How Do You Construct a Straight Line? 4 The Symmetries of a Line 6 Local (and Infinitesimal) Straightness 11 Chapter 2 Straightness on Spheres 15 Problem 2.1 What Is Straight on a Sphere? 15 Symmetries of Great Circles 20 *Every Geodesic Is a Great Circle 22 *Intrinsic Curvature 23 Chapter 3 What Is an Angle? 25 Problem 3.1 Vertical Angle Theorem (VAT) 25 Problem 3.2 What Is an Angle? 26 Hints for Three Different Proofs 29 Problem 3.3 Duality between Points and Lines 31 Chapter 4 Straightness on Cylinders and Cones 33 Problem 4.1 Straightness on Cylinders and Cones 34 Cones with Varying Cone Angles 36 Geodesics on Cylinders 39 Geodesics on Cones 40 Locally Isometric 41 Is "Shortest" Always "Straight"? 41 Relations to Differential Geometry 43 Chapter 5 Straightness on Hyperbolic Planes 45 A Short History of Hyperbolic Geometry 45 Constructions of Hyperbolic Planes 48 Hyperbolic Planes of Different Radii (Curvature) 55 Problem 5.1 What Is Straight in a Hyperbolic Plane? 57 *Problem 5.2 The Pseudosphere is Hyperbolic 57 Problem 5.3 Rotations and Reflections on Surfaces 60 Chapter 6 Triangles and Congruencies 63 *Geodesics Are Locally Unique 63 Problem 6.1 Properties of Geodesics 64 Problem 6.2 Isosceles Triangle Theorem (ITT) 65 Circles 66 Problem 6.3 Bisector Constructions 69 Problem 6.4 Side-Angle-Side (SAS) 69 Problem 6.5 Angle-Side-Angle (ASA) 74 Chapter 7 Area and Holonomy 79 Problem 7.1 The Area of a Triangle on a Sphere 80 Problem 7.2 Area of Hyperbolic Triangles 81 Problem 7.3 Sum of the Angles of a Triangle 85 Introducing Parallel Transport and Holonomy 85 Problem 7.4 The Holonomy of a Small Triangle 89 The Gauss–Bonnet Formula for Triangles 91 *Problem 7.5 Gauss–Bonnet Formula for Polygons 92 *Gauss–Bonnet Formula for Polygons on Surfaces 95 Chapter 8 Parallel Transport 99 Problem 8.1 Euclid's Exterior Angle Theorem (EEAT) 99 Problem 8.2 Symmetries of Parallel Transported Lines 101 Problem 8.3 Transversals through a Midpoint 104 Problem 8.4 What is "Parallel"? 105 Chapter 9 SSS, ASS, SAA, and AAA 109 Problem 9.1 Side-Side-Side (SSS) 109 Problem 9.2 Angle-Side-Side (ASS) 111 Problem 9.3 Side-Angle-Angle (SAA) 113 Problem 9.4 Angle-Angle-Angle (AAA) 115 Chapter 10 Parallel Postulates 117 Parallel Lines on the Plane Are Special 117 Problem 10.1 Parallel Transport on the Plane 118 Problem 10.2 Parallel Postulates Not Involving (Non-) Intersecting Lines 120 Equidistant Curves on Spheres and Hyperbolic Planes 122 Problem 10.3 Parallel Postulates Involving (Non-) Intersecting Lines 123 Problem 10.4 EFP and PPP on Sphere and Hyperbolic Plane 126 Comparisons of Plane, Spheres, and Hyperbolic Planes 128 Some Historical Notes on the Parallel Postulates 130 Chapter 11 Isometries and Patterns 133 Problem 11.1 Isometries 133 Symmetries and Patterns 137 Problem 11.2 Examples of Patterns 141 Problem 11.3 Isometry Determined by Three Points 143 Problem 11.4 Classification of Isometries 143 Problem 11.5 Classification of Discrete Strip Patterns 146 Problem 11.6 Classification of Finite Plane Patterns 146 Problem 11.7 Regular Tilings with Polygons 147 *Geometric Meaning of Abstract Group Terminology 149 Chapter 12 Dissection Theory 151 What Is Dissection Theory? 151 Problem 12.1 Dissect Plane Triangle and Parallelogram 153 Dissection Theory on Spheres and Hyperbolic Planes 154 Problem 12.2 Khayyam Quadrilaterals 155 Problem 12.3 Dissect Spherical and Hyperbolic Triangles and Khayyam Parallelograms 156 *Problem 12.4 Spherical Polygons Dissect to Lunes 157 Chapter 13 Square Roots, Pythagoras, and Similar Triangles 161 Square Roots 161 Problem 13.1 A Rectangle Dissects into a Square 163 Baudhayana's Sulbasutram 168 Problem 13.2 Equivalence of Squares 172 Any Polygon Can Be Dissected into a Square 173 Problem 13.3 Similar Triangles 174 Three-Dimensional Dissections and Hilbert's Third Problem 175 Chapter 14 Circles in the Plane 177 Problem 14.1 Angles and Power Points of Plane Circles 177 Problem 14.2 Inversions in Circles 180 *Problem 14.3 Applications of Inversions 183 Chapter 15 Projections of a Sphere onto a Plane 187 Problem 15.1 Charts Must Distort 188 Problem 15.2 Gnomic Projection 188 Problem 15.3 Cylindrical Projection 189 Problem 15.4 Stereographic Projection 190 Chapter 16 Projections (Models) of Hyperbolic Planes 193 *Problem 16.1 The Upper Half Plane Model 194 *Problem 16.2 Upper Half Plane Is Model of Annular Hyperbolic Plane 198 Problem 16.3 Properties of Hyperbolic Geodesics 200 Problem 16.4 Hyperbolic Ideal Triangles 201 Problem 16.5 Poincaré Disk Model 203 Problem 16.6 Projective Disk Model 205 Chapter 17 Geometric 2-Manifolds and Coverings 207 *Problem 17.1 Geodesics on Cylinders and Cones 208 n-Sheeted Coverings of a Cylinder 209 n-Sheeted (Branched) Coverings of a Cone 210 Problem 17.2 Flat Torus and Flat Klein Bottle 212 *Problem 17.3 Universal Covering of Flat 2-Manifolds 217 Problem 17.4 Spherical 2-Manifolds 219 *Coverings of a Sphere 222 Problem 17.5 Hyperbolic Manifolds 224 Problem 17.6 Area, Euler Number, and Gauss–Bonnet 228 *Triangles on Geometric Manifolds 230 Problem 17.7 Can the Bug Tell Which Manifold? 231 Chapter 18 Geometric Solutions of Quadratic and Cubic Equations 233 Problem 18.1 Quadratic Equations 234 Problem 18.2 Conic Sections and Cube Roots 238 Problem 18.3 Roots of Cubic Equations 242 Problem 18.4 Algebraic Solution of Cubics 246 So What Does This All Point To? 248 Chapter 19 Trigonometry and Duality 251 Problem 19.1 Circumference of a Circle 251 Problem 19.2 Law of Cosines 253 Problem 19.3 Law of Sines 257 Duality on a Sphere 258 Problem 19.4 The Dual of a Small Triangle 260 *Problem 19.5 Trigonometry with Congruences 261 Duality on the Projective Plane 261 *Problem 19.6 Properties on the Projective Plane 262 Perspective Drawings and Vision 263 Chapter 20 3-Spheres and Hyperbolic 3-Spaces 265 Problem 20.1 Explain 3-Space to 2-D Person 266 *Problem 20.2 A 3-Sphere in 4-Space 269 *Problem 20.3 Hyperbolic 3-Space, Upper Half Space 272 *Problem 20.4 Disjoint Equidistant Great Circles 274 *Problem 20.5 Hyperbolic and Spherical Symmetries 276 Problem 20.6 Triangles in 3-Dimensional Spaces 277 Chapter 21 Polyhedra 279 Definitions and Terminology 279 Problem 21.1 Measure of a Solid Angle 280 Problem 21.2 Edges and Face Angles 281 Problem 21.3 Edges and Dihedral Angles 283 Problem 21.4 Other Tetrahedra Congruence Theorems 283 Problem 21.5 The Five Regular Polyhedra 284 Chapter 22 3-Manifolds — the Shape of Space 287 Space as an Oriented Geometric 3-Manifold 288 Problem 22.1 Is Our Universe Non-Euclidean? 291 Problem 22.2 Euclidean 3-Manifolds 293 Problem 22.3 Dodecahedral 3-Manifolds 296 Problem 22.4 Some Other Geometric 3-Manifolds 299 Cosmic Background Radiation 300 Problem 22.5 Circle Patterns Show the Shape of Space 303 Appendix A Euclid's Definitions, Postulates, and Common Notions 305 Definitions 305 Postulates 308 Common Notions 308 Appendix B Square Roots in the Sulbasutram 309 Introduction 309 Construction of the Savi´e¸a for the Square Root of Two 311 Fractions in the Sulbasutram 316 Comparing with the Divide-and-Average (D&A) Method 317 Conclusions 319 Annotated Bibliography 321 AT Ancient Texts 321 CG Computers and Geometry 324 DG Differential Geometry 325 Di Dissections 327 DS Dimensions and Scale 328 GC Geometry in Different Cultures 329 Hi History 330 MP Models, Polyhedra 333 Na Nature 334 NE Non-Euclidean Geometries (Mostly Hyperbolic) 335 Ph Philosophy 337 RN Real Numbers 338 SE Surveys and General Expositions 338 SG Symmetry and Groups 339 SP Spherical and Projective Geometry 341 TG Teaching Geometry 341 Tp Topology 342 Tx Geometry Texts 343 Un The Physical Universe 345 Z Miscellaneous 346 Index 347