Table of Contents

Preface to the Dover Edition xi

Preface xiii

1 Other Geometries: A Computational Introduction 1

1.1 Spherical Geometry 1

1.2 Hyperbolic Geometry 9

1.3 Other Geometries 21

2 The Neutral Geometry of the Triangle 29

2.1 Introduction 29

2.2 Preliminaries 34

2.3 Propositions 1 through 28 46

2.4 Postulate 5 Revisited 81

3 Nonneutral Euclidean Geometry 87

3.1 Parallelism 87

3.2 Area 99

3.3 The Theorem of Pythagoras 112

3.4 Consequences of the Theorem of Pythagoras 119

3.5 Proportion and Similarity 122

4 Circles and Regular Polygons 133

4.1 The Neutral Geometry of the Circle 133

4.2 The Nonneutral Euclidean Geometry of the Circle 141

4.3 Regular Polygons 150

4.4 Circle Circumference and Area 155

4.5 Impossible Constructions 165

5 Toward Projective Geometry 177

5.1 Division of Line Segments 177

5.2 Collinearity and Concurrence 184

5.3 The Projective Plane 191

6 Planar Symmetries 197

6.1 Translations, Rotations, and Fixed Points 197

6.2 Reflections 203

6.3 Glide Reflections 210

6.4 The Main Theorems 216

6.5 Symmetries of Polygons 219

6.6 Frieze Patterns 223

6.7 Wallpaper Designs 228

7 Inversions 247

7.1 Inversions as Transformations 247

7.2 Inversions to the Rescue 255

7.3 Inversions as Hyperbolic Motions 259

8 Symmetry in Space 269

8.1 Regular and Semiregular Polyhedra 269

8.2 Rotational Symmetries of Regular Polyhedra 281

8.3 Monstrous Moonshine 288

9 Informal Topology 295

10 Graphs 305

10.1 Nodes and Arcs 305

10.2 Traversability 308

10.3 Colorings 314

10.4 Planarity 317

10.5 Graph Homeomorphisms 326

11 Surfaces 333

11.1 Polygonal Presentations 333

11.2 Closed Surfaces 346

11.3 Operations on Surfaces 358

11.4 Bordered Surfaces 367

12 Knots and Links 379

12.1 Equivalence of Knots and Links 379

12.2 Labelings 385

12.3 The Jones Polynomial 393

Appendix A A Brief Introduction to The Geometer's Sketchpad® 405

Appendix B Summary of Propositions 409

Appendix C George D. Birkhoff's Axiomatization of Euclidean Geometry 415

Appendix D The University of Chicago School Mathematics Project's Geometrical Axioms 417

Appendix E David Hilbert's Axiomatization of Euclidean Geometry 421

Appendix F Permutations 425

Appendix G Modular Arithmetic 429

Solutions and Hints to Selected Problems 433

Bibliography 447

Index 451