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 GaussBonnet Formula for Triangles 91
*Problem 7.5 GaussBonnet Formula for Polygons 92
*GaussBonnet 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 GaussBonnet 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