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