Contents

Preface xiii

Chapter Dependencies xvii

How to Use This Book xviii

Message to the Reader xx

Chapter 1 Straightness and Symmetry

- Problem 1. When Do You Call a Line Straight?, 1
- How Do You Construct a Straight Line?, 3
- The Symmetries of a Line, 6
- Local (and Infinitesimal) Straightness , 10
Chapter 2 Straightness on a Sphere

- Problem 2. What Is Straight on a Sphere?, 14
- Symmetries of Great Circles, 18
- Relationships with Differential Geometry, 20
Chapter 3 What Is an Angle?

- Problem 3. Vertical Angle Theorem (VAT), 22
- Problem 4. What Is an Angle?, 23
- Hints for Three Different Proofs, 26
Chapter 4 Straightness on Cylinder and Cone

- Problem 5. Intrinsic Straight Lines on Cones and Cylinders, 28
- Cones with Varying Cone Angles, 30
- Geodesics on Cylinders, 34
- Geodesics on Cones, 35
- Locally Isometric, 36
- Problem 5a. Covering Spaces and Global Properties of Geodesics, 37
- n-Sheeted Coverings of a Cylinder, 37
- n-Sheeted Coverings of a Cone, 39
- Covering Space of the Flat Torus, 42
- Coverings and the Sphere, 43
- Is "Shortest" Always "Straight"?, 45
Chapter 5 SAS and ASA

- Problem 6. Side-Angle-Side (SAS) , 48
- Problem 7. Angle-Side-Angle (ASA) , 52
- Addendum on the Use of Covering Spaces, 56
Chapter 6 Area, Parallel Transport, and Holonomy

- Problem 8. The Area of a Triangle on a Sphere, 57
- Introducing Parallel Transport and Holonomy, 58
- Problem 9. The Holonomy of a Small Triangle, 60
- The Gauss-Bonnet Formula for Small Triangles, 62
- Problem 10. The Gauss-Bonnet Formula for Polygons on a Sphere , 63
- Problem 11. Dissection of Polygons into Triangles, 64
- Gauss-Bonnet Formula on Surfaces, 65
Chapter 7 ITT, SSS, and ASS

- Problem 12. Isosceles Triangle Theorem (ITT), 67
- Circles, 68
- Triangles on Cone and Cylinder, 70
- Problem 13. Side-Side-Side (SSS) , 70
- Problem 14. Angle-Side-Side (ASS), 72
- Right-Leg-Hypotenuse (RLH), 74
Chapter 8 Parallel Transport

- Problem 15. Euclid's Exterior Angle Theorem (EEAT), 75
- Problem 16. Symmetries of Parallel Transported Lines, 77
- Problem 17. Transversals through a Midpoint, 79
Chapter 9 SAA and AAA

- Problem 18. Side-Angle-Angle (SAA), 81
- Problem 19. Angle-Angle-Angle (AAA), 83
Chapter 10 Parallel Postulates

- Parallel Lines on the Plane Are Special, 85
- Problem 20. Parallel Transport on the Plane, 86
- Parallel Circles on a Sphere, 88
- Parallel Postulates, 88
- Problem 21. Parallel Postulates on the Plane , 90
- Problem 22. Parallel Postulates on a Sphere, 90
- Parallelism in Spherical and Hyperbolic Geometry, 91
- Problem 23. Sum of the Angles of a Triangle , 93
Chapter 11 3-Spheres in 4-Space

- Problem 24. Explain 3-Space to 2-Dimensional Person, 94
- Terminology, 97
- Problem 25. Intersecting Great Circles in the 3-Sphere, 98
- Problem 26. Triangles in the 3-Sphere, 100
- Problem 27. Disjoint Equidistant Great Circles, 101
- A Rotation That Moves Every Point, 102
- Symmetries of Great Circles and Great 2-Spheres, 102
- Problem 28. Is Our Universe a 3-Sphere?, 104
Chapter 12 Dissection Theory

- Problem 29. Dissecting Plane Triangles, 108
- Problem 30. Dissecting Parallelograms, 108
- Dissection Theory on Spheres, 109
- Problem 31. Khayyam Quadrilaterals, 110
- Problem 32. Dissecting Spherical Triangles, 110
- Problem 33. Dissecting Khayyam Parallelograms, 110
- P roblem34. Spherical Polygons Dissect into Biangles , 111
Chapter 13 Square Roots, Pythagoras, and Similar Triangles

- Square Roots, 112
- Problem 35. A Rectangle Dissects into a Square , 113
- Baudhayana's Sulbasutram, 118
- Problem 36. Equivalence of Squares, 122
- Any Polygon Can Be Dissected into a Square, 123
- Problem 37. Similar Triangles, 124
- Three-Dimensional Dissections, 125
Chapter 14 Geometric Solutions of Quadratic and Cubic Equations

- Problem 38. Quadratic Equations, 126
- Problem 39. Conic Sections and Cube Roots, 131
- Problem 40. Roots of Cubic Equations, 134
- Problem 41. Algebraic Solution of Cubics, 137
- So What Does This All Point To?, 139
Chapter 15 Projections of a Sphere onto a Plane

- Problem 42. Gnomic Projection, 142
- Problem 43. Cylindrical Projection, 143
- Problem 44. Stereographic Projection, 144
Chapter 16 Duality and Trigonometry

- Problem 45. Circumference of a Circle, 146
- Problem 46. Law of Cosines, 147
- Problem 47. Law of Sines, 150
- Duality on a Sphere, 152
- Problem 48. The Dual of a Small Triangle, 154
- Problem 49. Trigonometry on Spherical Triangles, 154
- Duality on the Projective Plane, 154
- Problem 50. Properties on the Projective Plane, 155
- Perspective Drawings and Vision, 156
Chapter 17 Isometries and Patterns

- Definitions and Terminology, 157
- Problem 51. Examples of Patterns, 161
- Problem 52. Isometry Determined by Its Action on Three Points, 162
- Problem 53. Classification of Isometries on Plane and Sphere, 162
- Problem 54. Classification of Discrete Strip Patterns, 163
- Problem 55. Classification of Finite Plane Patterns, 163
- Geometric Meaning of Some Abstract Group Terminology, 164
Chapter 18 Polyhedra

- Definitions and Terminology, 166
- Problem 56. Measure of a Solid Angle, 167
- Problem 57. Edges and Face Angles, 168
- Problem 58. Edges and Dihedral Angles, 169
- Problem 59. Other Congruence Theorems for Tetrahedra, 170
- Problem 60. The Five Regular Polyhedra, 170
Appendix A Geometric Introduction to Differential Geometry

- The Universe
- Smooth Curves, 174
- Smooth Surfaces, Curvature, Geodesics, and Isometries, 175
- Theorems on Geodesics, 176
Bibliography

- A. Ancient Texts, 177
- B. Art and Design, 178
- C. Differential Geometry:, 179
- D. Dimensions and Scale, 179
- E. Fractals , 180
- F. Geometry in Different Cultures, 180
- G. History, 181
- H. Linear Algebra and Geometry, 182
- I. Models, Polyhedra, 183
- J. Nature, 183
- K. Non-Euclidean Geometries (Mostly Hyperbolic), 184
- L. Philosophy, 184
- M. Spherical and Projective Geometry, 185
- N. Symmetry and Groups, 185
- O. Surveys and General Expositions, 186
- P. Texts, 187
- Q. Miscellaneous, 188
Index 189