# 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