Table of Contents
1. What Is Straight?
Problem 1.1: When Do You Call a Line Straight? How Do You Construct a Straight Line? The Symmetries of a Line. Local (and Infinitesimal) Straightness.
2. Straightness on Spheres.
Problem 2.1: What Is Straight on a Sphere? Symmetries of Great Circles. Every Geodesic Is a Great Circle. Intrinsic Curvature.
3. What Is an Angle?
Problem 3.1: Vertical Angle Theorem (VAT). Problem 3.2: What Is an Angle? Hints for Three Different Proofs. Problem 3.3: Duality Between Points and Lines.
4. Straightness on Cylinders and Cones.
Problem 4.1: Straightness on Cylinders and Cones. Cones with Varying Cone Angles. Geodesics on Cylinders. Geodesics on Cones. Locally Isometric. Is "Shortest" Always "Straight"? Relations to Differential Geometry.
5. Straightness on Hyperbolic Planes.
A Short History of Hyperbolic Geometry. Constructions of Hyperbolic Planes. Hyperbolic Planes of Different Raddi (Curvature). Problem 5.1: What Is Straight in a Hyperbolic Plane? Problem 5.2: The Pseudosphere Is Hyperbolic. Problem 5.3: Rotations and Reflections on Surfaces.
6. Triangles and Congruencies.
Geodesics are Locally Unique. Problem 6.1: Properties of Geodesics. Problem 6.2: Isosceles Triangle Theorem (ITT). Circles. Problem 6.3: Bisector Constructions. Problem 6.4: Side-Angle-Side (SAS).Problem 6.5: Angle-Side-Angle (ASA).
7. Area and Holonomy.
Problem 7.1: The Area of a Triangle on a Sphere. Problem 7.2: Area of Hyperbolic Triangles. Problem 7.3: Sum of the Angles of a Triangle. Introduction Parallel Transport and Holonomy. Problem 7.4: The Holonomy of a Small Triangle. The Gauss-Bonnet Formula for Triangles. Problem 7.5: Gauss-Bonnet Formula for Polygons. Gauss-Bonnet Formula for Polygons on Surfaces.
8. Parallel Transport.
Problem 8.1: Euclid's Exterior Angle Theorem (EEAT). Problem 8.2: Symmetries of Parallel Transported Lines. Problem 8.3: Transversals through a Midpoint. Problem 8.4: What is "Parallel"?
9. SSS, ASS, SAA and AAA.
Problem 9.1: Side-Side-Side (SSS). Problem 9.2: Angle-Side-Side (ASS). Problem 9.3: Side-Angle-Angle (SAA). Problem 9.4: Angle-Angle-Angle (AAA).
10. Parallel Postulates.
Parallel Lines on the Plane are Special. Problem 10.1: Parallel Transport on the Plane. Problem 10.2: Parallel Postulates Not Involving (Non-) Intersecting Lines). Equidistant Curves on Spheres and Hyperbolic Planes. Problem 10.3: Parallel Postulates Involving (Non-) Intersecting Lines. Problem 10.4: EFP and PPP on Sphere and Hyperbolic Plane. Comparisons of Plane, Spheres, and Hyperbolic Planes. Some Historical Notes on the Parallel Postulates.
11. Isometries and Patterns.
Problem 11.1: Isometries. Symmetries and Patterns. Problem 11.2: Examples of Patterns. Problem 11.3: Isometry Determined by Three Points. Problem 11.4: Classification of Isometries. Problem 11.5: Classification of Discrete Strip Patterns. Problem 11.6: Classification of Finite Plane Patterns. Problem 11.7: Regular Tilings with Polygons. Geometric Meaning of Abstract Group Terminology.
12. Dissection Theory.
What is Dissection Theory? Problem 12.1: Dissect Plane Triangle and Parallelogram. Dissection Theory on Spheres and Hyperbolic Planes. Problem 12.2: Khayyam Quadrilaterals. Problem 12.3: Dissect Spherical and Hyperbolic Triangles and Khayyam Parallelograms. Problem 12.4: Spherical Polygons Dissect to Lunes.
13. Square Roots, Pythagoras and Similar Triangles.
Square Roots. Problem 13.1: A Rectangle Dissects into a Square. Baudhayana's Sulbasutram. Problem 13.2: Equivalence of Squares. Any Polygon Can Be Dissected into a Square. Problem 13.3: Similar Triangles. Three-Dimensional Dissections and Hilbert's Third Problem.
14. Circles in the Plane.
Problem 14.1: Angles and Power Points of Plane Circles. Problem 14.2: Inversions in Circles. Problem 14.3: Applications of Inversions.
15. Projection of a Sphere onto a Plane.
Problem 15.1: Charts Must Distort. Problem 15.2: Gnomic Projection. Problem 15.3: Cylindrical Projection. Problem 15.4: Stereographic Projection.
16. Projections (Models) of Hyperbolic Planes.
Problem 16.1: The Upper Half Plane Model. Problem 16.2: Upper Half Plane Is Model of Annular Hyperbolic Plane. Problem 16.3: Properties of Hyperbolic Geodesics. Problem 16.4: Hyperbolic Ideal Triangles. Problem 16.5: Poincaré Disk Model. Problem 16.6: Projective Disk Model.
17. Geometric 2-Manifolds and Coverings.
Problem 17.1: Geodesics on Cylinders and Cones. n-Sheeted Coverings of a Cylinder. n-Sheeted (Branched) Coverings of a Cone. Problem 17.2: Flat Torus and Flat Klein Bottle. Problem 17.3: Universal Covering of Flat 2-Manifolds. Problem 17.4: Spherical 2-Manifolds. Coverings of a Sphere. Problem 17.5: Hyperbolic Manifolds. Problem 17.6: Area, Euler Number, and Gauss-Bonnet. Triangles on Geometric Manifolds. Problem 17.7: Can the Bug Tell Which Manifold?
18. Geometric Solutions of Quadratic and Cubic Equations.
Problem 18.1: Quadratic Equations. Problem 18.2: Conic Sections and Cube Roots. Problem 18.3: Roots of Cubic Equations. Problem 18.4: Algebraic Solution of Cubics. So What Does This All Point To?
19. Trigonometry and Duality.
Problem 19.1: Circumference of a Circle. Problem 19.2: Law of Cosines. Problem 19.3: Law of Sines. Duality on a Sphere. Problem 19.4: The Dual of a Small Triangle. Problem 19.5: Trigonometry with Congruences. Duality on the Projective Plane. Problem 19.6: Properties on the Projective Plane. Perspective Drawings and Vision.
20. 3-Spheres and Hyperbolic 3-Spaces.
Problem 20.1: Explain 3-Space to 2-D Person. Problem 20.2: A 3-Sphere in 4-Space. Problem 20.3: Hyperbolic 3-Space, Upper Half Space. Problem 20.4: Disjoint Equidistant Great Circles. Problem 20.5: Hyperbolic and Spherical Symmetries. Problem 20.6: Triangles in 3-Dimensional Spaces.
Definitions and Terminology. Problem 21.1: Measure of a Solid Angle. Problem 21.2: Edges and Face Angles. Problem 21.3: Edges and Dihedral Angles. Problem 21.4: Other Tetrahedra Congruence Theorems. Problem 21.5: The Five Regular Polyhedra.
22. 3-ManifoldsThe Shape of Space.
Space as an Oriented Geometric 3-Manifold. Problem 22.1: Is Our Universe Non-Euclidean? Problem 22.2: Euclidean 3-Manifolds. Problem 22.3: Dodecahedral 3-Manifolds. Problem 22.4: Some Other Geometric 3-Manifolds. Cosmic Background Radiation. Problem 22.5: Circle Patterns Show the Shape of Space.
Appendix AEuclid's Definitions, Postulates, and Common Notions.
Definitions. Postulates. Common Notions.
Appendix BSquare Roots in the Sulbasutram.
Introduction. Construction of the Savisesa for the Square Root of Two. Fractions in the Sulbasutram. Comparing with the Divide-and-Average (D&A) Method. Conclusions.
AT: Ancient Texts. CG: Computers and Geometry. DG: Differential Geometry. Di: Dissections. DS: Dimensions and Scale. GC: Geometry in Different Cultures. Hi: History. MP: Models, Polyhedra. Na: Nature. NE: None-Euclideam Geometries (Mostly Hyperbolic). Ph: Philosophy. RN: Real Numbers. SE: Surveys and General Expositions. SG: Symmetry and Groups. SP: Spherical and Projective Geometry. TG: Teaching Geometry. Tp: Topology. Tx: Geometry Texts. Un: The Physical Universe. Z: Miscellaneous.