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ISBN13: 978-0130309532

ISBN10: 0130309532

Edition: 2ND 01

Copyright: 2001

Publisher: Prentice Hall, Inc.

Published: 2001

International: No

ISBN10: 0130309532

Edition: 2ND 01

Copyright: 2001

Publisher: Prentice Hall, Inc.

Published: 2001

International: No

**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.

**21. Polyhedra. **

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-Manifolds--The 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 A--Euclid's Definitions, Postulates, and Common Notions. **

Definitions. Postulates. Common Notions.

**Appendix B--Square 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.

**Annotated Bibliography. **

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.

Index.

ISBN10: 0130309532

Edition: 2ND 01

Copyright: 2001

Publisher: Prentice Hall, Inc.

Published: 2001

International: No

Table of Contents

**1. What Is Straight? **

**2. Straightness on Spheres. **

**3. What Is an Angle? **

**4. Straightness on Cylinders and Cones. **

**5. Straightness on Hyperbolic Planes. **

**6. Triangles and Congruencies. **

**7. Area and Holonomy. **

**8. Parallel Transport. **

**9. SSS, ASS, SAA and AAA. **

**10. Parallel Postulates. **

**11. Isometries and Patterns. **

**12. Dissection Theory. **

**13. Square Roots, Pythagoras and Similar Triangles. **

**14. Circles in the Plane. **

**15. Projection of a Sphere onto a Plane. **

**16. Projections (Models) of Hyperbolic Planes. **

**17. Geometric 2-Manifolds and Coverings. **

**18. Geometric Solutions of Quadratic and Cubic Equations. **

**19. Trigonometry and Duality. **

**20. 3-Spheres and Hyperbolic 3-Spaces. **

**21. Polyhedra. **

**22. 3-Manifolds--The Shape of Space. **

**Appendix A--Euclid's Definitions, Postulates, and Common Notions. **

Definitions. Postulates. Common Notions.

**Appendix B--Square Roots in the Sulbasutram. **

*Savisesa* for the Square Root of Two. Fractions in the Sulbasutram. Comparing with the Divide-and-Average (D&A) Method. Conclusions.

**Annotated Bibliography. **

Index.

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