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** ** The distinctive approach of Henderson and Taimina's volume stimulates readers to develop a broader, deeper, understanding of mathematics through active experience—including discovery, discussion, writing fundamental ideas and learning about the history of those ideas. A series of interesting, challenging problems encourage readers to gather and discuss their reasonings and understanding. The volume provides an understanding of the possible shapes of the physical universe. ** ** The authors provide extensive information on historical strands of geometry, straightness on cylinders and cones and hyperbolic planes, triangles and congruencies, area and holonomy, parallel transport, SSS, ASS, SAA, and AAA, parallel postulates, isometries and patterns, dissection theory, square roots, pythagoras and similar triangles, projections of a sphere onto a plane, inversions in circles, projections (models) of hyperbolic planes, trigonometry and duality, 3-spheres and hyperbolic 3-spaces and polyhedra. ** ** For mathematics educators and other who need to understand the meaning of geometry.

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The following ISBNs are associated with this title:

ISBN - 10:0131437488

ISBN - 13:9780131437487

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Table of Contents

**Preface.**

**How to Use this Book.**

**0. Historical Strands of Geometry.**

**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. Projections of a Sphere onto a Plane.**

**15. Circles.**

**16. Inversions in Circles.**

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

**18. Geometric 2-Manifolds.**

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

**20. Trigonometry and Duality.**

**21. Mechanisms.**

**22. 3-Spheres and Hyperbolic 3-Spaces.**

**23. Polyhedra.**

**24. 3-Manifolds—the Shape of Space.**

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

**Appendix B: Constructions of Hyperbolic Planes.**

**Bibliography.**

**Index.**

Editorial Reviews