The Shape of Space

Description

The Shape of Space, Third Edition maintains the standard of excellence set by the previous editions. This lighthearted textbook covers the basic geometry and topology of two- and three-dimensional spaces―stretching students’ minds as they learn to visualize new possibilities for the shape of our universe. Written by a master expositor, leading researcher in the field, and MacArthur Fellow, its informal exposition and engaging exercises appeal to an exceptionally broad audience, from liberal arts students to math undergraduate and graduate students looking for a clear intuitive understanding to supplement more formal texts, and even to laypeople seeking an entertaining self-study book to expand their understanding of space. Features of the Third Edition: Full-color figures throughout "Picture proofs" have replaced algebraic proofs Simpler handles-and-crosscaps approach to surfaces Updated discussion of cosmological applications Intuitive examples missing from many college and graduate school curricula About the Author: Jeffrey R. Weeks is a freelance geometer living in Canton, New York. With support from the U.S. National Science Foundation, the MacArthur Foundation and several science museums, his work spans pure mathematics, applications in cosmology and―closest to his heart―exposition for the general public. Contents Preface to the Third Edition Preface to the Second Edition Preface to the First Edition Acknowledgments PART I Surfaces and Three-Manifolds 1. Flatland 2. Gluing 3. Vocabulary 4. Orientability 5. Classification of Surfaces 6. Products 7. Flat Manifolds 8. Orientability vs. Two-Sidedness PART II Geometries on Surfaces 9. The Sphere 10. The Hyperbolic Plane 11. Geometries on Surfaces 12. The Gauss-Bonnet Formula and the Euler Number PART III Geometries on Three-Manifolds 13. Four-Dimensional Space 14. The Hypersphere 15. Hyperbolic Space 16. Geometries on Three-Manifolds I 17. Bundles 18. Geometries on Three-Manifolds II PART IV The Universe 19. The Universe 20. The History of Space Appendix A Answers Appendix B Bibliography Appendix C Conway’s ZIP Proof Index