### Synopses & Reviews

Maintaining the standard of excellence set by the previous edition, this textbook covers the basic geometry of two- and three-dimensional spaces. Written by a master expositor, leading researcher in the field, and MacArthur Fellow, it includes experiments to determine the true shape of the universe and contains illustrated examples and engaging exercises that teach mind-expanding ideas in an intuitive and informal way. Bridging the gap from geometry to the latest work in observational cosmology, the book illustrates the connection between geometry and the behaviour of the physical universe and explains how radiation remaining from the big bang may reveal the actual shape of the universe.

#### Book News Annotation:

This friendly textbook explores many of the possible answers to the question, "What is the universe shaped like?" It provides an overview of the basic geometry of two- and three-dimensional spaces, illustrating the connection between geometry, cosmology, and the behavior of the physical universe. Numerous illustrations, exercises, and thought-experiments are included. Technical language and mathematical equations are kept to a minimum. Weeks is a "freelance mathematician" and a MacArthur Fellow.

Annotation c. Book News, Inc., Portland, OR (booknews.com)

### Table of Contents

**Surfaces and Three-Manifolds **

Flatland

Gluing

Vocabulary

Orientability

Connected Sums

Products

Flat Manifolds

Orientability vs. Two-Sidedness

**Geometries on Surfaces **

The Sphere

The Hyperbolic Plane

Geometries on Surfaces

The GaussûBonnet Formula and the

Euler Number

**Geometries on Three-Manifolds **

Four-Dimensional Space

The Hypersphere

Hyperbolic Space

Geometries on Three-Manifolds I

Bundles

Geometries on Three-Manifolds II

**The Universe **

The Universe

The History of Space

Cosmic Crystallography

Circles in the Sky

Appendix A Answers

Appendix B Bibliography

Appendix C ConwayÆs ZIP Proof

Index