Machine generated contents note: 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. Connected Sums -- 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 -- 21. Cosmic Crystallography -- 22. Circles in the Sky -- Appendix A Answers -- Appendix B Bibliography -- Appendix C Conway's ZIP Proof.