Geometry and the Imagination

The book is overflowing with mathematical ideas, which are always explained clearly and elegantly, and above all, with penetrating insight. It is a joy to read, both for beginners and experienced mathematicians. Cover Title page Date-line Preface CONTENTS Chapter I THE SIMPLEST CURVES AND SURFACES § 1. Plane Curves § 2. The Cylinder, the Cone, the Conic Sections and Their Surfaces of Revolution § 3. The Second-Order Surfaces § 4. The Thread Construction of the Ellipsoid, and Confocal Quadrics APPENDICES TO CHAPTER I 1. The Pedal-Point Construction of the Conies 2. The Directrices of the Conies 3. The Movable Rod Model of the Hyperboloid Chapter II REGULAR SYSTEMS OF POINTS § 5. Plane Lattices § 6. Plane Lattices in the Theory of Numbers § 7. Lattices in Three and More than Three Dimensions § 8. Crystals as Regular Systems of Points § 9. Regular Systems of Points and Discontinuous Groups of Motions § 10. Plane Motions and their Composition; Classification of the Discontinuous Groups of Motions in the Plane § 11. The Discontinuous Groups of Plane Motions with Infinite Unit Cells § 12. The Crystallographic Groups of Motions in the Plane. Regular Systems of Points and Pointers. Division of the Plane into Congruent Cells § 13. Crystallographic Classes and Groups of Motions in Space. Groups and Systems of Points with Bilateral Symmetry § 14. The Regular Polyhedra Chapter III PROJECTIVE CONFIGURATIONS § 15. Preliminary Remarks about Plane Configurations § 16. The Configurations (7$_3$) and (8$_3$) § 17. The Configurations (9$_3$) § 18. Perspective, Ideal Elements, and the Principle of Duality in the Plane § 19. Ideal Elements and the Principle of Duality in Space. Desargues' Theorem and the Desargues Configuration (10$_3$) § 20. Comparison of Pascal's and Desargues Theorems § 21. Preliminary Remarks on Configurations in Space § 22. Reye's Configuration § 23. Regular Polyhedra in Three and Four Dimensions, and their Projections § 24. Enumerative Methods of Geometry § 25. Schlafli's Double-Six Chapter IV DIFFERENTIAL GEOMETRY § 26. Plane Curves § 27. Space Curves § 28. Curvature of Surfaces. Elliptic, Hyperbolic, and Parabolic Points. Lines of Curvature and Asymptotic Lines. Umbilical Points, Minimal Surfaces, Monkey Saddles § 29. The Spherical Image and Gaussian Curvature § 30. Developable Surfaces, Ruled Surfaces § 31. The Twisting of Space Curves § 32. Eleven Properties of the Sphere § 33. Bendings Leaving a Surface Invariant § 34. Elliptic Geometry § 35. Hyperbolic Geometry, and its Relation to Euclidean and to Elliptic Geometry § 36. Stereographic Projection and Circle-Preserving Transformations. Poincare's Model of the Hyperbolic Plane § 37. Methods of Mapping, Isometric, Area-Preserving, Geodesic, Continuous and Conformal Mappings § 38. Geometrical Function Theory. Riemann's Mapping Theorem. Conformal Mapping in Space § 39. Conformal Mappings of Curved Surfaces. Minimal Surfaces. Plateau's Problem Chapter V KINEMATICS § 40. Linkages § 41. Continuous Rigid Motions of Plane Figures § 42. An Instrument for Constructing the Ellipse and its Roulettes § 43. Continuous Motions in Space Chapter VI TOPOLOGY § 44. Polyhedra § 45. Surfaces § 46. One-Sided Surfaces § 47. The Projective Plane as a Closed Surface § 48. Standard Forms for the Surfaces of Finite Connectivity § 49. Topological Mappings of a Surface onto Itself. Fixed Points. Classes of Mappings. The Universal Covering Surface of the Torus § 50. Conformal Mapping of the Torus § 51. The Problem of Contiguous Regions, The Thread Problem, and the Color Problem APPENDICES TO CHAPTER VI 1. The Projective Plane in Four-Dimensional Space 2. The Euclidean Plane in Four-Dimensional Space Index