# Geometry

### and the

# Imagination

## D.Hilbert and S.Cohn-Vossen

PREFACE

**Chapter 1**

THE SIMPLEST CURVES AND SURFACES

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 Conics 2. The Directrices of the Conics 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. Schliifli'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 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

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