Geometry

and the

Imagination

D.Hilbert and S.Cohn-Vossen

PREFACE

Chapter 1
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 (7₃) and (8₃) 
17.	The Configurations (9₃)
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 (10₃)
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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