Geometry and the ImaginationThis remarkable book endures as a true masterpiece of mathematical exposition. 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. Geometry and the Imagination is full of interesting facts, many of which you wish you had known before. The book begins with examples of the simplest curves and surfaces, including thread constructions of certain quadrics and other surfaces. The chapter on regular systems of points leads to the crystallographic groups and the regular polyhedra in $\mathbb{R}^3$. In this chapter, they also discuss plane lattices. By considering unit lattices, and throwing in a small amount of number theory when necessary, they effortlessly derive Leibniz's series: $\pi/4 = 1 - 1/3 + 1/5 - 1/7 + - \ldots$. In the section on lattices in three and more dimensions, the authors consider sphere-packing problems, including the famous Kepler problem. One of the most remarkable chapters is ``Projective Configurations''. In a short introductory section, Hilbert and Cohn-Vossen give perhaps the most concise and lucid description of why a general geometer would care about projective geometry and why such an ostensibly plain setup is truly rich in structure and ideas. The chapter on kinematics includes a nice discussion of linkages and the geometry of configurations of points and rods that are connected and, perhaps, constrained in some way. This topic in geometry has become increasingly important in recent times, especially in applications to robotics. This is another example of a simple situation that leads to a rich geometry. It would be hard to overestimate the continuing influence Hilbert-Cohn-Vossen's book has had on mathematicians of this century. It surely belongs in the pantheon of great mathematics books. |
Contents
2 The Cylinder the Cone the Conic Sections and Their | 7 |
4 The Thread Construction of the Ellipsoid and Confocal | 19 |
APPENDICES T0 CHAPTER I | 25 |
CHAPTER II | 32 |
7 Lattices in Three and More than Three Dimensions | 44 |
Crystals as Regular Systems of Points | 52 |
Plane Motions and their Composition Classification of | 59 |
CHAPTER III | 94 |
Curvature of Surfaces Elliptic Hyperbolic and Parabolic Points Lines of Curvature and Asymptotic Lines Um bilical Points Minimal Surfaces Monkey... | 183 |
The Spherical Image and Gaussian Curvature | 193 |
Developable Surfaces Ruled Surfaces | 205 |
The Twisting of Space Curves | 211 |
Eleven Properties of the Sphere | 215 |
Bendings Leaving a Surface Invariant | 232 |
Elliptic Geometry | 235 |
Hyperbolic Geometry and its Relation to Euclidean and to Elliptic Geometry | 242 |
Preliminary Remarks about Plane Configurations | 95 |
The Configurations 73 and 83 | 98 |
17 The Configurations 93 | 109 |
Perspective Ideal Elements and the Principle of Duality in the Plane | 112 |
Ideal Elements and the Principle of Duality in Space Desargues Theorem and the Desargues Configuration | 119 |
Comparison of Pascals and Desargues Theorems | 128 |
Preliminary Remarks on Configurations in Space | 133 |
Reyes Configuration | 134 |
23 Regular Polyhedra in Three and Four Dimensions and their Projections | 143 |
Enumerative Methods of Geometry | 157 |
CHAPTER IV | 171 |
PlaneCurves | 172 |
Space Curves | 178 |
Stereographic Projection and CirclePreserving Trans formations Poincarés Model of the Hyperbolic Plane | 248 |
Methods of Mapping Isometric AreaPreserving Geo | 260 |
39 Conformal Mappings of Curved Surfaces Minimal Sur | 268 |
Continuous Rigid Motions of Plane Figures | 275 |
An Instrument for Constructing the Ellipse and its Roul | 283 |
CHAPTER VI | 289 |
Surfaces | 295 |
OneSided Surfaces | 302 |
The Projective Plane as a Closed Surface | 313 |
Topological Mappings of a Surface onto Itself Fixed | 324 |
Conformal Mapping of the Torus | 330 |
The Projective Plane in FourDimensional Space | 340 |
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Common terms and phrases
angles axes axioms axis boundary called centers of rotation centers of similitude circle circular closed curve cone configuration conformal mapping congruent conics connectivity consider construction contains corresponding cube cylinder defined definite Desargues discontinuous groups distance edges ellipse ellipsoid elliptic geometry equal equivalent Euclidean plane example faces figure find finite first five fixed point focal follows four Gaussian curvature geodesic geodesic lines given Hence heptahedron hexagon hyperbolic plane hyperboloid incidence infinitely lattice points mean curvature minimal surface move obtained octahedron parabolic points parallel pass perpendicular point of intersection pointers polygon polyhedron position projective plane proved quadrics radius reflections region rigid motions ruled surfaces satisfied segment sides space curve sphere spherical image square straight lines sufficient surface of revolution symmetry systems of points tangent plane theorem three-dimensional tion topological torus translation triangles umbilical points unit cell vertex vertices