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Geometry and the Imagination

AMS Chelsea Publishing: An Imprint of the American Mathematical Society
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Softcover ISBN: 978-1-4704-6302-1
Product Code: CHEL/87.S
List Price: $49.00 MAA Member Price:$44.10
AMS Member Price: $39.20 Electronic ISBN: 978-1-4704-6386-1 Product Code: CHEL/87.E List Price:$49.00
MAA Member Price: $44.10 AMS Member Price:$39.20
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List Price: $73.50 MAA Member Price:$66.15
AMS Member Price: $58.80 Click above image for expanded view Geometry and the Imagination AMS Chelsea Publishing: An Imprint of the American Mathematical Society Available Formats:  Softcover ISBN: 978-1-4704-6302-1 Product Code: CHEL/87.S  List Price:$49.00 MAA Member Price: $44.10 AMS Member Price:$39.20
 Electronic ISBN: 978-1-4704-6386-1 Product Code: CHEL/87.E
 List Price: $49.00 MAA Member Price:$44.10 AMS Member Price: $39.20 Bundle Print and Electronic Formats and Save! This product is available for purchase as a bundle. Purchasing as a bundle enables you to save on the electronic version.  List Price:$73.50 MAA Member Price: $66.15 AMS Member Price:$58.80
• Book Details

AMS Chelsea Publishing
Volume: 871952; 358 pp
MSC: Primary 00; 01;

This remarkable book has endured as a true masterpiece of mathematical exposition. There are few mathematics books that are still so widely read and continue to have so much to offer—even after more than half a century has passed! 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.

“Hilbert and Cohn-Vossen” is full of interesting facts, many of which you wish you had known before. It's also likely that you have heard those facts before, but surely wondered where they could be found. 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. Here, we see regular polyhedra again, from a different perspective. One of the high points of the chapter is the discussion of Schlafli's Double-Six, which leads to the description of the 27 lines on the general smooth cubic surface. As is true throughout the book, the magnificent drawings in this chapter immeasurably help the reader.

A particularly intriguing section in the chapter on differential geometry is Eleven Properties of the Sphere. Which eleven properties of such a ubiquitous mathematical object caught their discerning eye and why? Many mathematicians are familiar with the plaster models of surfaces found in many mathematics departments. The book includes pictures of some of the models that are found in the Göttingen collection. Furthermore, the mysterious lines that mark these surfaces are finally explained!

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.

• Front Cover
• Cover Page
• PREFACE
• CHAPTER I: THE SIMPLEST CURVES AND SURFACES
• § 1. Plane Curves
• § 2. The Cylinder, the Cone, the Conic Sections andTheir Surfaces of Revolution
• § 3. The Second-Order Surfaces
• APPENDICES TO CHAPTER I
• I. 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 theDiscontinuous Groups of Motions in the Plane
• § 11. The Discontinuous Groups of Plane Motionswith Infinite Unit Cells
• § 12. The Crystallographic Groups of Motions in the Plane.Regular Systems of Points and Pointers. Division of thePlane into Congruent Cells
• § 13. The 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 ( 8s)
• § 17. The Configurations (93 )
• § 18. Perspective, Ideal Elements, and the Principle ofDuality 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. Schlafli's Double-Six
• CHAPTER IV: DIFFERENTIAL GEOMETRY
• § 26. Plane Curves
• § 27. Space Curves
• § 28. Curvature of Surfaces. Elliptic, Hyperbolic, and ParabolicPoints. 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 toElliptic 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
• § 41. Continuous Rigid Motions of Plane Figures
• § 42. An Instrument for Constructing the Ellipse and its Roulettes 1
• § 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
• Back Cover

• Reviews

• This book is a masterpiece — a delightful classic that should never go out of print.

MAA Reviews
• [This] superb introduction to modern geometry was co-authored by David Hilbert, one of the greatest mathematicians of the 20th century.

Steven Strogatz, Cornell University
• A fascinating tour of the 20th century mathematical zoo … Anyone who would like to see proof of the fact that a sphere with a hole can always be bent (no matter how small the hole), learn the theorems about Klein's bottle—a bottle with no edges, no inside, and no outside—and meet other strange creatures of modern geometry, will be delighted with Hilbert and Cohn-Vossen's book.

Scientific American
• Should provide stimulus and inspiration to every student and teacher of geometry.

Nature
• Students, particularly, would benefit very much by reading this book … they will experience the sensation of being taken into the friendly confidence of a great mathematician and being shown the real significance of things.

Science Progress
• A person with a minimum of formal training can follow the reasoning … an important [book].

The Mathematics Teacher
• Request Review Copy
• Get Permissions
Volume: 871952; 358 pp
MSC: Primary 00; 01;

This remarkable book has endured as a true masterpiece of mathematical exposition. There are few mathematics books that are still so widely read and continue to have so much to offer—even after more than half a century has passed! 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.

“Hilbert and Cohn-Vossen” is full of interesting facts, many of which you wish you had known before. It's also likely that you have heard those facts before, but surely wondered where they could be found. 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. Here, we see regular polyhedra again, from a different perspective. One of the high points of the chapter is the discussion of Schlafli's Double-Six, which leads to the description of the 27 lines on the general smooth cubic surface. As is true throughout the book, the magnificent drawings in this chapter immeasurably help the reader.

A particularly intriguing section in the chapter on differential geometry is Eleven Properties of the Sphere. Which eleven properties of such a ubiquitous mathematical object caught their discerning eye and why? Many mathematicians are familiar with the plaster models of surfaces found in many mathematics departments. The book includes pictures of some of the models that are found in the Göttingen collection. Furthermore, the mysterious lines that mark these surfaces are finally explained!

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.

• Front Cover
• Cover Page
• PREFACE
• CHAPTER I: THE SIMPLEST CURVES AND SURFACES
• § 1. Plane Curves
• § 2. The Cylinder, the Cone, the Conic Sections andTheir Surfaces of Revolution
• § 3. The Second-Order Surfaces
• APPENDICES TO CHAPTER I
• I. 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 theDiscontinuous Groups of Motions in the Plane
• § 11. The Discontinuous Groups of Plane Motionswith Infinite Unit Cells
• § 12. The Crystallographic Groups of Motions in the Plane.Regular Systems of Points and Pointers. Division of thePlane into Congruent Cells
• § 13. The 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 ( 8s)
• § 17. The Configurations (93 )
• § 18. Perspective, Ideal Elements, and the Principle ofDuality 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. Schlafli's Double-Six
• CHAPTER IV: DIFFERENTIAL GEOMETRY
• § 26. Plane Curves
• § 27. Space Curves
• § 28. Curvature of Surfaces. Elliptic, Hyperbolic, and ParabolicPoints. 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 toElliptic 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
• § 41. Continuous Rigid Motions of Plane Figures
• § 42. An Instrument for Constructing the Ellipse and its Roulettes 1
• § 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
• Back Cover
• This book is a masterpiece — a delightful classic that should never go out of print.

MAA Reviews
• [This] superb introduction to modern geometry was co-authored by David Hilbert, one of the greatest mathematicians of the 20th century.

Steven Strogatz, Cornell University
• A fascinating tour of the 20th century mathematical zoo … Anyone who would like to see proof of the fact that a sphere with a hole can always be bent (no matter how small the hole), learn the theorems about Klein's bottle—a bottle with no edges, no inside, and no outside—and meet other strange creatures of modern geometry, will be delighted with Hilbert and Cohn-Vossen's book.

Scientific American
• Should provide stimulus and inspiration to every student and teacher of geometry.

Nature
• Students, particularly, would benefit very much by reading this book … they will experience the sensation of being taken into the friendly confidence of a great mathematician and being shown the real significance of things.

Science Progress
• A person with a minimum of formal training can follow the reasoning … an important [book].

The Mathematics Teacher
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