**Geometry and the Imagination by David Hilbert, ISBN-13: 978-1470463021**

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- Publisher: American Mathematical Society (October 30, 2020)
- Language: English
- 357 pages
- ISBN-10: 1470463024
- ISBN-13: 978-1470463021

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.\n\n“Hilbert and Cohn\-Vossen” is full of interesting facts, many of which you wish you had known before. It\x27s 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 R3. 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\x27s series: π\/4=1-1\/3+1\/5-1\/7+-…. In the section on lattices in three and more dimensions, the authors consider sphere\-packing problems, including the famous Kepler problem.\n\nOne 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\x27s 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.\n\nA 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!\n\nThe 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.\n\nIt would be hard to overestimate the continuing influence Hilbert\-Cohn\-Vossen\x27s book has had on mathematicians of this century. It surely belongs in the “pantheon” of great mathematics books.

**Table of Contents:**

PREFACE •••••••••.••••••.••••••••••••••••.••••••••••••• iii

CHAPTER I

THE SIMPLEST CURVES AND SURFACES

§ 1. Plane Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

§ 2. The Cylinder, the Cone, the Conic Sections and Their

Surfaces of Revolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

§ 3. The Second-Order Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . 12

§ 4. The Thread Construction of the Ellipsoid, and Confocal

Quadrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

APPENDICES TO CHAPTER I

1. The Pedal-Point Construction of the Conics. . . . . . . . . . . . 25

2. The Directrices of the Conics. . . . . . . . . . . . . . . . . . . . . . . . 27

3. The Movable Rod Model of the Hyperboloid. . . . . . . . . . . . 29

CHAPTER II

REGULAR SYSTEMS OF POINTS

§ 5. Plane Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

§ 6. Plane Lattices in the Theory of Numbers. . . . . . . . . . . . . . . 37

§ 7. Lattices in Three and More than Three Dimensions ….. 44

§ 8. Crystals as Regular Systems of Points. . . . . . . . . . . . . . . . . 52

§ 9. Regular Systems of Points and Discontinuous Groups of

Motions ………………………………….. 56

§ 10. Plane Motions and their Composition; Classification of the

Discontinuous Groups of Motions in the Plane. . . . . . . 59

§ 11. The Discontinuous Groups of Plane Motions with Infinite

Unit Cells ………………………………… 64

§ 12. The Crystallographic Groups of Motions in the Plane.

Regular Systems of Points and Pointers. Division of

the Plane into Congruent Cells . . . . . . . . . . . . . . . . . . . . 70

§ 13. Crystallographic Classes and Groups of Motions in Space.

Groups and Systems of Points with Bilateral Symmetry 81

§ 14. The Regular Polyhedra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

CHAPTER III

PROJECTIVE CONFIGURATIONS

§ 15. Preliminary Remarks about Plane Configurations . . . . . . . 95

§ 16. The Configurations (73 ) and (83 ) • • • • • • • • • • • • • • • • • • • • • 98

§ 17. The Configurations (9a ) ………………………..1 02

§ 18. Perspective, Ideal Elements, and the Principle of Duality

in the Plane ………………………………. 112

§ 19. Ideal Elements and the Principle of Duality in Space.

Desargues’ Theorem and the Desargues Configuration

(10a) ……………………………………. 119

§ 20. Comparison of Pascal’s and Desargues Theorems ……..1 28

§ 21. Preliminary Remarks on Configurations in Space ……..1 33

§ 22. Reye’s Configuration …………………………..1 34

§ 23. Regular Polyhedra in Three and Four Dimensions, and

their Projections ……………………………1 43

§ 24. Enumerative Methods of Geometry ………………..1 57

§ 25. Schlafli’s Double-Six …………………………..1 64

CHAPTER IV

DIFFERENTIAL GEOMETRY

§ 26. Plane Curves ……………………….. ,, ……..1 72

§ 27. Space Curves ……………………………….. 178

§ 28. Curvature of Surfaces. Elliptic, Hyperbolic, and Parabolic

Points. Lines of Curvature and Asymptotic Lines. Umbilical

Points, Minimal Surfaces, Monkey Saddles ….. 183

§ 29. The Spherical Image and Gaussian Curvature ………..1 93

§ 30. Developable Surfaces, Ruled Surfaces ……………..2 04

§ 31. The Twisting of Space Curves …………………… 211

§ 32. Eleven Properties of the Sphere ………………….2 15

§ 33. Ben dings Leaving a Surf ace Invariant …………….. 232

§ 34. Elliptic Geometry ……………………………. 235

§ 35. Hyperbolic Geometry, and its Relation to Euclidean and to

Elliptic Geometry …………………………..2 42

§ 36. Stereographic Projection and Circle-Preserving Transformations.

Poincare’s Model of the Hyperbolic Plane .248

§ 37. Methods of Mapping, Isometric, Area-Preserving, Geodesic,

Continuous and Conformal Mappings ……….2 60

§ 38. Geometrical Function Theory. Riemann’s Mapping Theorem.

Conformal Mapping in Space ……………..2 63

§ 39. Conformal Mappings of Curved Surfaces. Minimal Surf

aces. Plateau’s Problem …………………….. 268

CHAPTER V

**KINEMATICS**

§ 40. Linkages …………………………………… 272

§ 41. Continuous Rigid Motions of Plane Figures …………. 275

§ 42. An Instrument for Constructing the Ellipse and its Roulettes

……………………………………..2 83

§ 43. Continuous Motions in Space ……………………2 85

CHAPTER VI

TOPOLOGY

§ 44. Polyhedra …………………………………..2 90

§ 45. Surfaces …………………………………… 295

§ 46. One-Sided Surfaces ……………………………3 02

§ 47. The Projective Plane as a Closed Surface …………..3 13

§ 49. Topological Mappings of a Surface onto Itself. Fixed

Points. Classes of Mappings. The Universal Covering

Surface of the Torus …………………………3 24

§ 50. Conformal Mapping of the Torus …………………3 30

§ 51. The Problem of Contiguous Regions, The Thread Problem,

and the Color Problem ……………………….3 33

APPENDICES TO CHAPTER VI

1. The Projective Plane in Four-Dimensional Space …….3 40

2. The Euclidean Plane in Four-Dimensional Space ……..3 41

INDEX ……….•………….•…..•………•…•…• 345

* David Hilbert *(1862 – 1943) received his PhD from the University of Königsberg, Prussia (now Kaliningrad, Russia) in 1884. He remained there until 1895, after which he was appointed Professor of Mathematics at the University of Göttingen. He held this professorship for most of his life. Hilbert is recognized as one of the most influential mathematicians of the 19th and early 20th centuries. His own discoveries alone would have given him that honour, yet it was his leadership in the field of mathematics throughout his later life that distinguishes him. Hilbert’s name is given to Infinite-Dimensional space, called Hilbert space, used as a conception for the mathematical analysis of the kinetic gas theory and the theory of radiations.

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