Geometry and the Imagination (CHEL/87.H) (AMS Chelsea Publishing) (Hardcover)

by David Hilbert (Author), S. Cohn-Vossen (Author) "The simplest surface is the plane..." (more)
Key Phrases: generating parallelogram, geometrical function theory, hyperbolic rigid motions, New York, Number Theory (more...)

Editorial Reviews

Review
" ... sensation of being taken into the friendly confidence of a great mathematician and being shown the real significance of things." -- Science Progress

"A fascinating tour of the 20th century mathematical zoo ... " -- Scientific American

"Should provide stimulus and inspiration to every student and teacher of geometry." -- Nature

A person with a minimum of formal training can follow the reasoning ... an important [book]. -- Mathematics Teacher

Should provide stimulus and inspiration to every student and teacher of geometry. -- Nature

Product Description
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--after more than half a century! 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, or had 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.

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Product Details

• Hardcover: 357 pages
• Publisher: American Mathematical Society (October 1, 1999)
• Language: English
• ISBN-10: 0821819984
• ISBN-13: 978-0821819982
• Product Dimensions: 9.2 x 6.1 x 0.9 inches
• Shipping Weight: 1.5 pounds (View shipping rates and policies)
• Average Customer Review: 5.0 out of 5 stars See all reviews (8 customer reviews)
• Amazon.com Sales Rank: #132,961 in Books (See Bestsellers in Books)

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Customer Reviews

Most Helpful Customer Reviews

38 of 38 people found the following review helpful:

5.0 out of 5 stars A glimpse of mathematics as Hilbert saw it, November 8, 2001

By	Colin McLarty (Chardon, OH USA) - See all my reviews

The leading mathematician of the 20th century, David Hilbert liked to quote "an old French mathematician" saying "A mathematical theory should not be considered complete until you have made it so clear that you can explain it to the first man you meet on the street". By that standard, this book by Hilbert was the first to complete several branches of geometry: for example, plane projective geometry and projective duality, regular polyhedra in 4 dimensions, elliptic and hyperbolic non-Euclidean geometries, topology of surfaces, curves in space, Gaussian curvature of surfaces (esp. that fact that you cannot bend a sphere without stretching some part of it, but you can if there is just one hole however small), and how lattices in the plane relate to number theory.

It is beautiful geometry, beautifully described. Besides the relatively recent topics he handles classics like conic sections, ruled surfaces, crystal groups, and 3 dimensional polyhedra. In line with Hilbert's thinking, the results and the descriptions are beautiful because they are so clear.

More than that, this book is an accessible look at how Hilbert saw mathematics. In the preface he denounces "the superstition that mathematics is but a continuation ... of juggling with numbers". Ironically, some people today will tell you Hilbert thought math was precisely juggling with formal symbols. That is a misunderstanding of Hilbert's logical strategy of "formalism" which he created to avoid various criticisms of set theory. This book is the only written work where Hilbert actually applied that strategy by dividing proofs up into intuitive and infinitary/set-theoretic parts. Alongside many thoroughly intuitive proofs, Hilbert gives several extensively intuitive proofs which also require detailed calculation with the infinite sets of real of complex numbers. In those cases Hilbert says "we would use analysis to show ..." and then he wraps up the proof without actually giving the analytic part.

If you find it terribly easy to absorb Hilbert's THEORY OF ALGEBRAIC NUMBER FIELDS and also Hilbert and Courant METHODS OF MATHEMATICAL PHYSICS, then of course you'll get a fuller idea of his math by reading them--but only if you find it very easy. Hilbert did. And that ease is a part of how he saw the subject. I do not mean he found the results easily but he easily grasped them once found. And you'll have to read both, and a lot more, to see the sweep of his view. For Hilbert the lectures in GEOMETRY AND THE IMAGINATION were among the crowns of his career. He showed the wide scope of geometry and finally completed the proofs of recent, advanced results from all around it. He made them so clear he could explain them to you or me.

28 of 29 people found the following review helpful:

5.0 out of 5 stars A masterpiece!, August 5, 1998

By	henrique fleming (Sao Paulo, SP Brazil) - See all my reviews

This review is from: Geometry and the Imagination (Hardcover)

This is one of the best books on Mathematics ever written. The author is arguably the best mathematician of the century. Here he treats geometry, including topology, in an elementary, though profound, way, with no formalism. A work of art. Books like this shouldn't ever become "out-of-print".

17 of 18 people found the following review helpful:

5.0 out of 5 stars Beautiful, Rewarding, and Deep., July 21, 2003

By	Peter Renz (Brookline, MA United States) - See all my reviews (REAL NAME)

I have some 47 books in the geometry section of my shelves. If I had to discard 40 of these, Geometry and the Imagination would be among the 7 remaining.

Geometry is the study of relationships between shapes, and this book helps you see how shapes fit together. Ultimately, you must make the connections in your mind using your mind's eye. The illustrations and text help you make these connections. This is a book that requires effort and delivers rewards.