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Geometry and the Imagination (AMS Chelsea Publishing) Hardcover – October 1, 1999


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Editorial Reviews

Review

This book is a masterpiece -- a delightful classic that should never go out of print. --Mathematical Association of America

[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

Language Notes

Text: English (translation)
Original Language: German
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Product Details

  • Series: AMS Chelsea Publishing
  • Hardcover: 357 pages
  • Publisher: American Mathematical Society (October 1, 1999)
  • Language: English
  • ISBN-10: 0821819984
  • ISBN-13: 978-0821819982
  • Product Dimensions: 0.8 x 6 x 9 inches
  • Shipping Weight: 1.2 pounds (View shipping rates and policies)
  • Average Customer Review: 5.0 out of 5 stars  See all reviews (12 customer reviews)
  • Amazon Best Sellers Rank: #401,133 in Books (See Top 100 in Books)

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60 of 60 people found the following review helpful By Colin McLarty on November 8, 2001
Format: Hardcover
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.
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35 of 39 people found the following review helpful By henrique fleming on August 5, 1998
Format: 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".
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25 of 27 people found the following review helpful By Peter Renz on July 21, 2003
Format: Hardcover
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.
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4 of 4 people found the following review helpful By Carles V on July 14, 2012
Format: Hardcover Verified Purchase
The purpose of this book is very well described by the author in the preface.

There Hilbert says "...In this book, it is our purpose to give a presentation of geometry, as it stands today, in its visual, intuitive aspects. With the aid of visual imagination we can illuminate the manifold facts and problems of geometry, and beyond this, it is possible in many cases to depict the geometric outline ot the methods of investigation and proof, without necessarily entering into the details connected with the strict definitions of concepts and with the actual calculations."

A little further, he says "...This book was written to bring about a greater enjoyment of mathematics, by making it easier for the reader to penetrate to the essence of mathematics without having to weight himself down under a laborious course of studies."

As a reader of this book, I can say that the key words are "visual imagination" and "enjoyment of mathematics".
The purpose described by Hilbert is completely (and excellently) achieved.

The book is a masterpiece, written by one of the masters of Mathematics.

In an elegant and clear style, Hilbert explains the most beautiful geometrical concepts.
When reading it, you feel as if Hilbert was sit down besides you, just talking about geometry to you, (maybe with the aid of a sheet of paper and a pencil), and you can grasp the genius of the Göttingen Professor.
He does not use practically any formula or mathematical expression, however his prose is full of mathematical insights, geometrical facts, stimulating images and delicious "expository" proofs.
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16 of 21 people found the following review helpful By Marvin J. Greenberg on December 23, 2006
Format: Hardcover Verified Purchase
I agree that this book, co-authored by the co-greatest mathematician of the first quarter of the twentieth century, is a masterpiece to be treasured and kept in print, as other reviewers have stated.

However: The Preface states: "This book was written to bring about a greater enjoyment of mathematics, by making it easier for the reader to penetrate to the essence of mathematics without having to weight himself down under a laborious course of studies."

All I can say is that if you read this and find it "easy," then you have terrific mathematical talent! Yes, the drawings and the intuitive descriptions are helpful, but much of the book is so obscure that I have been told that one of the world's leading geometers is working on an annotated edition explaining what the authors were talking about. On topics which I had already studied elsewhere, I found the presentation illuminating.

I still recommend this book.
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