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Geometry and the Imagination Hardcover – January 1, 1952


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Product Details

  • Hardcover: 357 pages
  • Publisher: Chelsea Publishing; First American Edition edition (1952)
  • Language: English
  • ISBN-10: 0828400873
  • ISBN-13: 978-0828400879
  • Product Dimensions: 9.8 x 1.4 x 0.1 inches
  • Shipping Weight: 1.7 pounds
  • Average Customer Review: 5.0 out of 5 stars  See all reviews (11 customer reviews)
  • Amazon Best Sellers Rank: #3,271,783 in Books (See Top 100 in Books)

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A work of art.
henrique fleming
Everyone acquainted with math knows this book's content and accessible style has made it a classic.
Alan M. Wolsky
In an elegant and clear style, Hilbert explains the most beautiful geometrical concepts.
Carles V

Most Helpful Customer Reviews

55 of 55 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 38 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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22 of 24 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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15 of 19 people found the following review helpful By Marvin J. Greenberg on December 23, 2006
Format: Hardcover
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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12 of 15 people found the following review helpful By Viktor Blasjo on January 11, 2007
Format: Hardcover
This is a marvellous book. I will illustrate by one sample from each chapter (except chapter 1 on "the simplest curves and surfaces" which is the least exciting chapter).

Chapter 2 on "regular system of points" contains a beautiful derivation of Leibnitz' series pi/4=1-1/3+1/5-1/7+... If we draw a large circle centred at the origin then of course a good measure of its area is the number of integer points it contains. Now, for any such point, x^2+y^2 is an integer less than r^2. So the number of such points can be obtained by going through all integers less than r^2 and counting how many times it can be written as the sum of two squares. But this is a classical problem in number theory and the solution is known. So this number theoretic result essentially tells us the area of a large circle, so it implies an expression for pi, namely Leibnitz' series.

Chapter 3 is on projective geometry. We go through many projective configurations that are not seen very often today, but still the classics are the best, such as Desargues' theorem. If we have a triangular pyramid and cut it with two planes to get two triangles then the three points of intersection of the extensions of corresponding sides will or course be on a line (the intersection of the two planes), which is the three-dimensional Desargues' theorem. But by projecting the triangles onto one of the walls of the pyramid we get two projectively related plane triangles and the theorem holds for them also. All we have to do to prove the plane Desargues' theorem is to prove that all such configurations can be obtained in his way (i.e. that one can always erect an appropriate pyramid based on two projectively related plane triangles) which is practically obvious.

Chapter 4 is on differential geometry.
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