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The Poincare Conjecture: In Search of the Shape of the Universe Paperback – Bargain Price, December 26, 2007

ISBN-13: 860-1419975334 ISBN-10: 0802716547 Edition: 1st

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Editorial Reviews

From Publishers Weekly

The reclusive Russian mathematician Grigory Perelman became a minor media celebrity last summer when he refused the prestigious Fields medal, awarded every four years to a mathematician under the age of 40. Perelman had succeeded in solving the Poincaré conjecture, named for 19th-century French mathematician Henri Poincaré, and which contemporary cosmologists believe has implications for our understanding of the shape of the universe. O'Shea, a professor of mathematics at Mount Holyoke College, begins his account of the long and contentious search for a solution to the puzzle by looking at how we came to understand the shape of the Earth, beginning with the Greeks, in particular Pythagoras and Plato. Writing for generalist science buffs, O'Shea gives a brief course in geometry and in topology and the topological structures called manifolds that are the basis of Poincaré's puzzle. Inexplicably, however, O'Shea doesn't give readers a formal statement of the conjecture itself until well into the book. O'Shea describes mind-bending structures in topology as clearly as most of us can describe a cube, but readers will need to do a little Wikipedia-ing first to find out just what it is they're reading about. Illus. (Mar.)
Copyright © Reed Business Information, a division of Reed Elsevier Inc. All rights reserved. --This text refers to an out of print or unavailable edition of this title.

From Booklist

Euclid's Elements is historically the most popular mathematics book ever written, but one thing about it nagged its readers: its postulate that every line has exactly one line parallel to it. Doubt about the postulate's truth is O'Shea's starting point for this accessible if challenging presentation of a famous problem ultimately rooted in the parallel postulate. The great mathematician Henri Poincare (1854-1912) spent years investigating the implications of non-Euclidian space. Aided by diagrams and analogies, O'Shea, a professional mathematician, explains non-Euclidian spaces, populated by objects technically called manifolds and n-spheres (n means the number of dimensions), which leads to Poincare's conjecture, verbatim: "Is it possible that the fundamental group of a manifold could be the identity, but that the manifold might not be homeomorphic to the three-dimensional sphere?" Readers defeated by such language, despite O'Shea's valiant nonnumerical clarity, can yet digest the author's connection of the conjecture to the shape of the universe, the biographical portraits that animate his text, and the drama of the conjecture's proof, announced in 2006. Gilbert Taylor
Copyright © American Library Association. All rights reserved --This text refers to an out of print or unavailable edition of this title.

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Product Details

  • Paperback: 304 pages
  • Publisher: Walker & Company; 1st edition (December 26, 2007)
  • Language: English
  • ISBN-10: 0802716547
  • ASIN: B001PTG4IC
  • Product Dimensions: 5.6 x 1 x 8.2 inches
  • Shipping Weight: 10.4 ounces
  • Average Customer Review: 4.4 out of 5 stars  See all reviews (32 customer reviews)
  • Amazon Best Sellers Rank: #1,900,323 in Books (See Top 100 in Books)

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Customer Reviews

I enjoyed reading this book very much and it really opens up my mind.
The main story is very well organized with the focus on topology nicely combined with relevant and interesting historical facts.
Marek Petrik
The Perfect Rigor is the story about Grigory Perelman, a mathematical genius who recently solved the Poincare Conjecture.

Most Helpful Customer Reviews

79 of 82 people found the following review helpful By Nim Sudo on May 8, 2007
Format: Hardcover
The Poincare conjecture was one of the most beautiful and important unsolved problems in mathematics for the last century. It has recently been solved, in a remarkable story, with the final breakthrough due to Perelman, who was awarded the Fields medal for his work but declined to accept it. The Poincare conjecture concerns the possible shapes of three-dimensional spaces, such as the universe that we live in. This book explains what the Poincare conjecture says, and tells the history of its formulation and proof. There are no equations in the main text (and only a couple in the endnotes), so in principle anyone can read this.

The book does a nice job of motivating the Poincare conjecture, by first discussing the possibilities for the shape of the two-dimensional surface of the earth (before we had explored the whole earth and figured out that it is a sphere), and then discussing the possibilities for the shape of the three-dimensional universe (which is currently unknown). The book also does a good job of explaining what modern geometry is about and how this has drastically changed since Euclid.

There were three things about the book that I didn't like. (Bear in mind that I do topology for a living so I am maybe being too critical here.) First, there is a lot of history, not only of the people who worked on the Poincare conjecture, but also of the institutions and political environment in which they worked. A lot of this seemed to me to have little relevance to the Poincare conjecture and didn't hold my interest. Second, in between these historical asides, the mathematical sections often rush through too much material, in not enough detail to be really understandable to a lay reader. Third, the pictures were subpar.
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80 of 87 people found the following review helpful By Steve Koss VINE VOICE on July 21, 2007
Format: Hardcover
Describing some of Henri Poincare's work, author Donal O'Shea writes on page 132 that Poincare "produced an infinite family of closed three-dimensional manifolds that are not homeomorphic to one another and showed that one can have nonhomeomorphic manifolds with the same Betti numbers and, in fact, with the same Betti numbers as a sphere." Oh, so you don't know what a Betti number is? Well, O'Shea's Glossary of Terms describes it as "an integer counting that number of inequivalent manifolds of a given dimension in a manifold that do not bound a submanifold of one dimension higher." If this is not quite your cup of mathematical tea (as it is not quite mine, despite my B.S. in mathematics from a highly-regarded engineering school), then THE POINCARE CONJECTURE might just be a full teapot that you want to skip.

A fundamental rule of nonfiction is to identify and write for your intended audience, and it is difficult to imagine who O'Shea saw here as his audience. THE POINCARE CONJECTURE addresses a mathematically famous but publicly obscure hypothesis from the general field of topology, the study of shapes and curvature. Filled to overflowing with historical background, dating back to Euclid's original five postulates for plane geometry, the main body of O'Shea's book is exclusively textual. Even the hypercritical Ricci flow equation, the main vehicle through which Grigory Perelman achieved his landmark proof, is relegated in abbreviated form to the footnotes. All of this, combined with O'Shea's opening chapters attempting to explain topological manifolds, suggests a book targeted at nonmathematicians. Yet as the excerpt above demonstrates, the author seems too often not to have found the necessary nonmathematical explication to reach successfully a nonprofessional mathematical audience.
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9 of 9 people found the following review helpful By J.D. WHITE on March 26, 2008
Format: Paperback
This book feels as if the author tried to edit it himself, complete with embarassingly frequent mistakes in grammar and punctuation, not to mention horribly botched illustrations.

While several of the reviewers here have stated that they weren't satisfied with the mathematical "meatiness" of this book, I represent the lay side that found plenty of challenge following the concepts here (most of which I was seeing for the first time). As such, the histories were welcome asides to the often very long, hard to follow, and dubiously worded (AND poorly illustrated) technical paragraphs.

Still, for someone who used this book as an introduction to topology, it was a fascinating parts. If it ever sees another edition that allows for decent editing and proofreading, I imagine I would tack a fourth star onto the review.
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11 of 12 people found the following review helpful By Simmoril on June 3, 2007
Format: Hardcover
I myself am not a mathematician, but I am a fan of mathematics in general. I had been following Grigori Perelman's work in the news ever since he gave his lectures at MIT, and had been awaiting a book to cover this amazing story.

The book itself does quite a bit of leg work covering the history behind the Poincare Conjecture and the lives of the key contributors (Gauss, Riemann, Poincare, Klein, etc.). In the first few chapters, the author gives the reader a 'crash course' in topology (as well as talking about how the field of topology came to be), and in the last few chapters, talks about the failed attempts at proving the conjecture by various mathematicians, and finally, of course, the successful attempt by Perelman.

While making my way through this book, it felt like the author was attempting to do too much in too small a space. At exactly 200 pages of text (the last 90 pages or so are footnotes, appendices, and index), it's pretty much a featherweight when you consider the material the author is trying to cover. The book at once tries to be a history lesson, a treatise on the importance of mathematics (or learning in general depending on your interpretation), a short tutorial on topology, and a brief outline of the conjecture and it's proof. Each of these topics is covered in varying detail at the expense of brevity in others. I was shocked to see that talk about Perelman's proof occupied a scant 10 or so pages at the end of the book.

The history lesson, albeit very well researched and nicely written, shouldn't have been the main focus of the book.
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