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 enlarge | Author: Donal O'shea
Publisher: Walker & Company
Category: Book
List Price: $15.95
Buy New: $5.88
You Save: $10.07 (63%)
New (30) Used (20) Collectible (1) from $4.62
Rating:  20 reviews
Sales Rank: 274666
Media: Paperback
Edition: 1st
Pages: 304
Number Of Items: 1
Shipping Weight (lbs): 0.6
Dimensions (in): 8.2 x 5.4 x 0.9
ISBN: 0802716547
Dewey Decimal Number: 500
EAN: 9780802716545
Publication Date: December 26, 2007
Availability: Usually ships in 1-2 business days
Condition: Hardcover
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| Customer Reviews:
 Interesting and Enjoyable Read January 28, 2008
Lawcool (NY)
1 out of 1 found this review helpful
I enjoyed reading this book very much and it really opens up my mind. For example, I did not know proving the Poincare conjecture has implications on finding the shape of our universe before I read this book. The author does a good job in introducing the ideas of topology, its history and origin, and of course the Poincare conjecture, to his readers. As a casual reader (I am not a mathematician), I found the level of details and explanation in the book just right to give its readers the basic and intuitive understanding of the mathematics behind. Now I am interested to read more about modern topology!
A must read! January 22, 2008
Ralph L. Ades (Northridge, CA USA)
1 out of 2 found this review helpful
A must read! One of the best book I have ever read! This book taught me more topology than the two university courses. A clear and precisely written history and exposition of one the most important ideas in science.
What is the Shape of the Universe? A Three-sphere? May 2, 2008
Man Kam Tam (Calexico, CA USA)
0 out of 2 found this review helpful
Donal O'Shea's "The Poincare Conjecture: In Search of the Shape of the Universe" is about Henri Poincare's conjecture, which is "central to our understanding of ourselves and the universe in which we live." The book is written for "the curious individual who remembers a little high school geometry." The book traces "the history of geometry, the discovery of non-Euclidean geometry, and the birth of topology and differential geometry through five millennia..."
What is the shape of the universe? With the proof of Poincare conjecture, we have a "method" to find out whether the universe is three-sphere or not. The method is "by using a complete atlas to check whether every closed loop could be shrunk to a point."
"... Space and matter are intimately related, and the assertion that the universe has an infinite amount of matter causes serious theoretical problems ... The universe could have a boundary of some kind ... Regarding the size and shape of the universe, we are almost in precisely the same position that Columbus was in 1492 ... there was no complete atlas of the Earth in Columbus's time, there is no complete atlas of the universe today. If we left the Earth on a very fast spaceship, headed out in a fixed direction ... after a very long time, most cosmologists and mathematicians believe, we would come back close to where we started."
"... a two-dimensional manifold is a mathematical object that shares a key property with the surface of our earth [... all regions can mapped onto on a piece of paper] ... The corresponding mathematical object that models our universe is a three-dimensional manifold, or thee-manifold. It is a set in which every point belongs to a region that can be mapped onto the points inside a clear aquarium or shoebox. In other words, the region around any point looks like space rather than a plane ... an atlas is a collection of maps that is complete in the sense that every point belongs to some region that is covered by one of the maps. A three-manifold is the object that is covered by all the maps in an atlas ... A three-dimensional manifold is called compact or finite if there is an atlas of it that is finite ... The very simplest finite three-manifold is the three-dimensional sphere, or three sphere."
"Over the last century, many individuals have devoted their life's work to furthering our understanding of three-manifold. But ... all efforts ... [arrive] at an answer: Among all those three-manifolds, is there anyone that is different from the three-sphere and that has the property that every path can be shrunk to a point? If there is no such manifold, then we could say for sure whether our universe is a three-sphere by using a complete atlas to check whether every closed loop could be shrunk to a point. The Poincare conjecture states that there is no such manifold. ... the Poincare conjecture is the assertion that any compact three-manifold on which any closed path can be shrunk to a point, is the same topologically as (... homeomorphic to) the three-sphere..."
"If the manifold is simply connected ( ... every loop can be shrunk to a point), ... Perelman proves that the Ricci flow [analogous to the diffusion of heat]... will eventually smooth out the extremes of curvature, giving a manifold with constant positive curvature homeomorphic the original manifold. Arguments that have been known for a long time show that a simply connected manifold with constant positive curvature is necessary the three-dimensional sphere. Therefore, Perelman's work proves the Poincare conjecture."
A Magnificent Story July 1, 2007
Frank Morgan (Williams College)
O'Shea's beautiful and sweeping account of a thousand years of mathematics and mathematicians climaxing in Perelman's recent proof of the Poincare conjecture gives a thrilling sense of the grandeur of humanity's progress and potential. It is a magnificent story, and that's how he tells it.
beautiful mathematics May 9, 2007
Nim Sudo
39 out of 40 found this review helpful
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. Many of them looked like they were drawn with MacPaint, and are reproduced so small and dark that they are hard to make out. At least one picture is mathematically incorrect: it shows a disc of paper with a wedge cut out being folded to produce a spherical cap, but really one would get a cone instead. This mistake is unfortunate since it contradicts the whole point of the chapter in which it appears. In short, if I were writing this book, I would want to trim the history, remove some of the mathematics, explain the rest of the mathematics in more detail, and improve the pictures.
Perelman's papers finishing off the Poincare conjecture were sketchy, and a lot of work by other mathematicians was required to turn his papers into a detailed proof. There was some (in my opinion silly and unfortunate) controversy in the media regarding how much credit should go to various people for this. The book does not go into this controversy, which I think is a good thing (although it gives some hints without fully explaining the situation). There is also no discussion of why Perelman made the unusual decision to decline the Fields medal. Maybe no one really knows.
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