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The Poincare Conjecture: In Search of the Shape of the Universe Paperback – January 2, 2008

by Donal O'Shea (Author)
4.5 4.5 out of 5 stars 84 ratings
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"O'Shea tells the fascinating story of this mathematical mystery and its solution by the eccentric Mr. Perelman."-Wall Street Journal
In 1904, Henri Poincaré, a giant among mathematicians who transformed the fledging area of topology into a powerful field essential to all mathematics and physics, posed the Poincaré conjecture, a tantalizing puzzle that speaks to the possible shape of the universe. For more than a century, the conjecture resisted attempts to prove or disprove it. As Donal O'Shea reveals in his elegant narrative, Poincaré's conjecture opens a door to the history of geometry, from the Pythagoreans of ancient Greece to the celebrated geniuses of the nineteenth-century German academy and, ultimately, to a fascinating array of personalities-Poincaré and Bernhard Riemann, William Thurston and Richard Hamilton, and the eccentric genius who appears to have solved it, Grigory Perelman. The solution seems certain to open up new corners of the mathematical universe.

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  1. Print length
    304 pages
  2. Language
    English
  3. Publisher
    Walker Books
  4. Publication date
    January 2, 2008
  5. Dimensions
    5.56 x 0.85 x 8.33 inches
  6. ISBN-10
    0802716547
  7. ISBN-13
    978-0802716545
  8. See all details
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Editorial Reviews

Review

“O'Shea inspires readers to note the beauty, application, and humanity involved with this mathematical journey.” ―Library Journal

“O'Shea describes mind-bending structures in topology as clearly as most of us can "describe a cube…” ―Publishers Weekly

“Accessible…. valiant nonnumerical clarity…” ―Booklist

“Fascinating….[O'Shea] does a good job of explaining the mathematics involved in solving the conjecture…” ―Wall St Journal

“A layman's guide to this mathematical odyssey is long overdue, and this one will appeal to math whizzes and interested novices alike.” ―Discover magazine

“O'Shea shows that, just like chasing ‘sensual passions,' the single-minded, relentless pursuit of proof can be a creative process.” ―Chicago Tribune

“O'Shea tells the whole story in this book, neatly interweaving his main theme with the history of ideas about our planet and universe. There is good coverage of all the main personalities involved, each one set in the social and academic context of his time.” ―New York Sun

“Donal O'Shea has written a truly marvelous book. Not only does he explain the long-unsolved, beautiful Poincaré conjecture, he also makes clear how the Russian mathematician Grigory Perelman finally solved it. Around this drama O'Shea weaves a tapestry of elementary topology and astonishing concepts, such as the Ricci flow, that have contributed to Perelman's brilliant achievement. One can't read The Poincaré Conjecture without an overwhelming awe at the infinite depths and richness of a mathematical realm not made by us.” ―Martin Gardner, author of The Annotated Alice and Aha! Insight

“In The Poincaré Conjecture, Mr. O'Shea tells the fascinating story of this mathematical mystery and its solution by the eccentric Mr. Perelman . . . Mr. O'Shea does a good job of explaining the mathematics involved in solving the conjecture . . . [He] avoids cliché (we're spared the usual reference to coffee cups turning into doughnuts as an explanation of how surfaces might stretch without closing holes), and he tries to keep things lively.” ―Amir D. Aczel, The Wall Street Journal

“The history of the Poincaré conjecture is the story of one of the most important areas of modern mathematics. Donal O'Shea tells that story in a delightful and informative way--the concepts, the issues, and the people who made everything happen. I recommend it highly.” ―Keith Devlin, Stanford University, author of The Millennium Problems

About the Author

Donal O'Shea is a professor of mathematics and the dean of faculty and vice president for academic affairs of Mount Holyoke College in Massachusetts. This is his first book for a general audience. He lives in South Hadley, Massachusetts.

Product details

  • Publisher ‏ : ‎ Walker Books; First Edition (January 2, 2008)
  • Language ‏ : ‎ English
  • Paperback ‏ : ‎ 304 pages
  • ISBN-10 ‏ : ‎ 0802716547
  • ISBN-13 ‏ : ‎ 978-0802716545
  • Item Weight ‏ : ‎ 10.4 ounces
  • Dimensions ‏ : ‎ 5.56 x 0.85 x 8.33 inches
  • Customer Reviews:
    4.5 4.5 out of 5 stars 84 ratings

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Donal O'Shea

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Customer reviews

4.5 out of 5 stars
4.5 out of 5
84 global ratings

Customers say

Customers find the book well-written and mixed with mathematical history and ideas. They also appreciate the great explanations of tricky concepts.

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Writing styleMathematical concepts
10 customers mention "Writing style"10 positive0 negative

Customers find the writing style well written, historical, and beautiful. They also say the complex subject is explained beautifully. Readers recommend the book for scientific minds and enjoy the history of math.

"...It also has some more technical details, but both books are good reading for a mathematically educated reader." Read more

"Amazing story, O'Shea does a nice job of mixing mathematical history and ideas to give outsiders a glimpse of what Poincare and Perelman were..." Read more

"This book is well written. A very complicated subject written in terms that average intellectualpeople can understand...." Read more

"An excellent book with a ''smooth'' introduction in a very complex albeit fcsinating mathematical field." Read more

5 customers mention "Mathematical concepts"5 positive0 negative

Customers find the mathematical concepts in the book great, fascinating, and accessible. They also recommend it for scientific minds.

"...This book capure the beauty of mathematics and makes them accessible to the general reader" Read more

"Well written. Good on explaining topological concepts." Read more

"This was a fascinating and accessible exploration of topology in general carried by the history of the field." Read more

"A complex subject explained beautifully. Recommended for scientific minds." Read more

Top reviews from the United States

Jaume Puigbo Vila
5.0 out of 5 stars The first solution to one of the Millenium problems
Reviewed in the United States on August 1, 2007
Verified Purchase
Lee Carlson's review casts some doubt about the validity of Perelman's proof, but this is not what the mathematical community of experts is saying. Even the people who have filled in the details of Perelman's proof agree that all the merit is his. As this book shows, Morgan clearly states in his address in the ICM in Madrid that Perelman proved the Poincaré's conjecture and much more (Thurston's conjecture) and introduced new methods that will be used by many mathematicians in the coming years.

O'Shea's book is a good complement to Szpiro's. O'Shea is more encompassing and starts the history of the conjecture going back as far as Babylonic mathematics. It only gives the biography of Poincaré in page 111 and misses some of the details of the controversy provoked by Yau and explained in detail in an article in New Yorker and also in the book by Szpiro. It also has some more technical details, but both books are good reading for a mathematically educated reader.
11 people found this helpful
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Man Kam Tam
5.0 out of 5 stars What is the Shape of the Universe? A Three-sphere?
Reviewed in the United States on May 2, 2008
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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."
6 people found this helpful
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Amazon Customer
4.0 out of 5 stars Amazing history, O'Shea does a nice job of mixing ...
Reviewed in the United States on December 25, 2017
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Amazing story, O'Shea does a nice job of mixing mathematical history and ideas to give outsiders a glimpse of what Poincare and Perelman were getting at. At times the history details felt too far from the big picture and lazy (some big block quotes in the middle of the book), and there are bits of unnecessary editorializing, but overall quite enjoyed it.
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John Murphy
5.0 out of 5 stars mindblowing even if you don't get all of the math
Reviewed in the United States on November 6, 2023
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the historical perspective is every bit as good as the math perspective. Some parts were over my head, but that shouldn't deter any reader who's interested.
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Tom Gray
5.0 out of 5 stars I was able to understand the purpose of the field of topology much better than with the efforts of the professors who tried ...
Reviewed in the United States on July 25, 2015
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This book performed the impossible. With it, I was able to understand the purpose of the field of topology much better than with the efforts of the professors who tried to teach me it. I don't understand why mathematicians explain the beauty of mathematics in such obscure ways. This book capure the beauty of mathematics and makes them accessible to the general reader
3 people found this helpful
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Charlie
5.0 out of 5 stars The Poincare Conjecture
Reviewed in the United States on April 19, 2010
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This book is well written. A very complicated subject written in terms that average intellectual
people can understand. The subject matter deals with material that most people have never
contemplated. The Perfect Rigor is the story about Grigory Perelman, a mathematical genius who
recently solved the Poincare Conjecture. Anyone who reads The Poincare Conjecture will have a
great appreciation for the brilliance of Mr. Perelman.
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George Papaioannou
4.0 out of 5 stars Four Stars
Reviewed in the United States on April 16, 2018
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An excellent book with a ''smooth'' introduction in a very complex albeit fcsinating mathematical field.
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William Palin Cooke
5.0 out of 5 stars Well written historical account with clarity on mathematical concepts.
Reviewed in the United States on September 16, 2019
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Well written. Good on explaining topological concepts.
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Top reviews from other countries

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Michail L.
5.0 out of 5 stars An amazing story through time and space
Reviewed in France on September 1, 2022
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The story behind the latest scientific adventure that started with Poincaré in the early 1900s and ended with Perelman in the early 2000s. A great tale on the history of geometry from the Ionian Greeks to the modern geometry specialists.

A nice introduction in modern geometry and topology for the broad public.

The footnotes and book recommendations fill in the details for those that want to dive deep in the technical details.
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jdavidg
5.0 out of 5 stars fascinating fusion of geometry and history
Reviewed in Canada on November 12, 2011
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Fast paced read from Pythagoras to present.
Well woven yarn of both societal history and
the evolution of math, geometry and topology.
Having read "The Shape of Inner Space": Yau;
"The Fabric of the Cosmos": Greene; and
stumbled with "The Road to Reality": Penrose;
I found this book very easy to read and
enabled me to grasp a clearer concept of
topology: metrics, tensors, three spheres,
Riemannian geometry and manifolds, Ricci flow,
you get the picture.
A quick read with no math background will
fill you with awe. A more contemplative
approach will allow you to taste the beauty
of higher dimensional mathematics.
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Perseus
4.0 out of 5 stars Ein gutes Buch über ein komplexes Thema
Reviewed in Germany on April 5, 2013
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O'Shea hat ein gutes Buch zu einem überaus komplexen Thema abgeliefert. Da ich kein Laie bin, kann ich nur bedingt beurteilen, wie gut der Zugang sonst wäre. Ich habe allerdings schon Material aus dem Buch sehr gut verwendet, um Vorträge vor Laienpublikum (unter anderem sogar vor Schülern der 9. Klasse) zu halten. Die Resonanz war bis dato immer sehr gut.

O'Shea geht sehr detailliert auf die Geschichte der (algebraischen) Topologie ein. An einigen Stellen, wie ein anderer Rezensent das schon angemerkt hat, könnte man hier noch etwas raffen bzw. einen klareren roten Faden haben. Es ist an manchen Stellen nicht ganz ersichtlich, worauf O'Shea hinauswill.

Ein weiterer Kritikpunkt ist eher, dass das Buch zu kurz ist. Wegen mir hätte man noch viel mehr über Perelmans Beweis bzw. seine Beweistechniken schreiben können. Ich vermute, dass das allerdings den Rahmen deutlich sprengen würde.

Insgesamt ein gutes Buch --- ob es für ein ebenso breites Publikum geeignet ist wie andere Werke, bezweifle ich allerdings. Wer es an Eltern verschenken möchte: Einfach gemeinsam durchgehen. Hat meinen Eltern und mir viel Spaß gemacht, endlich mal zu verstehen, was der Sohn so den lieben Arbeitstag über treibt.
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Oskoreii
5.0 out of 5 stars Much needed
Reviewed in the United Kingdom on January 31, 2013
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This book delves into a topic that is much neglected in popular science: the curious story of the people, the cultures and the zeitgeists that shaped modern mathematics, and accessible information regarding the subject matter put into a scientific and historical context.
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Willy Van den Driessche
5.0 out of 5 stars Brilliant story
Reviewed in France on February 23, 2010
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This tells the story of the solution to the Poincaré conjecture by mr. Perelman. Te author does it's best to explain the context of the problem and even tries to explain what it is all about. He manages to do so with enough mathematical dept that there is something to learn and not too much to not shy away novices (like myself). It's not only history here and definitely the kind of books I love;
2 people found this helpful
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