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The Poincare Conjecture: In Search of the Shape of the Universe
Henri Poincaré was one of the greatest mathematicians of the late nineteenth and early twentieth century. He revolutionized the field of topology, which studies properties of geometric configurations that are unchanged by stretching or twisting. The Poincaré conjecture lies at the heart of modern geometry and topology, and even pertains to the possible shape of the universe. The conjecture states that there is only one shape possible for a finite universe in which every loop can be contracted to a single point. Poincaré's conjecture is one of the seven "millennium problems" that bring a one-million-dollar award for a solution. Grigory Perelman, a Russian mathematician, has offered a proof that is likely to win the Fields Medal, the mathematical equivalent of a Nobel prize, in August 2006. He also will almost certainly share a Clay Institute millennium award. In telling the vibrant story of The Poincaré Conjecture, Donal O'Shea makes accessible to general readers for the first time the meaning of the conjecture, and brings alive the field of mathematics and the achievements of generations of mathematicians whose work have led to Perelman's proof of this famous conjecture.
412 pages, Kindle Edition
First published January 1, 2007
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Displaying 1 - 30 of 88 reviews
April 22, 2015
My meeting with this book fell considerably short of love at first sight. Not saw it on sale yesterday at a Melbourne bookstore and asked if I thought it might be interesting. I picked it up, glanced at the less-than-brilliant cover and leafed through it for a minute or two; the writing seemed lackluster and the first anecdote I found was one I'd seen before. I was about to put it back when I reconsidered. It cost $10 and was evidently an easy read. I'd always wondered what the deal was with the mysterious Poincaré conjecture. Why not find out?
Well, I couldn't have more wrong: this is a truly excellent book. The bare bones of the story are easy to summarize. The Poincaré conjecture, formulated in 1900 by Henri Poincaré, states cryptically that "every simply connected, closed 3-manifold is homeomorphic to the 3-sphere". It remained an important unsolved problem for about a century, until it was proved correct by the reclusive Russian mathematician Grigori Perelman. Perelman was awarded two of the most prestigious prizes in mathematics, but turned them down.
On that description it doesn't sound very interesting, but the author makes it come alive; he's done a huge amount of background reading on both the mathematics and the history, and when he puts it in its historical context you see how fascinating it is. Well over half the book is a history of geometry, starting from its foundations in antiquity with the Babylonians, Pythagoras and Euclid. O'Shea, a cultured mathematician with an intense interest in the history of his subject, gives you plenty of material on the Greeks (did you know there's a mistake in the proof of Euclid's Proposition 1?), then traces how their work was passed through the Arabs to Renaissance Europe. En route, he finds a delightful way to explain to the non-mathematicians what a "3-sphere" is: it turns out to be the shape of the universe as described in Dante's Divine Comedy, two sets of concentric spheres mystically joined at their common surface. He illustrates with a famous picture from Doré:
As he progresses towards the present day, he finds opportunities to introduce the other terms that will eventually be used in the Conjecture, and the narrative starts to focus in on the key concepts: manifolds, connectedness, topology and, above all, non-Euclidean geometry. This is the clearest overview of the subject I've ever seen, and he has a whole bunch of stories and observations I hadn't come across before. One thing I found particularly remarkable was the long guerilla war waged by the 19th century German mathematicians against Kant's conceptions of geometry. I have had several discussions with philosophically knowledgeable people on this site about Einstein's claim to have refuted Kant. What I didn't realize was that it was just the final battle in a campaign that had gone on for a century. Gauss laid the groundwork, but thought it was so controversial that he couldn't publish: at least in Germany, it wasn't possible to openly say that Kant was wrong, and non-Euclidean geometries made perfectly good sense. But other great mathematicians - Riemann, Lobachevsky and Bolyai - found the same ideas, and they gradually came out in the open. Einstein finished it off: not only is it logically possible that the space we live in might be non-Euclidean, it actually happens to be true!
Another remarkable story from the end of this period is the intense rivalry between the German Klein (who, I learned, married Hegel's granddaughter) and the French Poincaré, a professional duel which so exhausted them that they both suffered nervous breakdowns as a result. O'Shea, who knows both French and German, includes lovely quotations from their correspondence. By the time we reach 1900 and the formulation of the Conjecture, it all makes perfect sense, and it's obvious why the problem captivated several generations of top mathematicians. I was worried that the last third would be anticlimactic, but my fears again turned out to be groundless. O'Shea hardly loses momentum at all as he goes into the finishing stretch, which involves explaining some horribly difficult mathematics; once again, he finds clever visual analogies to make the esoteric technique of Ricci flow seem reasonable and intuitive. It's obviously impossible to give us the details of Perelman's proof, but he successfully conveys both its general outline and the process which led to its acceptance by the world mathematical community.
At the end, there is the tantalizing mystery: why did Perelman turn down the huge prizes he'd won, and what was the even larger discovery he hinted at, which would make the Poincaré conjecture no more than a stepping stone? If this had been a novel, I would have groaned at the author's unsubtle attempt to set up a sequel, but oddly enough it happens to be real life. Stranger than fiction, you know.
Well, I couldn't have more wrong: this is a truly excellent book. The bare bones of the story are easy to summarize. The Poincaré conjecture, formulated in 1900 by Henri Poincaré, states cryptically that "every simply connected, closed 3-manifold is homeomorphic to the 3-sphere". It remained an important unsolved problem for about a century, until it was proved correct by the reclusive Russian mathematician Grigori Perelman. Perelman was awarded two of the most prestigious prizes in mathematics, but turned them down.
On that description it doesn't sound very interesting, but the author makes it come alive; he's done a huge amount of background reading on both the mathematics and the history, and when he puts it in its historical context you see how fascinating it is. Well over half the book is a history of geometry, starting from its foundations in antiquity with the Babylonians, Pythagoras and Euclid. O'Shea, a cultured mathematician with an intense interest in the history of his subject, gives you plenty of material on the Greeks (did you know there's a mistake in the proof of Euclid's Proposition 1?), then traces how their work was passed through the Arabs to Renaissance Europe. En route, he finds a delightful way to explain to the non-mathematicians what a "3-sphere" is: it turns out to be the shape of the universe as described in Dante's Divine Comedy, two sets of concentric spheres mystically joined at their common surface. He illustrates with a famous picture from Doré:
As he progresses towards the present day, he finds opportunities to introduce the other terms that will eventually be used in the Conjecture, and the narrative starts to focus in on the key concepts: manifolds, connectedness, topology and, above all, non-Euclidean geometry. This is the clearest overview of the subject I've ever seen, and he has a whole bunch of stories and observations I hadn't come across before. One thing I found particularly remarkable was the long guerilla war waged by the 19th century German mathematicians against Kant's conceptions of geometry. I have had several discussions with philosophically knowledgeable people on this site about Einstein's claim to have refuted Kant. What I didn't realize was that it was just the final battle in a campaign that had gone on for a century. Gauss laid the groundwork, but thought it was so controversial that he couldn't publish: at least in Germany, it wasn't possible to openly say that Kant was wrong, and non-Euclidean geometries made perfectly good sense. But other great mathematicians - Riemann, Lobachevsky and Bolyai - found the same ideas, and they gradually came out in the open. Einstein finished it off: not only is it logically possible that the space we live in might be non-Euclidean, it actually happens to be true!
Another remarkable story from the end of this period is the intense rivalry between the German Klein (who, I learned, married Hegel's granddaughter) and the French Poincaré, a professional duel which so exhausted them that they both suffered nervous breakdowns as a result. O'Shea, who knows both French and German, includes lovely quotations from their correspondence. By the time we reach 1900 and the formulation of the Conjecture, it all makes perfect sense, and it's obvious why the problem captivated several generations of top mathematicians. I was worried that the last third would be anticlimactic, but my fears again turned out to be groundless. O'Shea hardly loses momentum at all as he goes into the finishing stretch, which involves explaining some horribly difficult mathematics; once again, he finds clever visual analogies to make the esoteric technique of Ricci flow seem reasonable and intuitive. It's obviously impossible to give us the details of Perelman's proof, but he successfully conveys both its general outline and the process which led to its acceptance by the world mathematical community.
At the end, there is the tantalizing mystery: why did Perelman turn down the huge prizes he'd won, and what was the even larger discovery he hinted at, which would make the Poincaré conjecture no more than a stepping stone? If this had been a novel, I would have groaned at the author's unsubtle attempt to set up a sequel, but oddly enough it happens to be real life. Stranger than fiction, you know.
August 30, 2016
So – the shape of the universe. It’s a giant ball, right? Especially when you think of its beginning in a big bang. But that brings up the awkward question of what’s outside the ball. Space (universe) is not infinite. It’s believed to be finite, but without a boundary. It becomes easier to understand this if you consider two-dimensional beings living in a spherical (the two-dimensional surface of a ball) universe. Their universe is finite, but has no boundaries. There are no edges, and if they start off from one point and keep going in the same direction they’ll come back to where they started. Our universe is finite and without boundary in the same way. If you get on a spaceship and keep going in the same direction, eventually you’ll be back in the same neighborhood! This one is harder to imagine, isn’t it? In the case of two-dimensional people living on a sphere, we can see how it can be finite but without boundary because we can see how the sphere bends in a third dimension. But how is it for our three-dimensional universe? There’s no fourth dimension to bend in. Reading this book didn’t make it any easier for me to really understand how the universe can be finite but without a boundary. All I can do is quote the two-dimensional analogy, but I’m still a three-dimensional earthling.
But even assuming that the universe is finite and without boundary – is it a three-sphere? To go back to the two-sphere analogy, just because Magellan sailed in the same direction and came back to where he started doesn’t mean that the earth is a sphere. It can also be doughnut shaped, and the same would still happen. No one really knows what the shape of the universe is. There’s a lot of evidence for it being flat (whatever that means).
And the Poincare Conjecture: It says that a finite, no-boundary space that is “simply connected” is a three-sphere. This question is obviously of great interest both to mathematicians and to the physicists studying the geometry of the universe. (We still don’t know if the universe is “simply connected” or not. A ball is simply connect, but something like a doughnut is not simply connected.) Unlike Reimann’s Hypothesis, the Poincare Conjecture was finally proved after much heartbreak and agony – by an eccentric Russian mathematician named Gregori Perelman who didn’t even accept the award for it. The book tells the story of the conjecture and the man who proved it. Good pop-science and math history.
But even assuming that the universe is finite and without boundary – is it a three-sphere? To go back to the two-sphere analogy, just because Magellan sailed in the same direction and came back to where he started doesn’t mean that the earth is a sphere. It can also be doughnut shaped, and the same would still happen. No one really knows what the shape of the universe is. There’s a lot of evidence for it being flat (whatever that means).
And the Poincare Conjecture: It says that a finite, no-boundary space that is “simply connected” is a three-sphere. This question is obviously of great interest both to mathematicians and to the physicists studying the geometry of the universe. (We still don’t know if the universe is “simply connected” or not. A ball is simply connect, but something like a doughnut is not simply connected.) Unlike Reimann’s Hypothesis, the Poincare Conjecture was finally proved after much heartbreak and agony – by an eccentric Russian mathematician named Gregori Perelman who didn’t even accept the award for it. The book tells the story of the conjecture and the man who proved it. Good pop-science and math history.
December 21, 2011
There was some explanation earlier in the book, but later explanation was poor. I came away with little understanding of how the Poincare conjecture was solved. The book was a disappointment, but did provide a reference to book by Jeffrey Weeks that might offer better layman-level explanations of topological concepts.
September 14, 2015
Why is this book not more widely read? It's at least as good as books like Fermat's Last Theorem, with far more mathematical content. If any layman wants a glimpse into the world of top-level mathematics, I cannot recommend a better book.
May 8, 2025
There are very few popular science books with the depth to be educational for people with post-secondary education in STEM. This is one of them. Coming from an undergrad degree in engineering and (at the end of) a Ph.D. in physics, this book explained the prerequisite topology and geometry to understand the Poincaré Conjecture with a level of rigour and depth I've never seen outside a classroom or textbook. In building the mathematical foundation to understand the Poincaré Conjecture, Prof. O'Shea takes the reader through the history of geometry and topology, tracing a line from Euclid and the Pythagorians to the cutting edge of research in the early 2000s. This book is a triumph of popular science and the history of science. Now if only I could find another book like this...
July 16, 2008
I've been interested in the Millennium problems since I first read about them several years ago. It was exciting to read about the first one to be solved. I never took topology in college, though, so I have to admit that much of this went right over my head. If you wanted to know without reading all the math, yes, the Poincare conjecture turned out to be true. Pretty cool stuff!
August 15, 2007
As a recent grad student in mathematics I found this book incredibly interesting. It made me want to go on and get my Ph.D. in manifold theory.
February 5, 2014
This was a decent book, but a bit of a hard read.
Firstly, the book introduces many concepts by name, with some short descriptions, and then goes on to discuss them in some qualitative detail; how one concept leads to another; how concepts fail to connect. For me, at least, this was difficult to follow. Granted, in order to truly understand what is being discussed, you would need to understand the mathematics; perhaps this is just an insurmountable problem in trying to translate high-level and difficult mathematics into lay-language.
Secondly, there are too many sections where names and dates and attempted proofs of such-and-such a conjecture/theory/etc. are listed; in these sections it very much feels like the only people who would be able to pull much meaning would be already quite familiar with the topics. There is much more of this in the last third or quarter of the book.
The middle 85% of the book isn't about the Poincare Conjecture per se. In this, I would describe the book as the history of mathematicians and mathematics, from ancient times to today, as told from the point of view of the Poincare Conjecture. An analogy might be something like a book that details the life of some famous figure by telling the history of their family/ancestry and the times and events their family lived through.
Firstly, the book introduces many concepts by name, with some short descriptions, and then goes on to discuss them in some qualitative detail; how one concept leads to another; how concepts fail to connect. For me, at least, this was difficult to follow. Granted, in order to truly understand what is being discussed, you would need to understand the mathematics; perhaps this is just an insurmountable problem in trying to translate high-level and difficult mathematics into lay-language.
Secondly, there are too many sections where names and dates and attempted proofs of such-and-such a conjecture/theory/etc. are listed; in these sections it very much feels like the only people who would be able to pull much meaning would be already quite familiar with the topics. There is much more of this in the last third or quarter of the book.
The middle 85% of the book isn't about the Poincare Conjecture per se. In this, I would describe the book as the history of mathematicians and mathematics, from ancient times to today, as told from the point of view of the Poincare Conjecture. An analogy might be something like a book that details the life of some famous figure by telling the history of their family/ancestry and the times and events their family lived through.
February 15, 2010
This book was about as painful as reading the book of Genesis: its pages mostly comprise a chronological list of mathematicians ("and so-and-so's work begot so-and-so's thesis"...) interspersed with definitions sans explanation or example (a group, a ring, etc.). The highlights were the only occasional example of geometry in mathematical physics or when the author found time to elaborate a little more on an interesting property of a certain metric or surface structure.
In fact, the best part of the book is the final two sentences that state (for about the 5th time but with the most clarity) the thesis of the book as defined by its title. Good grief.
In fact, the best part of the book is the final two sentences that state (for about the 5th time but with the most clarity) the thesis of the book as defined by its title. Good grief.
June 5, 2016
The conjecture from which the title comes doesn't make an appearance until 136 pages into this 200 page book. Poincare himself is only present for about 1/10th of the book. It's more of a very brief history of geometry and topology than a treatment of the problem.
Also, the resolutions of the images in the book are so poor it's as if the publisher printed out jpegs and made Xeroxes of them.
Also, the resolutions of the images in the book are so poor it's as if the publisher printed out jpegs and made Xeroxes of them.
August 5, 2008
p. 47: "absolute precision buys the freedom to dream meaningfully."
June 21, 2020
The fact is I would need infinitive (sets of) lifes to read all the books I want and (another set of) infinitive lifes to put into practice everything I read in all the books I would achieve to read in those other (infinite sets of) lifes (certainly, an infinite number of books). And yet, I would need an infinite memory to recall all the things I learn from them and correct, maybe, all the infinite (sets of) mistakes I would make during my infinite learning. If infinite books available, I might not be able to start anew with the first book, but having enough (infinite) time, who knows? Life would be infinite, even if memory would not.
I finished this book with a feeling of satisfaction, with the great pleasure of having touched, albeit with the points of my fingers, the fascinating world of topology and geometry and while I want to learn more I get the feeling I will not have enough time in this life to grasp this incredible world it's been opened to me to understand all the nuancies, not even the most simple ones. It is a very sad moment to realise this life is simply much too short to discover all the beauty hidden behind the walls of ignorance.
This is a fascinating book casting the search for a solution for an unsurmountable (until Perelman arrived, of course) and difficult (extremely difficult) problem, for performing the task of solving an open question (well, rather a conjecture) posed by Poincaré (one of the greatest mathematicians in history) in the last page of his last work on topology ('analysis situs'): the Poincaré's conjecture.
I knew little about topology and geometry before reading the book and after reading the book I want more (as a physicist I have the right to say I was ignorant before reading the book, but I remain ignorant as well after the reading and this is quite disatisfying).
So many brilliant minds failed and then, out of the blue (well, a blue which is not at all such looking at the brilliant background of the 'solver' and his career), one clear mind (Perelman, of course) came from the cold (Russia) with modesty and right attitude, a bold mind who, after solving such Conjecture, went back to his cave from where he came from to never show up, after saying something similar to 'hey world, look what I leave for the future of maths. Just some notes for you to read. By the way, I got solved the Poincaré's conjecture, but please leave me alone. I was just playing Sudoku. Thank you very much'.
I loved and read with great pleasure the way the author presented the very difficult concepts and math topics to later give a sucint explanation of what was solved by Perelman actually. I loved to read the historical overview surrounding the lifes and circumstances, the diffculties and disappointments the many great mathematicians suffere, the context and the background, human and mathematical, until a Russian mathematician came to fill in the void. If this is not a fascinating story then you really don't have a sense for beauty and the misterious ways you may need to arrive at it.
At times, reading was difficult. Mathematical concepts are not easy to explain for the layman, but the author achieves, when necessary, almost always, to use the correct explanation, find the correct example or comparison, to use the right words a clear mind would need to use and an average mind would need to understand. Enlightning and absolutely recommendable.
I finished this book with a feeling of satisfaction, with the great pleasure of having touched, albeit with the points of my fingers, the fascinating world of topology and geometry and while I want to learn more I get the feeling I will not have enough time in this life to grasp this incredible world it's been opened to me to understand all the nuancies, not even the most simple ones. It is a very sad moment to realise this life is simply much too short to discover all the beauty hidden behind the walls of ignorance.
This is a fascinating book casting the search for a solution for an unsurmountable (until Perelman arrived, of course) and difficult (extremely difficult) problem, for performing the task of solving an open question (well, rather a conjecture) posed by Poincaré (one of the greatest mathematicians in history) in the last page of his last work on topology ('analysis situs'): the Poincaré's conjecture.
I knew little about topology and geometry before reading the book and after reading the book I want more (as a physicist I have the right to say I was ignorant before reading the book, but I remain ignorant as well after the reading and this is quite disatisfying).
So many brilliant minds failed and then, out of the blue (well, a blue which is not at all such looking at the brilliant background of the 'solver' and his career), one clear mind (Perelman, of course) came from the cold (Russia) with modesty and right attitude, a bold mind who, after solving such Conjecture, went back to his cave from where he came from to never show up, after saying something similar to 'hey world, look what I leave for the future of maths. Just some notes for you to read. By the way, I got solved the Poincaré's conjecture, but please leave me alone. I was just playing Sudoku. Thank you very much'.
I loved and read with great pleasure the way the author presented the very difficult concepts and math topics to later give a sucint explanation of what was solved by Perelman actually. I loved to read the historical overview surrounding the lifes and circumstances, the diffculties and disappointments the many great mathematicians suffere, the context and the background, human and mathematical, until a Russian mathematician came to fill in the void. If this is not a fascinating story then you really don't have a sense for beauty and the misterious ways you may need to arrive at it.
At times, reading was difficult. Mathematical concepts are not easy to explain for the layman, but the author achieves, when necessary, almost always, to use the correct explanation, find the correct example or comparison, to use the right words a clear mind would need to use and an average mind would need to understand. Enlightning and absolutely recommendable.
Want to read
December 3, 2022Dewey 514.2
May 7, 2013
Henri Poincaré enuncio' una congettura: "E' possibile che il gruppo fondamentale di una varieta' sia l'identita', ma che la varieta' in questione non sia omeomorfa alla sfera tridimensionale?"
A chi non ha studiato matematica, tante delle parole presenti nella congettura di Poincaré - siamo a cavallo tra il 1800 e il 1900, non dicono niente. Ma sfido chiunque a non rintracciare in esse un qualche fascino.
Questa congettura e' legata ad una domanda che, se possibile, e' ancora piu' affascinante: "qual e' la forma dell'universo?". Alla fine del libro si dice che la maggior parte degli studi attualmente esistenti dimostrerebbe che l'universo e' piatto, ovvero la sua curvatura e' praticamente nulla; di sicuro non e' mai negativa.
Il libro di O'Shea e' la storia del pensiero matematico da Euclide fino al russo Perelman che, nel 2002, ha dimostrato la congettura del matematico francese. Una storia che scalda per il sapore romantico di qualche passo; che appassiona per le sfide che racconta.
Una cavalcata impressionante che coinvolge per i toni con cui l'autore descrive le scoperte susseguitesi dall'enunciato, attraverso i passi fatti nella topologia e nella geometria negli anni, fino alla recentissima dimostrazione.
E' una storia bella. Che parla di come i matematici si siano lasciati nei secoli il testimone: anche i piu' eruditi matematici babilonesi non sarebbero stati in grado di risolvere problemi che oggi vengono affrontati con estrema facilita' da chi ha imparato la matematica nelle scuole superiori. Noi abbiamo adesso strumenti che prima erano impensabili; e non si parla di computer o calcolatrici, ma di progressi fatti anche grazie ai primi passi mossi proprio da quei babilonesi. Noi lasceremo alle generazioni future i mezzi per spingersi dove noi, ora, nemmeno possiamo immaginare.
La matematica non e' fredda. I numeri sono davvero come una poesia perche', proprio come le poesie, sono scoperti e scritti dalla passione [vedi Bruno D'Amore in Matematica, stupore e poesia]
A chi non ha studiato matematica, tante delle parole presenti nella congettura di Poincaré - siamo a cavallo tra il 1800 e il 1900, non dicono niente. Ma sfido chiunque a non rintracciare in esse un qualche fascino.
Questa congettura e' legata ad una domanda che, se possibile, e' ancora piu' affascinante: "qual e' la forma dell'universo?". Alla fine del libro si dice che la maggior parte degli studi attualmente esistenti dimostrerebbe che l'universo e' piatto, ovvero la sua curvatura e' praticamente nulla; di sicuro non e' mai negativa.
Il libro di O'Shea e' la storia del pensiero matematico da Euclide fino al russo Perelman che, nel 2002, ha dimostrato la congettura del matematico francese. Una storia che scalda per il sapore romantico di qualche passo; che appassiona per le sfide che racconta.
Una cavalcata impressionante che coinvolge per i toni con cui l'autore descrive le scoperte susseguitesi dall'enunciato, attraverso i passi fatti nella topologia e nella geometria negli anni, fino alla recentissima dimostrazione.
E' una storia bella. Che parla di come i matematici si siano lasciati nei secoli il testimone: anche i piu' eruditi matematici babilonesi non sarebbero stati in grado di risolvere problemi che oggi vengono affrontati con estrema facilita' da chi ha imparato la matematica nelle scuole superiori. Noi abbiamo adesso strumenti che prima erano impensabili; e non si parla di computer o calcolatrici, ma di progressi fatti anche grazie ai primi passi mossi proprio da quei babilonesi. Noi lasceremo alle generazioni future i mezzi per spingersi dove noi, ora, nemmeno possiamo immaginare.
La matematica non e' fredda. I numeri sono davvero come una poesia perche', proprio come le poesie, sono scoperti e scritti dalla passione [vedi Bruno D'Amore in Matematica, stupore e poesia]
May 9, 2018
The goal of this book, as stated, was to outline the buildup to and eventual solution of the Poincare conjecture in a way that non-mathematicians would be able to follow. As someone who studied some graduate math, I feel comfortable saying that O'Shea partially achieved his goal. Definitely my passing familiarity with some of the topics mentioned made my reading of the book easier. Since even with that understanding I simply passed by some of the content, there are parts that a non-mathematician will have to skim.
However, there are a lot of good details on the personalities and biographies of the various mathematicians in the story of this conjecture, and great inclusion the interplay of the relevant historical events with the growth and decline of various academic institutions. Tracking one major concept in mathematics to write about the history of mathematics generally is overall a good formula, pun intended.
Best of all was the introduction of the timeline in the reference at the end, because it clearly laid out the author's plan for the book ("The narrative of develops the following three interlocking themes..."). It's as if he included a notecard on which he'd planned his book. Although this might have been better expanded into a full-fledged introduction chapter.
One major complaint I have is the use of endnotes rather than footnotes because of the way they are used in this book. First, about half of the endnotes actually contain comments that might be interesting or enlightening in reading, which is when I feel they should be included as footnotes, so they might be read without needed to flip back and forth. Second, as the other half are just the citations, flipping to the back hoping for more information and only finding a citation is very distracting, and could be avoided with footnotes (quick glance - nothing to note - return back up the page).
However, there are a lot of good details on the personalities and biographies of the various mathematicians in the story of this conjecture, and great inclusion the interplay of the relevant historical events with the growth and decline of various academic institutions. Tracking one major concept in mathematics to write about the history of mathematics generally is overall a good formula, pun intended.
Best of all was the introduction of the timeline in the reference at the end, because it clearly laid out the author's plan for the book ("The narrative of develops the following three interlocking themes..."). It's as if he included a notecard on which he'd planned his book. Although this might have been better expanded into a full-fledged introduction chapter.
One major complaint I have is the use of endnotes rather than footnotes because of the way they are used in this book. First, about half of the endnotes actually contain comments that might be interesting or enlightening in reading, which is when I feel they should be included as footnotes, so they might be read without needed to flip back and forth. Second, as the other half are just the citations, flipping to the back hoping for more information and only finding a citation is very distracting, and could be avoided with footnotes (quick glance - nothing to note - return back up the page).
November 15, 2010
Quando Grigori Perelman rifiutò il milione di dollari che il Clay Institute gli aveva assegnato per la dimostrazione della Congettura di Poincaré, la notizia raggiunse le prime pagine di tutti i giornali. Non che la gente sapesse che diavolo fosse questa congettura, a dire il vero; ma l'idea di tutti quei soldi li stuzzicava. Fortunatamente ci sono stati alcuni matematici che hanno pensato non tanto di raccontare la dimostrazione quanto di riuscire a dare uno sguardo generale sui temi trattati, per dare almeno un'idea di quello di cui si stava parlando. Donal O'Shea ci è riuscito benissimo con questo suo libro: dopo l'incipit molto americano ero un po' prevenuto, ma lo stile del resto dell'opera è molto chiaro, e conduce man mano il lettore a capire il contesto in cui il problema nacque e fiorì, comprese le implicazioni con la relatività generale; il tutto con un ampio apparato di note utili per chi volesse saperne di più. In fin dei conti la congettura di Poincaré parla anche del nostro universo: afferma infatti che se il nostro universo non è infinito e si comporta come pensiamo faccia allora è in un certo senso l'equivalente quadridimensionale di una sfera. Servirà a qualcosa? Probabilmente no, ma la matematica non si preoccupa certo della cosa. La traduzione è scorrevole, ma in qualche punto (non matematico, a dire il vero) mi ha dato l'idea di essere stata tirata un po' via, come nelle "poesie in cinque versi" che probabilmente sono limerick. Troppa semplicità fa male...
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February 5, 2025Notice that the subtitle is "in search of the shape of the universe," not "the shape of the universe". The book is about what mathematicians have been able to determine about the shape and that is not very specific: it's all connected, it's finite, it has no boundary. So even though it's finite and you could go anywhere in it, you would never get to the end of it. A counter-intuitive idea, but the book presents some comprehensible possibilities of how that might work. I very much enjoyed it for that, but mathematics is hard to put into words and sometimes you run into things like the following.
"He reinterpreted the numbers that Betti had considered, nowadays called Betti numbers, by introducing equations between submanifolds of a manifold, called homologies on a manifold that expressed the relation of bounding within the manifold."
If that isn't clear, don't blame yourself. Ludwig Wittgenstein said the following and if you substitute the word "universe" for "thought" I think you get a good description of the human perspective on this problem.
In order to be able to set a limit to thought, we should have to find both sides of the limit thinkable (i.e. we should have to be able to think what cannot be thought).
"He reinterpreted the numbers that Betti had considered, nowadays called Betti numbers, by introducing equations between submanifolds of a manifold, called homologies on a manifold that expressed the relation of bounding within the manifold."
If that isn't clear, don't blame yourself. Ludwig Wittgenstein said the following and if you substitute the word "universe" for "thought" I think you get a good description of the human perspective on this problem.
In order to be able to set a limit to thought, we should have to find both sides of the limit thinkable (i.e. we should have to be able to think what cannot be thought).
February 28, 2015
The first quarter of the book gives you a lot of cool info on the shape of the universe, Grigori Perelman, and Poincare. But that's it! Because after that, it begins throwing some really unnecessary paragraphs (and chapters) on the biography of the people.
That's what I hate about this book: unnecessary information. Some sections are completely unneeded and even most of the times, it's somewhere you don't expect it to be, right in the middle of a serious work!
Perhaps it's because I'm a math major myself and I was looking for a couple of formulas and equations to explain it all. Anyways, I've picked up a topology class this term and I'll figure it out more intellectually and more exact.
That's what I hate about this book: unnecessary information. Some sections are completely unneeded and even most of the times, it's somewhere you don't expect it to be, right in the middle of a serious work!
Perhaps it's because I'm a math major myself and I was looking for a couple of formulas and equations to explain it all. Anyways, I've picked up a topology class this term and I'll figure it out more intellectually and more exact.
April 21, 2012
Saggio interessante sulla risoluzione di uno dei 7 millennium problems (si vince un milione di dollari per la soluzione di ognuno). Lo scienziato che ha risolto la congettura, il russo Grigori Perelman, ha rifiutato sia il premio in denaro, sia la medaglia Fields, per la quale ogni matematico sulla faccia della terra penso sia disposto a uccidere.
Il libro è abbastanza divulgativo, ma ci vogliono nozioni di topologia per comprendere appieno di che cosa si stia parlando.
Il libro è abbastanza divulgativo, ma ci vogliono nozioni di topologia per comprendere appieno di che cosa si stia parlando.
April 4, 2013
I enjoy books about mathematics. Not a daunting read, easily understood and very clear explainations.
Takes some imagination and thinking to get ones mind around the concepts discussed but all in all an awesome book. One of my favorite when it comes to popular science.
Its kind of like a "history of topology", "story of a frustrating problem and the journey to its solution" and discussion between you and the author about what topology really is about all wrapped into one book.
Takes some imagination and thinking to get ones mind around the concepts discussed but all in all an awesome book. One of my favorite when it comes to popular science.
Its kind of like a "history of topology", "story of a frustrating problem and the journey to its solution" and discussion between you and the author about what topology really is about all wrapped into one book.
May 19, 2022
Careful (and not too formula-heavy) explication of some of the concepts in mathematical topology. Unfortunately, slightly past the halfway mark, the book devolves into an historical account of attempts at proving the Poincaré conjecture, with very little in the way of explanation, so by that point I was completely confused. Still a worthwhile book, and those whose minds are more adept at abstraction that mine is, would doubtless enjoy it.
Three and a half stars.
Three and a half stars.
September 20, 2008
I originally purchased this book to learn more about Gregory Perelman and the fields medal he turned down, but over the course of the book you get such a detailed explanation of the history of math, that I spent just as much time in wikipedia as I did reading this book. Fantastic read, for every type of math fan out there, of every level of proficiency.
May 4, 2009
Fun book. Made me want to read an introductory book on topology...
October 18, 2010
The universe is a infinite topologically bounded three dimensional manifold. Did I say that right?
April 27, 2015
Puts the wanker into Poincaré conjecture. Too many cute biographical details, not enough Ricci flow.
February 11, 2018
Amazingly traces the development of topology over the years and it's culmination in the Poincare Conjecture. A must-read for any math enthusiast.
August 19, 2018
sometimes the concepts were explained in depth, sometimes no attempt was made. but very fascinating
October 29, 2018
A very good introduction to the problem: there is no way to know the shape of the earth without a satellite. And the same for the universe.
June 26, 2019
Too technical for a layman audience.
April 6, 2020
Book is good but I believe later chapters require careful reading since notions of Riemann 's geometry are not the easiest to understand.
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