Good historical review, 17 Dec 2008
Unlike John Derbyshire's books on the history of mathematics, this book does not contain much mathematics apart from concepts in topology, but it does contain a good history of the developments leading up to the conjecture in 1904, and the attempts since to provide a proof. The book is intended for a non-specialist audience, but it helps if you have a mathematical background, at least to A level, and are interested in the development of ideas in their historical context. There is not much material on the conjecture itself since to include it would take it out of the scope of a popular book, but one gets the flavour of the idea and its usefulness beyond merely an abtruse mathematical concept. The book is well-written and entertaining and recommended if you don't want a mathematical text-book approach to the subject.

What about mathematics ?, 16 Dec 2008
Only few people have an idea on what mathematicians active at the frontiers of research try to achieve.

If you want to get a glimpse: read this book. No mathematical background is needed (if you nevertheless do have some background then the notes contain interesting material). The book describes the development of mathematical theories (in words, not in formulas) starting with Euclid's fifth postulate, via non-euclidian geometries to the Poincaré Conjecture and its solution by Grigory Perelman in 2003. It does so by introducing the people involved, describing their characters and their ways of living and presenting their contribution to the theory being developed.

The book also gives a clear account of the interaction of mathematics with other sciences.

The book is well written.

I recommend the book to every-one who wants to know what mathematicians do and why that is of importance. By the way the book is also interesting for people with a mathematical background who want to know what the Poincaré Conjecture is about.




A bit too historic, 09 Aug 2008
I have read a book called `The Poincare Conjecture'. I understood the book, I even enjoyed the book; yet I am none the wiser on the actual Conjecture from where I set out.

I am not sure why D. O'Shea avoided the hard bits. There is no risk of an unsuspecting member of the public picking this up in an airport bookstore before boarding a long haul expecting a Tom Clancy novel.

This book is too focused on historical topics behind the Conjecture and associated topics in topology that make light reading. If you are seeking to learn more about the specifics of the 'Poincare Conjecture' this is not good enough.

A great deal of this book, say 50%, is centered on the evolution of Euclidian to non-Euclidian Geometry. Only about 2/3's of the way in are we actually dealing with the conjecture.
A massive amount of this book focuses on the golden period of mathematics at Göttingen. A fairer title for this book would have been 'Topics in the history of Geometry'.

In fairness, the mathematics behind Perelman's solution are pretty much inaccessible and even the conjecture itself is difficult to understand properly.

This book will not satisfy anyone who is seriously interested in the conjecture nor will it deepen anyone's understanding who wishes to understand it more.

Not quite what I expected..., 22 Aug 2007
There are some magnificent books about mathematics, and in particular on the history of some mathematical breakthrough, like that of Simon Singh on fermat last theorem, which I read more than once. (Simon Singh has a physics degree, if I recall).
This book of Donald O'Shea is not very well written. First of all, I think the book is not well structured. He doesn't conduct the story simply from a to b, he retains himself in too many subjects a bit off topic, not relevant, or doesn't seem quite pertinent to the main subject, which is the poincaré conjecture, (although some are interesting); what's the relevance of the second world war, or the history of united states mathematics and it's universities. He turns back and forth some times, like forgetting something behind. The prose is unpleasant, except maybe in the lasts chapters. The author spent several chapters in the beginning, talking about the shape of the earth, coulumbus travels, history of maps, defining manifolds of dimension 2, pitagoras and euclid elements, euclidian geometry, the fifth postulate, and suddenly jumps over almost every pertinent concept to understand the poincaré conjecture and the solution by perelman. That is, if he starts the book writing to a public with no knowledge on mathematics, he ends it as writing to a professional mathematician. Everyone that buys a book of this sort, obviously knows what a surface is (or even what is a manifold, or have some knowledge on calculus) don't see the point in explaining that. On the other hand, in the end of the book he says something like: "the complements of two knots could be homeomorphic without the knots being isotopic to each other or their mirror image" with no explanation whatsoever. Let me detail a bit more: for example, in page 131 alone O'Shea introduces several fundamental concepts in topology, see how he does:

about "betti numbers", and "homologies":
-Betti associated numbers with manifolds and poincaré reinterpreted this numbers by introducing equations between submanifolds of a manifold called homologies on a manifold that expressed the relation of bounding within the manifold;

about the "fundamental group":
-Poincaré associated a completely new algebraic object with each manifold which e called the fundamental group.
Sure, as I know from the beginning, that all this terms are associated with topology somehow!

In spite of being a mathematician, Donald O'Shea doesn't seem to think like one, he presents concepts, and tries to define them, in a confusing way. There are some mistakes, but not serious: "..a spherical piece of cloth that would fit perfectly on the top of your head. (...) The cloth would have to have less area inside a circle of fixed radius than there would be on a bedsheet."(page 96) Defines at least 2 times wrongly the number pi, as: "the ratio of the diameter of a circle to its radius"(page 208). These are 2 examples. Distractions of course, but nevertheless, doesn't look nice for a mathematician.
If you want to know the recent story about the poincaré conjecture and some facts about perelman's solution, you just need to read the last 3 chapters. And of course, you won't get any clear idea how perelman did it!
Many facts revealed, for example, in the article "Manifold Destiny" published in The New Yorker, important as they are to understand the circumstances of the solution, and all the complications that emerged around it, are simply ignored!!
The book has one good thing though, has lots of references, articles, books and websites.
For a mathematician who took a whole sabbatical to investigate and write this 200 page story, Donald O'Shea, in my view, did quite a miserable job.

Good historical review, 17 Dec 2008
Unlike John Derbyshire's books on the history of mathematics, this book does not contain much mathematics apart from concepts in topology, but it does contain a good history of the developments leading up to the conjecture in 1904, and the attempts since to provide a proof. The book is intended for a non-specialist audience, but it helps if you have a mathematical background, at least to A level, and are interested in the development of ideas in their historical context. There is not much material on the conjecture itself since to include it would take it out of the scope of a popular book, but one gets the flavour of the idea and its usefulness beyond merely an abtruse mathematical concept. The book is well-written and entertaining and recommended if you don't want a mathematical text-book approach to the subject.

What about mathematics ?, 16 Dec 2008
Only few people have an idea on what mathematicians active at the frontiers of research try to achieve.

If you want to get a glimpse: read this book. No mathematical background is needed (if you nevertheless do have some background then the notes contain interesting material). The book describes the development of mathematical theories (in words, not in formulas) starting with Euclid's fifth postulate, via non-euclidian geometries to the Poincaré Conjecture and its solution by Grigory Perelman in 2003. It does so by introducing the people involved, describing their characters and their ways of living and presenting their contribution to the theory being developed.

The book also gives a clear account of the interaction of mathematics with other sciences.

The book is well written.

I recommend the book to every-one who wants to know what mathematicians do and why that is of importance. By the way the book is also interesting for people with a mathematical background who want to know what the Poincaré Conjecture is about.




A bit too historic, 09 Aug 2008
I have read a book called `The Poincare Conjecture'. I understood the book, I even enjoyed the book; yet I am none the wiser on the actual Conjecture from where I set out.

I am not sure why D. O'Shea avoided the hard bits. There is no risk of an unsuspecting member of the public picking this up in an airport bookstore before boarding a long haul expecting a Tom Clancy novel.

This book is too focused on historical topics behind the Conjecture and associated topics in topology that make light reading. If you are seeking to learn more about the specifics of the 'Poincare Conjecture' this is not good enough.

A great deal of this book, say 50%, is centered on the evolution of Euclidian to non-Euclidian Geometry. Only about 2/3's of the way in are we actually dealing with the conjecture.
A massive amount of this book focuses on the golden period of mathematics at Göttingen. A fairer title for this book would have been 'Topics in the history of Geometry'.

In fairness, the mathematics behind Perelman's solution are pretty much inaccessible and even the conjecture itself is difficult to understand properly.

This book will not satisfy anyone who is seriously interested in the conjecture nor will it deepen anyone's understanding who wishes to understand it more.

Not quite what I expected..., 22 Aug 2007
There are some magnificent books about mathematics, and in particular on the history of some mathematical breakthrough, like that of Simon Singh on fermat last theorem, which I read more than once. (Simon Singh has a physics degree, if I recall).
This book of Donald O'Shea is not very well written. First of all, I think the book is not well structured. He doesn't conduct the story simply from a to b, he retains himself in too many subjects a bit off topic, not relevant, or doesn't seem quite pertinent to the main subject, which is the poincaré conjecture, (although some are interesting); what's the relevance of the second world war, or the history of united states mathematics and it's universities. He turns back and forth some times, like forgetting something behind. The prose is unpleasant, except maybe in the lasts chapters. The author spent several chapters in the beginning, talking about the shape of the earth, coulumbus travels, history of maps, defining manifolds of dimension 2, pitagoras and euclid elements, euclidian geometry, the fifth postulate, and suddenly jumps over almost every pertinent concept to understand the poincaré conjecture and the solution by perelman. That is, if he starts the book writing to a public with no knowledge on mathematics, he ends it as writing to a professional mathematician. Everyone that buys a book of this sort, obviously knows what a surface is (or even what is a manifold, or have some knowledge on calculus) don't see the point in explaining that. On the other hand, in the end of the book he says something like: "the complements of two knots could be homeomorphic without the knots being isotopic to each other or their mirror image" with no explanation whatsoever. Let me detail a bit more: for example, in page 131 alone O'Shea introduces several fundamental concepts in topology, see how he does:

about "betti numbers", and "homologies":
-Betti associated numbers with manifolds and poincaré reinterpreted this numbers by introducing equations between submanifolds of a manifold called homologies on a manifold that expressed the relation of bounding within the manifold;

about the "fundamental group":
-Poincaré associated a completely new algebraic object with each manifold which e called the fundamental group.
Sure, as I know from the beginning, that all this terms are associated with topology somehow!

In spite of being a mathematician, Donald O'Shea doesn't seem to think like one, he presents concepts, and tries to define them, in a confusing way. There are some mistakes, but not serious: "..a spherical piece of cloth that would fit perfectly on the top of your head. (...) The cloth would have to have less area inside a circle of fixed radius than there would be on a bedsheet."(page 96) Defines at least 2 times wrongly the number pi, as: "the ratio of the diameter of a circle to its radius"(page 208). These are 2 examples. Distractions of course, but nevertheless, doesn't look nice for a mathematician.
If you want to know the recent story about the poincaré conjecture and some facts about perelman's solution, you just need to read the last 3 chapters. And of course, you won't get any clear idea how perelman did it!
Many facts revealed, for example, in the article "Manifold Destiny" published in The New Yorker, important as they are to understand the circumstances of the solution, and all the complications that emerged around it, are simply ignored!!
The book has one good thing though, has lots of references, articles, books and websites.
For a mathematician who took a whole sabbatical to investigate and write this 200 page story, Donald O'Shea, in my view, did quite a miserable job.

Well done.., 22 Jul 1999
Bold has a gem of a book here. It's only a little bit over a hundred pages, but it's packed full of the great geometry problems that occupied the minds of the world's greatest thinkers for the past 2000 years.

The title describes the book perfectly. These really are "Famous Problems from Geometry" and he does indeed explain how to solve them.

The book has four major sections/chapters. He discusses in detail the three problems from antiquity (one section each): squaring a circle, doubling a cube, and trisecting an angle. Furthermore, he spends significant time with constructions of regular polygons (the fourth section) - which ones can be constructed and why. He also discusses which ones cannot be constructed and why.

The reader will be expected to understand concepts from Modern Algebra, particularly the concept of a Field. While Bold does spend time explaining what a Field is, his definition is quick and is assumed to be more of a refresher for someone who has already learned about them. Bold also has a section on Complex Numbers where he derives one of the formulas used later in the book. Again - this section is assumed to be a refresher on Complex Numbers. High School Geometry or Algebra students would have significant trouble understanding his explanations and proofs.

Bold provides problems for the reader to work along the way. These are problems that logically lead to the proof of the problem being studied. The problems are good. As a third year college student majoring in mathematics, I found the explanations/solutions to be sometimes hard to follow. He assumes a great deal about the reader's level of proficiency in math and in geometry. As a result, he liberally skips steps in proofs that are assumed to be "obvious."

If you're expecting simple proofs to these problems, you're not going to find them. If they were simple, they wouldn't have taken 2000 years to solve. But they are explained clearly here in terms that anyone with a college degree should be able to understand.

Overall, a superb book. A must have for anyone interested in the famous problems from the history of Geometry.

Good historical review, 17 Dec 2008
Unlike John Derbyshire's books on the history of mathematics, this book does not contain much mathematics apart from concepts in topology, but it does contain a good history of the developments leading up to the conjecture in 1904, and the attempts since to provide a proof. The book is intended for a non-specialist audience, but it helps if you have a mathematical background, at least to A level, and are interested in the development of ideas in their historical context. There is not much material on the conjecture itself since to include it would take it out of the scope of a popular book, but one gets the flavour of the idea and its usefulness beyond merely an abtruse mathematical concept. The book is well-written and entertaining and recommended if you don't want a mathematical text-book approach to the subject.

What about mathematics ?, 16 Dec 2008
Only few people have an idea on what mathematicians active at the frontiers of research try to achieve.

If you want to get a glimpse: read this book. No mathematical background is needed (if you nevertheless do have some background then the notes contain interesting material). The book describes the development of mathematical theories (in words, not in formulas) starting with Euclid's fifth postulate, via non-euclidian geometries to the Poincaré Conjecture and its solution by Grigory Perelman in 2003. It does so by introducing the people involved, describing their characters and their ways of living and presenting their contribution to the theory being developed.

The book also gives a clear account of the interaction of mathematics with other sciences.

The book is well written.

I recommend the book to every-one who wants to know what mathematicians do and why that is of importance. By the way the book is also interesting for people with a mathematical background who want to know what the Poincaré Conjecture is about.




A bit too historic, 09 Aug 2008
I have read a book called `The Poincare Conjecture'. I understood the book, I even enjoyed the book; yet I am none the wiser on the actual Conjecture from where I set out.

I am not sure why D. O'Shea avoided the hard bits. There is no risk of an unsuspecting member of the public picking this up in an airport bookstore before boarding a long haul expecting a Tom Clancy novel.

This book is too focused on historical topics behind the Conjecture and associated topics in topology that make light reading. If you are seeking to learn more about the specifics of the 'Poincare Conjecture' this is not good enough.

A great deal of this book, say 50%, is centered on the evolution of Euclidian to non-Euclidian Geometry. Only about 2/3's of the way in are we actually dealing with the conjecture.
A massive amount of this book focuses on the golden period of mathematics at Göttingen. A fairer title for this book would have been 'Topics in the history of Geometry'.

In fairness, the mathematics behind Perelman's solution are pretty much inaccessible and even the conjecture itself is difficult to understand properly.

This book will not satisfy anyone who is seriously interested in the conjecture nor will it deepen anyone's understanding who wishes to understand it more.

Not quite what I expected..., 22 Aug 2007
There are some magnificent books about mathematics, and in particular on the history of some mathematical breakthrough, like that of Simon Singh on fermat last theorem, which I read more than once. (Simon Singh has a physics degree, if I recall).
This book of Donald O'Shea is not very well written. First of all, I think the book is not well structured. He doesn't conduct the story simply from a to b, he retains himself in too many subjects a bit off topic, not relevant, or doesn't seem quite pertinent to the main subject, which is the poincaré conjecture, (although some are interesting); what's the relevance of the second world war, or the history of united states mathematics and it's universities. He turns back and forth some times, like forgetting something behind. The prose is unpleasant, except maybe in the lasts chapters. The author spent several chapters in the beginning, talking about the shape of the earth, coulumbus travels, history of maps, defining manifolds of dimension 2, pitagoras and euclid elements, euclidian geometry, the fifth postulate, and suddenly jumps over almost every pertinent concept to understand the poincaré conjecture and the solution by perelman. That is, if he starts the book writing to a public with no knowledge on mathematics, he ends it as writing to a professional mathematician. Everyone that buys a book of this sort, obviously knows what a surface is (or even what is a manifold, or have some knowledge on calculus) don't see the point in explaining that. On the other hand, in the end of the book he says something like: "the complements of two knots could be homeomorphic without the knots being isotopic to each other or their mirror image" with no explanation whatsoever. Let me detail a bit more: for example, in page 131 alone O'Shea introduces several fundamental concepts in topology, see how he does:

about "betti numbers", and "homologies":
-Betti associated numbers with manifolds and poincaré reinterpreted this numbers by introducing equations between submanifolds of a manifold called homologies on a manifold that expressed the relation of bounding within the manifold;

about the "fundamental group":
-Poincaré associated a completely new algebraic object with each manifold which e called the fundamental group.
Sure, as I know from the beginning, that all this terms are associated with topology somehow!

In spite of being a mathematician, Donald O'Shea doesn't seem to think like one, he presents concepts, and tries to define them, in a confusing way. There are some mistakes, but not serious: "..a spherical piece of cloth that would fit perfectly on the top of your head. (...) The cloth would have to have less area inside a circle of fixed radius than there would be on a bedsheet."(page 96) Defines at least 2 times wrongly the number pi, as: "the ratio of the diameter of a circle to its radius"(page 208). These are 2 examples. Distractions of course, but nevertheless, doesn't look nice for a mathematician.
If you want to know the recent story about the poincaré conjecture and some facts about perelman's solution, you just need to read the last 3 chapters. And of course, you won't get any clear idea how perelman did it!
Many facts revealed, for example, in the article "Manifold Destiny" published in The New Yorker, important as they are to understand the circumstances of the solution, and all the complications that emerged around it, are simply ignored!!
The book has one good thing though, has lots of references, articles, books and websites.
For a mathematician who took a whole sabbatical to investigate and write this 200 page story, Donald O'Shea, in my view, did quite a miserable job.

Well done.., 22 Jul 1999
Bold has a gem of a book here. It's only a little bit over a hundred pages, but it's packed full of the great geometry problems that occupied the minds of the world's greatest thinkers for the past 2000 years.

The title describes the book perfectly. These really are "Famous Problems from Geometry" and he does indeed explain how to solve them.

The book has four major sections/chapters. He discusses in detail the three problems from antiquity (one section each): squaring a circle, doubling a cube, and trisecting an angle. Furthermore, he spends significant time with constructions of regular polygons (the fourth section) - which ones can be constructed and why. He also discusses which ones cannot be constructed and why.

The reader will be expected to understand concepts from Modern Algebra, particularly the concept of a Field. While Bold does spend time explaining what a Field is, his definition is quick and is assumed to be more of a refresher for someone who has already learned about them. Bold also has a section on Complex Numbers where he derives one of the formulas used later in the book. Again - this section is assumed to be a refresher on Complex Numbers. High School Geometry or Algebra students would have significant trouble understanding his explanations and proofs.

Bold provides problems for the reader to work along the way. These are problems that logically lead to the proof of the problem being studied. The problems are good. As a third year college student majoring in mathematics, I found the explanations/solutions to be sometimes hard to follow. He assumes a great deal about the reader's level of proficiency in math and in geometry. As a result, he liberally skips steps in proofs that are assumed to be "obvious."

If you're expecting simple proofs to these problems, you're not going to find them. If they were simple, they wouldn't have taken 2000 years to solve. But they are explained clearly here in terms that anyone with a college degree should be able to understand.

Overall, a superb book. A must have for anyone interested in the famous problems from the history of Geometry.

Excellent course book, 30 Dec 2007
This is the book that was used in the (now defunct) M435 Topology course at the Open University. Thus I came to know and love this book.

I've taken a look at a few topology texts and they're all more or less difficult to get to grips with. This one is the most accessible of those.

If you need to understand topology for any reason, or you're studying it and it's not on your list of course works, then get it.

Clear and easy to follow, 30 Apr 2007
This is an excellent introduction to basic analytic topology and metric spaces. The fundamental concepts are clearly presented and the theory is developed so that it is easy to follow; but the book is also concise and compact (not topologically, I think!!).

I doubt that you could find a better starting place if you want to learn about metric spaces or basic analytic topology.

Clear and rigorous exposition of elementary material., 20 Dec 2006
Complements nicely the metric and topological spaces course lectured in the first year at Cambridge and develops compactness in the proper manner and finishes cleanly with the Arzela-Ascoli theorem. Perhaps provides far too few applications of the contraction mapping theorem and occasionally hides important and well known results in starred exercises which are generally interesting and well worth doing -- equivalence of norms on R^n, one-point compactification and various counterexamples to unreasonable assumptions of continuity to name just a few.

The author also seems to have completely forgotten the p-adic numbers when developing metric spaces which is a huge loss.

The book largely misses off the more fun aspects of topology which are explored in their beauty in, for example, Henle's "A combinatorial introduction to topology".

Overall, I have yet to find a better book for anyone to begin study of topology from but this book is not enough on its own for the reader to truly enjoy the subject.

A comprehenisve book that is easy to read, 01 Feb 2001
This is a great book for students taking credits in metric and topological spaces, for example the G13MTS course at Nottingham. It is easy to read, and quite small, yet covers nearly all the introductary material very well. Also contains problems to solve if you want some extra practice.

Extremely comprehensive self contained text, 06 Dec 1999
Having scouted the shelves at the Nottingham University library, this was the only text I found which gave an introduction into both Metric and Topological Spaces without the prerequisite for deep previous knowledge. The book starts with a recap of some results from real analysis and develops these comprehensively by first looking at properties of Metric spaces and then the more general topological space scenario. It's a very self contained text with good use of examples and counterexamples to assist learning.

Good historical review, 17 Dec 2008
Unlike John Derbyshire's books on the history of mathematics, this book does not contain much mathematics apart from concepts in topology, but it does contain a good history of the developments leading up to the conjecture in 1904, and the attempts since to provide a proof. The book is intended for a non-specialist audience, but it helps if you have a mathematical background, at least to A level, and are interested in the development of ideas in their historical context. There is not much material on the conjecture itself since to include it would take it out of the scope of a popular book, but one gets the flavour of the idea and its usefulness beyond merely an abtruse mathematical concept. The book is well-written and entertaining and recommended if you don't want a mathematical text-book approach to the subject.

What about mathematics ?, 16 Dec 2008
Only few people have an idea on what mathematicians active at the frontiers of research try to achieve.

If you want to get a glimpse: read this book. No mathematical background is needed (if you nevertheless do have some background then the notes contain interesting material). The book describes the development of mathematical theories (in words, not in formulas) starting with Euclid's fifth postulate, via non-euclidian geometries to the Poincaré Conjecture and its solution by Grigory Perelman in 2003. It does so by introducing the people involved, describing their characters and their ways of living and presenting their contribution to the theory being developed.

The book also gives a clear account of the interaction of mathematics with other sciences.

The book is well written.

I recommend the book to every-one who wants to know what mathematicians do and why that is of importance. By the way the book is also interesting for people with a mathematical background who want to know what the Poincaré Conjecture is about.




A bit too historic, 09 Aug 2008
I have read a book called `The Poincare Conjecture'. I understood the book, I even enjoyed the book; yet I am none the wiser on the actual Conjecture from where I set out.

I am not sure why D. O'Shea avoided the hard bits. There is no risk of an unsuspecting member of the public picking this up in an airport bookstore before boarding a long haul expecting a Tom Clancy novel.

This book is too focused on historical topics behind the Conjecture and associated topics in topology that make light reading. If you are seeking to learn more about the specifics of the 'Poincare Conjecture' this is not good enough.

A great deal of this book, say 50%, is centered on the evolution of Euclidian to non-Euclidian Geometry. Only about 2/3's of the way in are we actually dealing with the conjecture.
A massive amount of this book focuses on the golden period of mathematics at Göttingen. A fairer title for this book would have been 'Topics in the history of Geometry'.

In fairness, the mathematics behind Perelman's solution are pretty much inaccessible and even the conjecture itself is difficult to understand properly.

This book will not satisfy anyone who is seriously interested in the conjecture nor will it deepen anyone's understanding who wishes to understand it more.

Not quite what I expected..., 22 Aug 2007
There are some magnificent books about mathematics, and in particular on the history of some mathematical breakthrough, like that of Simon Singh on fermat last theorem, which I read more than once. (Simon Singh has a physics degree, if I recall).
This book of Donald O'Shea is not very well written. First of all, I think the book is not well structured. He doesn't conduct the story simply from a to b, he retains himself in too many subjects a bit off topic, not relevant, or doesn't seem quite pertinent to the main subject, which is the poincaré conjecture, (although some are interesting); what's the relevance of the second world war, or the history of united states mathematics and it's universities. He turns back and forth some times, like forgetting something behind. The prose is unpleasant, except maybe in the lasts chapters. The author spent several chapters in the beginning, talking about the shape of the earth, coulumbus travels, history of maps, defining manifolds of dimension 2, pitagoras and euclid elements, euclidian geometry, the fifth postulate, and suddenly jumps over almost every pertinent concept to understand the poincaré conjecture and the solution by perelman. That is, if he starts the book writing to a public with no knowledge on mathematics, he ends it as writing to a professional mathematician. Everyone that buys a book of this sort, obviously knows what a surface is (or even what is a manifold, or have some knowledge on calculus) don't see the point in explaining that. On the other hand, in the end of the book he says something like: "the complements of two knots could be homeomorphic without the knots being isotopic to each other or their mirror image" with no explanation whatsoever. Let me detail a bit more: for example, in page 131 alone O'Shea introduces several fundamental concepts in topology, see how he does:

about "betti numbers", and "homologies":
-Betti associated numbers with manifolds and poincaré reinterpreted this numbers by introducing equations between submanifolds of a manifold called homologies on a manifold that expressed the relation of bounding within the manifold;

about the "fundamental group":
-Poincaré associated a completely new algebraic object with each manifold which e called the fundamental group.
Sure, as I know from the beginning, that all this terms are associated with topology somehow!

In spite of being a mathematician, Donald O'Shea doesn't seem to think like one, he presents concepts, and tries to define them, in a confusing way. There are some mistakes, but not serious: "..a spherical piece of cloth that would fit perfectly on the top of your head. (...) The cloth would have to have less area inside a circle of fixed radius than there would be on a bedsheet."(page 96) Defines at least 2 times wrongly the number pi, as: "the ratio of the diameter of a circle to its radius"(page 208). These are 2 examples. Distractions of course, but nevertheless, doesn't look nice for a mathematician.
If you want to know the recent story about the poincaré conjecture and some facts about perelman's solution, you just need to read the last 3 chapters. And of course, you won't get any clear idea how perelman did it!
Many facts revealed, for example, in the article "Manifold Destiny" published in The New Yorker, important as they are to understand the circumstances of the solution, and all the complications that emerged around it, are simply ignored!!
The book has one good thing though, has lots of references, articles, books and websites.
For a mathematician who took a whole sabbatical to investigate and write this 200 page story, Donald O'Shea, in my view, did quite a miserable job.

Well done.., 22 Jul 1999
Bold has a gem of a book here. It's only a little bit over a hundred pages, but it's packed full of the great geometry problems that occupied the minds of the world's greatest thinkers for the past 2000 years.

The title describes the book perfectly. These really are "Famous Problems from Geometry" and he does indeed explain how to solve them.

The book has four major sections/chapters. He discusses in detail the three problems from antiquity (one section each): squaring a circle, doubling a cube, and trisecting an angle. Furthermore, he spends significant time with constructions of regular polygons (the fourth section) - which ones can be constructed and why. He also discusses which ones cannot be constructed and why.

The reader will be expected to understand concepts from Modern Algebra, particularly the concept of a Field. While Bold does spend time explaining what a Field is, his definition is quick and is assumed to be more of a refresher for someone who has already learned about them. Bold also has a section on Complex Numbers where he derives one of the formulas used later in the book. Again - this section is assumed to be a refresher on Complex Numbers. High School Geometry or Algebra students would have significant trouble understanding his explanations and proofs.

Bold provides problems for the reader to work along the way. These are problems that logically lead to the proof of the problem being studied. The problems are good. As a third year college student majoring in mathematics, I found the explanations/solutions to be sometimes hard to follow. He assumes a great deal about the reader's level of proficiency in math and in geometry. As a result, he liberally skips steps in proofs that are assumed to be "obvious."

If you're expecting simple proofs to these problems, you're not going to find them. If they were simple, they wouldn't have taken 2000 years to solve. But they are explained clearly here in terms that anyone with a college degree should be able to understand.

Overall, a superb book. A must have for anyone interested in the famous problems from the history of Geometry.

Excellent course book, 30 Dec 2007
This is the book that was used in the (now defunct) M435 Topology course at the Open University. Thus I came to know and love this book.

I've taken a look at a few topology texts and they're all more or less difficult to get to grips with. This one is the most accessible of those.

If you need to understand topology for any reason, or you're studying it and it's not on your list of course works, then get it.

Clear and easy to follow, 30 Apr 2007
This is an excellent introduction to basic analytic topology and metric spaces. The fundamental concepts are clearly presented and the theory is developed so that it is easy to follow; but the book is also concise and compact (not topologically, I think!!).

I doubt that you could find a better starting place if you want to learn about metric spaces or basic analytic topology.

Clear and rigorous exposition of elementary material., 20 Dec 2006
Complements nicely the metric and topological spaces course lectured in the first year at Cambridge and develops compactness in the proper manner and finishes cleanly with the Arzela-Ascoli theorem. Perhaps provides far too few applications of the contraction mapping theorem and occasionally hides important and well known results in starred exercises which are generally interesting and well worth doing -- equivalence of norms on R^n, one-point compactification and various counterexamples to unreasonable assumptions of continuity to name just a few.

The author also seems to have completely forgotten the p-adic numbers when developing metric spaces which is a huge loss.

The book largely misses off the more fun aspects of topology which are explored in their beauty in, for example, Henle's "A combinatorial introduction to topology".

Overall, I have yet to find a better book for anyone to begin study of topology from but this book is not enough on its own for the reader to truly enjoy the subject.

A comprehenisve book that is easy to read, 01 Feb 2001
This is a great book for students taking credits in metric and topological spaces, for example the G13MTS course at Nottingham. It is easy to read, and quite small, yet covers nearly all the introductary material very well. Also contains problems to solve if you want some extra practice.

Extremely comprehensive self contained text, 06 Dec 1999
Having scouted the shelves at the Nottingham University library, this was the only text I found which gave an introduction into both Metric and Topological Spaces without the prerequisite for deep previous knowledge. The book starts with a recap of some results from real analysis and develops these comprehensively by first looking at properties of Metric spaces and then the more general topological space scenario. It's a very self contained text with good use of examples and counterexamples to assist learning.

Good historical review, 17 Dec 2008
Unlike John Derbyshire's books on the history of mathematics, this book does not contain much mathematics apart from concepts in topology, but it does contain a good history of the developments leading up to the conjecture in 1904, and the attempts since to provide a proof. The book is intended for a non-specialist audience, but it helps if you have a mathematical background, at least to A level, and are interested in the development of ideas in their historical context. There is not much material on the conjecture itself since to include it would take it out of the scope of a popular book, but one gets the flavour of the idea and its usefulness beyond merely an abtruse mathematical concept. The book is well-written and entertaining and recommended if you don't want a mathematical text-book approach to the subject.

What about mathematics ?, 16 Dec 2008
Only few people have an idea on what mathematicians active at the frontiers of research try to achieve.

If you want to get a glimpse: read this book. No mathematical background is needed (if you nevertheless do have some background then the notes contain interesting material). The book describes the development of mathematical theories (in words, not in formulas) starting with Euclid's fifth postulate, via non-euclidian geometries to the Poincaré Conjecture and its solution by Grigory Perelman in 2003. It does so by introducing the people involved, describing their characters and their ways of living and presenting their contribution to the theory being developed.

The book also gives a clear account of the interaction of mathematics with other sciences.

The book is well written.

I recommend the book to every-one who wants to know what mathematicians do and why that is of importance. By the way the book is also interesting for people with a mathematical background who want to know what the Poincaré Conjecture is about.




A bit too historic, 09 Aug 2008
I have read a book called `The Poincare Conjecture'. I understood the book, I even enjoyed the book; yet I am none the wiser on the actual Conjecture from where I set out.

I am not sure why D. O'Shea avoided the hard bits. There is no risk of an unsuspecting member of the public picking this up in an airport bookstore before boarding a long haul expecting a Tom Clancy novel.

This book is too focused on historical topics behind the Conjecture and associated topics in topology that make light reading. If you are seeking to learn more about the specifics of the 'Poincare Conjecture' this is not good enough.

A great deal of this book, say 50%, is centered on the evolution of Euclidian to non-Euclidian Geometry. Only about 2/3's of the way in are we actually dealing with the conjecture.
A massive amount of this book focuses on the golden period of mathematics at Göttingen. A fairer title for this book would have been 'Topics in the history of Geometry'.

In fairness, the mathematics behind Perelman's solution are pretty much inaccessible and even the conjecture itself is difficult to understand properly.

This book will not satisfy anyone who is seriously interested in the conjecture nor will it deepen anyone's understanding who wishes to understand it more.

Not quite what I expected..., 22 Aug 2007
There are some magnificent books about mathematics, and in particular on the history of some mathematical breakthrough, like that of Simon Singh on fermat last theorem, which I read more than once. (Simon Singh has a physics degree, if I recall).
This book of Donald O'Shea is not very well written. First of all, I think the book is not well structured. He doesn't conduct the story simply from a to b, he retains himself in too many subjects a bit off topic, not relevant, or doesn't seem quite pertinent to the main subject, which is the poincaré conjecture, (although some are interesting); what's the relevance of the second world war, or the history of united states mathematics and it's universities. He turns back and forth some times, like forgetting something behind. The prose is unpleasant, except maybe in the lasts chapters. The author spent several chapters in the beginning, talking about the shape of the earth, coulumbus travels, history of maps, defining manifolds of dimension 2, pitagoras and euclid elements, euclidian geometry, the fifth postulate, and suddenly jumps over almost every pertinent concept to understand the poincaré conjecture and the solution by perelman. That is, if he starts the book writing to a public with no knowledge on mathematics, he ends it as writing to a professional mathematician. Everyone that buys a book of this sort, obviously knows what a surface is (or even what is a manifold, or have some knowledge on calculus) don't see the point in explaining that. On the other hand, in the end of the book he says something like: "the complements of two knots could be homeomorphic without the knots being isotopic to each other or their mirror image" with no explanation whatsoever. Let me detail a bit more: for example, in page 131 alone O'Shea introduces several fundamental concepts in topology, see how he does:

about "betti numbers", and "homologies":
-Betti associated numbers with manifolds and poincaré reinterpreted this numbers by introducing equations between submanifolds of a manifold called homologies on a manifold that expressed the relation of bounding within the manifold;

about the "fundamental group":
-Poincaré associated a completely new algebraic object with each manifold which e called the fundamental group.
Sure, as I know from the beginning, that all this terms are associated with topology somehow!

In spite of being a mathematician, Donald O'Shea doesn't seem to think like one, he presents concepts, and tries to define them, in a confusing way. There are some mistakes, but not serious: "..a spherical piece of cloth that would fit perfectly on the top of your head. (...) The cloth would have to have less area inside a circle of fixed radius than there would be on a bedsheet."(page 96) Defines at least 2 times wrongly the number pi, as: "the ratio of the diameter of a circle to its radius"(page 208). These are 2 examples. Distractions of course, but nevertheless, doesn't look nice for a mathematician.
If you want to know the recent story about the poincaré conjecture and some facts about perelman's solution, you just need to read the last 3 chapters. And of course, you won't get any clear idea how perelman did it!
Many facts revealed, for example, in the article "Manifold Destiny" published in The New Yorker, important as they are to understand the circumstances of the solution, and all the complications that emerged around it, are simply ignored!!
The book has one good thing though, has lots of references, articles, books and websites.
For a mathematician who took a whole sabbatical to investigate and write this 200 page story, Donald O'Shea, in my view, did quite a miserable job.

Good historical review, 17 Dec 2008
Unlike John Derbyshire's books on the history of mathematics, this book does not contain much mathematics apart from concepts in topology, but it does contain a good history of the developments leading up to the conjecture in 1904, and the attempts since to provide a proof. The book is intended for a non-specialist audience, but it helps if you have a mathematical background, at least to A level, and are interested in the development of ideas in their historical context. There is not much material on the conjecture itself since to include it would take it out of the scope of a popular book, but one gets the flavour of the idea and its usefulness beyond merely an abtruse mathematical concept. The book is well-written and entertaining and recommended if you don't want a mathematical text-book approach to the subject.

What about mathematics ?, 16 Dec 2008
Only few people have an idea on what mathematicians active at the frontiers of research try to achieve.

If you want to get a glimpse: read this book. No mathematical background is needed (if you nevertheless do have some background then the notes contain interesting material). The book describes the development of mathematical theories (in words, not in formulas) starting with Euclid's fifth postulate, via non-euclidian geometries to the Poincaré Conjecture and its solution by Grigory Perelman in 2003. It does so by introducing the people involved, describing their characters and their ways of living and presenting their contribution to the theory being developed.

The book also gives a clear account of the interaction of mathematics with other sciences.

The book is well written.

I recommend the book to every-one who wants to know what mathematicians do and why that is of importance. By the way the book is also interesting for people with a mathematical background who want to know what the Poincaré Conjecture is about.




A bit too historic, 09 Aug 2008
I have read a book called `The Poincare Conjecture'. I understood the book, I even enjoyed the book; yet I am none the wiser on the actual Conjecture from where I set out.

I am not sure why D. O'Shea avoided the hard bits. There is no risk of an unsuspecting member of the public picking this up in an airport bookstore before boarding a long haul expecting a Tom Clancy novel.

This book is too focused on historical topics behind the Conjecture and associated topics in topology that make light reading. If you are seeking to learn more about the specifics of the 'Poincare Conjecture' this is not good enough.

A great deal of this book, say 50%, is centered on the evolution of Euclidian to non-Euclidian Geometry. Only about 2/3's of the way in are we actually dealing with the conjecture.
A massive amount of this book focuses on the golden period of mathematics at Göttingen. A fairer title for this book would have been 'Topics in the history of Geometry'.

In fairness, the mathematics behind Perelman's solution are pretty much inaccessible and even the conjecture itself is difficult to understand properly.

This book will not satisfy anyone who is seriously interested in the conjecture nor will it deepen anyone's understanding who wishes to understand it more.

Not quite what I expected..., 22 Aug 2007
There are some magnificent books about mathematics, and in particular on the history of some mathematical breakthrough, like that of Simon Singh on fermat last theorem, which I read more than once. (Simon Singh has a physics degree, if I recall).
This book of Donald O'Shea is not very well written. First of all, I think the book is not well structured. He doesn't conduct the story simply from a to b, he retains himself in too many subjects a bit off topic, not relevant, or doesn't seem quite pertinent to the main subject, which is the poincaré conjecture, (although some are interesting); what's the relevance of the second world war, or the history of united states mathematics and it's universities. He turns back and forth some times, like forgetting something behind. The prose is unpleasant, except maybe in the lasts chapters. The author spent several chapters in the beginning, talking about the shape of the earth, coulumbus travels, history of maps, defining manifolds of dimension 2, pitagoras and euclid elements, euclidian geometry, the fifth postulate, and suddenly jumps over almost every pertinent concept to understand the poincaré conjecture and the solution by perelman. That is, if he starts the book writing to a public with no knowledge on mathematics, he ends it as writing to a professional mathematician. Everyone that buys a book of this sort, obviously knows what a surface is (or even what is a manifold, or have some knowledge on calculus) don't see the point in explaining that. On the other hand, in the end of the book he says something like: "the complements of two knots could be homeomorphic without the knots being isotopic to each other or their mirror image" with no explanation whatsoever. Let me detail a bit more: for example, in page 131 alone O'Shea introduces several fundamental concepts in topology, see how he does:

about "betti numbers", and "homologies":
-Betti associated numbers with manifolds and poincaré reinterpreted this numbers by introducing equations between submanifolds of a manifold called homologies on a manifold that expressed the relation of bounding within the manifold;

about the "fundamental group":
-Poincaré associated a completely new algebraic object with each manifold which e called the fundamental group.
Sure, as I know from the beginning, that all this terms are associated with topology somehow!

In spite of being a mathematician, Donald O'Shea doesn't seem to think like one, he presents concepts, and tries to define them, in a confusing way. There are some mistakes, but not serious: "..a spherical piece of cloth that would fit perfectly on the top of your head. (...) The cloth would have to have less area inside a circle of fixed radius than there would be on a bedsheet."(page 96) Defines at least 2 times wrongly the number pi, as: "the ratio of the diameter of a circle to its radius"(page 208). These are 2 examples. Distractions of course, but nevertheless, doesn't look nice for a mathematician.
If you want to know the recent story about the poincaré conjecture and some facts about perelman's solution, you just need to read the last 3 chapters. And of course, you won't get any clear idea how perelman did it!
Many facts revealed, for example, in the article "Manifold Destiny" published in The New Yorker, important as they are to understand the circumstances of the solution, and all the complications that emerged around it, are simply ignored!!
The book has one good thing though, has lots of references, articles, books and websites.
For a mathematician who took a whole sabbatical to investigate and write this 200 page story, Donald O'Shea, in my view, did quite a miserable job.

Well done.., 22 Jul 1999
Bold has a gem of a book here. It's only a little bit over a hundred pages, but it's packed full of the great geometry problems that occupied the minds of the world's greatest thinkers for the past 2000 years.

The title describes the book perfectly. These really are "Famous Problems from Geometry" and he does indeed explain how to solve them.

The book has four major sections/chapters. He discusses in detail the three problems from antiquity (one section each): squaring a circle, doubling a cube, and trisecting an angle. Furthermore, he spends significant time with constructions of regular polygons (the fourth section) - which ones can be constructed and why. He also discusses which ones cannot be constructed and why.

The reader will be expected to understand concepts from Modern Algebra, particularly the concept of a Field. While Bold does spend time explaining what a Field is, his definition is quick and is assumed to be more of a refresher for someone who has already learned about them. Bold also has a section on Complex Numbers where he derives one of the formulas used later in the book. Again - this section is assumed to be a refresher on Complex Numbers. High School Geometry or Algebra students would have significant trouble understanding his explanations and proofs.

Bold provides problems for the reader to work along the way. These are problems that logically lead to the proof of the problem being studied. The problems are good. As a third year college student majoring in mathematics, I found the explanations/solutions to be sometimes hard to follow. He assumes a great deal about the reader's level of proficiency in math and in geometry. As a result, he liberally skips steps in proofs that are assumed to be "obvious."

If you're expecting simple proofs to these problems, you're not going to find them. If they were simple, they wouldn't have taken 2000 years to solve. But they are explained clearly here in terms that anyone with a college degree should be able to understand.

Overall, a superb book. A must have for anyone interested in the famous problems from the history of Geometry.

Excellent course book, 30 Dec 2007
This is the book that was used in the (now defunct) M435 Topology course at the Open University. Thus I came to know and love this book.

I've taken a look at a few topology texts and they're all more or less difficult to get to grips with. This one is the most accessible of those.

If you need to understand topology for any reason, or you're studying it and it's not on your list of course works, then get it.

Clear and easy to follow, 30 Apr 2007
This is an excellent introduction to basic analytic topology and metric spaces. The fundamental concepts are clearly presented and the theory is developed so that it is easy to follow; but the book is also concise and compact (not topologically, I think!!).

I doubt that you could find a better starting place if you want to learn about metric spaces or basic analytic topology.

Clear and rigorous exposition of elementary material., 20 Dec 2006
Complements nicely the metric and topological spaces course lectured in the first year at Cambridge and develops compactness in the proper manner and finishes cleanly with the Arzela-Ascoli theorem. Perhaps provides far too few applications of the contraction mapping theorem and occasionally hides important and well known results in starred exercises which are generally interesting and well worth doing -- equivalence of norms on R^n, one-point compactification and various counterexamples to unreasonable assumptions of continuity to name just a few.

The author also seems to have completely forgotten the p-adic numbers when developing metric spaces which is a huge loss.

The book largely misses off the more fun aspects of topology which are explored in their beauty in, for example, Henle's "A combinatorial introduction to topology".

Overall, I have yet to find a better book for anyone to begin study of topology from but this book is not enough on its own for the reader to truly enjoy the subject.

A comprehenisve book that is easy to read, 01 Feb 2001
This is a great book for students taking credits in metric and topological spaces, for example the G13MTS course at Nottingham. It is easy to read, and quite small, yet covers nearly all the introductary material very well. Also contains problems to solve if you want some extra practice.

Extremely comprehensive self contained text, 06 Dec 1999
Having scouted the shelves at the Nottingham University library, this was the only text I found which gave an introduction into both Metric and Topological Spaces without the prerequisite for deep previous knowledge. The book starts with a recap of some results from real analysis and develops these comprehensively by first looking at properties of Metric spaces and then the more general topological space scenario. It's a very self contained text with good use of examples and counterexamples to assist learning.

Good historical review, 17 Dec 2008
Unlike John Derbyshire's books on the history of mathematics, this book does not contain much mathematics apart from concepts in topology, but it does contain a good history of the developments leading up to the conjecture in 1904, and the attempts since to provide a proof. The book is intended for a non-specialist audience, but it helps if you have a mathematical background, at least to A level, and are interested in the development of ideas in their historical context. There is not much material on the conjecture itself since to include it would take it out of the scope of a popular book, but one gets the flavour of the idea and its usefulness beyond merely an abtruse mathematical concept. The book is well-written and entertaining and recommended if you don't want a mathematical text-book approach to the subject.

What about mathematics ?, 16 Dec 2008
Only few people have an idea on what mathematicians active at the frontiers of research try to achieve.

If you want to get a glimpse: read this book. No mathematical background is needed (if you nevertheless do have some background then the notes contain interesting material). The book describes the development of mathematical theories (in words, not in formulas) starting with Euclid's fifth postulate, via non-euclidian geometries to the Poincaré Conjecture and its solution by Grigory Perelman in 2003. It does so by introducing the people involved, describing their characters and their ways of living and presenting their contribution to the theory being developed.

The book also gives a clear account of the interaction of mathematics with other sciences.

The book is well written.

I recommend the book to every-one who wants to know what mathematicians do and why that is of importance. By the way the book is also interesting for people with a mathematical background who want to know what the Poincaré Conjecture is about.




A bit too historic, 09 Aug 2008
I have read a book called `The Poincare Conjecture'. I understood the book, I even enjoyed the book; yet I am none the wiser on the actual Conjecture from where I set out.

I am not sure why D. O'Shea avoided the hard bits. There is no risk of an unsuspecting member of the public picking this up in an airport bookstore before boarding a long haul expecting a Tom Clancy novel.

This book is too focused on historical topics behind the Conjecture and associated topics in topology that make light reading. If you are seeking to learn more about the specifics of the 'Poincare Conjecture' this is not good enough.

A great deal of this book, say 50%, is centered on the evolution of Euclidian to non-Euclidian Geometry. Only about 2/3's of the way in are we actually dealing with the conjecture.
A massive amount of this book focuses on the golden period of mathematics at Göttingen. A fairer title for this book would have been 'Topics in the history of Geometry'.

In fairness, the mathematics behind Perelman's solution are pretty much inaccessible and even the conjecture itself is difficult to understand properly.

This book will not satisfy anyone who is seriously interested in the conjecture nor will it deepen anyone's understanding who wishes to understand it more.

Not quite what I expected..., 22 Aug 2007
There are some magnificent books about mathematics, and in particular on the history of some mathematical breakthrough, like that of Simon Singh on fermat last theorem, which I read more than once. (Simon Singh has a physics degree, if I recall).
This book of Donald O'Shea is not very well written. First of all, I think the book is not well structured. He doesn't conduct the story simply from a to b, he retains himself in too many subjects a bit off topic, not relevant, or doesn't seem quite pertinent to the main subject, which is the poincaré conjecture, (although some are interesting); what's the relevance of the second world war, or the history of united states mathematics and it's universities. He turns back and forth some times, like forgetting something behind. The prose is unpleasant, except maybe in the lasts chapters. The author spent several chapters in the beginning, talking about the shape of the earth, coulumbus travels, history of maps, defining manifolds of dimension 2, pitagoras and euclid elements, euclidian geometry, the fifth postulate, and suddenly jumps over almost every pertinent concept to understand the poincaré conjecture and the solution by perelman. That is, if he starts the book writing to a public with no knowledge on mathematics, he ends it as writing to a professional mathematician. Everyone that buys a book of this sort, obviously knows what a surface is (or even what is a manifold, or have some knowledge on calculus) don't see the point in explaining that. On the other hand, in the end of the book he says something like: "the complements of two knots could be homeomorphic without the knots being isotopic to each other or their mirror image" with no explanation whatsoever. Let me detail a bit more: for example, in page 131 alone O'Shea introduces several fundamental concepts in topology, see how he does:

about "betti numbers", and "homologies":
-Betti associated numbers with manifolds and poincaré reinterpreted this numbers by introducing equations between submanifolds of a manifold called homologies on a manifold that expressed the relation of bounding within the manifold;

about the "fundamental group":
-Poincaré associated a completely new algebraic object with each manifold which e called the fundamental group.
Sure, as I know from the beginning, that all this terms are associated with topology somehow!

In spite of being a mathematician, Donald O'Shea doesn't seem to think like one, he presents concepts, and tries to define them, in a confusing way. There are some mistakes, but not serious: "..a spherical piece of cloth that would fit perfectly on the top of your head. (...) The cloth would have to have less area inside a circle of fixed radius than there would be on a bedsheet."(page 96) Defines at least 2 times wrongly the number pi, as: "the ratio of the diameter of a circle to its radius"(page 208). These are 2 examples. Distractions of course, but nevertheless, doesn't look nice for a mathematician.
If you want to know the recent story about the poincaré conjecture and some facts about perelman's solution, you just need to read the last 3 chapters. And of course, you won't get any clear idea how perelman did it!
Many facts revealed, for example, in the article "Manifold Destiny" published in The New Yorker, important as they are to understand the circumstances of the solution, and all the complications that emerged around it, are simply ignored!!
The book has one good thing though, has lots of references, articles, books and websites.
For a mathematician who took a whole sabbatical to investigate and write this 200 page story, Donald O'Shea, in my view, did quite a miserable job.

Probably the greatest work in graph theory of all time, 08 Oct 2007
rigorous yet accessible - this book is the entire stimulus for my lov of graph theory, and indeed my research project.

Good historical review, 17 Dec 2008
Unlike John Derbyshire's books on the history of mathematics, this book does not contain much mathematics apart from concepts in topology, but it does contain a good history of the developments leading up to the conjecture in 1904, and the attempts since to provide a proof. The book is intended for a non-specialist audience, but it helps if you have a mathematical background, at least to A level, and are interested in the development of ideas in their historical context. There is not much material on the conjecture itself since to include it would take it out of the scope of a popular book, but one gets the flavour of the idea and its usefulness beyond merely an abtruse mathematical concept. The book is well-written and entertaining and recommended if you don't want a mathematical text-book approach to the subject.

What about mathematics ?, 16 Dec 2008
Only few people have an idea on what mathematicians active at the frontiers of research try to achieve.

If you want to get a glimpse: read this book. No mathematical background is needed (if you nevertheless do have some background then the notes contain interesting material). The book describes the development of mathematical theories (in words, not in formulas) starting with Euclid's fifth postulate, via non-euclidian geometries to the Poincaré Conjecture and its solution by Grigory Perelman in 2003. It does so by introducing the people involved, describing their characters and their ways of living and presenting their contribution to the theory being developed.

The book also gives a clear account of the interaction of mathematics with other sciences.

The book is well written.

I recommend the book to every-one who wants to know what mathematicians do and why that is of importance. By the way the book is also interesting for people with a mathematical background who want to know what the Poincaré Conjecture is about.




A bit too historic, 09 Aug 2008
I have read a book called `The Poincare Conjecture'. I understood the book, I even enjoyed the book; yet I am none the wiser on the actual Conjecture from where I set out.

I am not sure why D. O'Shea avoided the hard bits. There is no risk of an unsuspecting member of the public picking this up in an airport bookstore before boarding a long haul expecting a Tom Clancy novel.

This book is too focused on historical topics behind the Conjecture and associated topics in topology that make light reading. If you are seeking to learn more about the specifics of the 'Poincare Conjecture' this is not good enough.

A great deal of this book, say 50%, is centered on the evolution of Euclidian to non-Euclidian Geometry. Only about 2/3's of the way in are we actually dealing with the conjecture.
A massive amount of this book focuses on the golden period of mathematics at Göttingen. A fairer title for this book would have been 'Topics in the history of Geometry'.

In fairness, the mathematics behind Perelman's solution are pretty much inaccessible and even the conjecture itself is difficult to understand properly.

This book will not satisfy anyone who is seriously interested in the conjecture nor will it deepen anyone's understanding who wishes to understand it more.

Not quite what I expected..., 22 Aug 2007
There are some magnificent books about mathematics, and in particular on the history of some mathematical breakthrough, like that of Simon Singh on fermat last theorem, which I read more than once. (Simon Singh has a physics degree, if I recall).
This book of Donald O'Shea is not very well written. First of all, I think the book is not well structured. He doesn't conduct the story simply from a to b, he retains himself in too many subjects a bit off topic, not relevant, or doesn't seem quite pertinent to the main subject, which is the poincaré conjecture, (although some are interesting); what's the relevance of the second world war, or the history of united states mathematics and it's universities. He turns back and forth some times, like forgetting something behind. The prose is unpleasant, except maybe in the lasts chapters. The author spent several chapters in the beginning, talking about the shape of the earth, coulumbus travels, history of maps, defining manifolds of dimension 2, pitagoras and euclid elements, euclidian geometry, the fifth postulate, and suddenly jumps over almost every pertinent concept to understand the poincaré conjecture and the solution by perelman. That is, if he starts the book writing to a public with no knowledge on mathematics, he ends it as writing to a professional mathematician. Everyone that buys a book of this sort, obviously knows what a surface is (or even what is a manifold, or have some knowledge on calculus) don't see the point in explaining that. On the other hand, in the end of the book he says something like: "the complements of two knots could be homeomorphic without the knots being isotopic to each other or their mirror image" with no explanation whatsoever. Let me detail a bit more: for example, in page 131 alone O'Shea introduces several fundamental concepts in topology, see how he does:

about "betti numbers", and "homologies":
-Betti associated numbers with manifolds and poincaré reinterpreted this numbers by introducing equations between submanifolds of a manifold called homologies on a manifold that expressed the relation of bounding within the manifold;

about the "fundamental group":
-Poincaré associated a completely new algebraic object with each manifold which e called the fundamental group.
Sure, as I know from the beginning, that all this terms are associated with topology somehow!

In spite of being a mathematician, Donald O'Shea doesn't seem to think like one, he presents concepts, and tries to define them, in a confusing way. There are some mistakes, but not serious: "..a spherical piece of cloth that would fit perfectly on the top of your head. (...) The cloth would have to have less area inside a circle of fixed radius than there would be on a bedsheet."(page 96) Defines at least 2 times wrongly the number pi, as: "the ratio of the diameter of a circle to its radius"(page 208). These are 2 examples. Distractions of course, but nevertheless, doesn't look nice for a mathematician.
If you want to know the recent story about the poincaré conjecture and some facts about perelman's solution, you just need to read the last 3 chapters. And of course, you won't get any clear idea how perelman did it!
Many facts revealed, for example, in the article "Manifold Destiny" published in The New Yorker, important as they are to understand the circumstances of the solution, and all the complications that emerged around it, are simply ignored!!
The book has one good thing though, has lots of references, articles, books and websites.
For a mathematician who took a whole sabbatical to investigate and write this 200 page story, Donald O'Shea, in my view, did quite a miserable job.

Well done.., 22 Jul 1999
Bold has a gem of a book here. It's only a little bit over a hundred pages, but it's packed full of the great geometry problems that occupied the minds of the world's greatest thinkers for the past 2000 years.

The title describes the book perfectly. These really are "Famous Problems from Geometry" and he does indeed explain how to solve them.

The book has four major sections/chapters. He discusses in detail the three problems from antiquity (one section each): squaring a circle, doubling a cube, and trisecting an angle. Furthermore, he spends significant time with constructions of regular polygons (the fourth section) - which ones can be constructed and why. He also discusses which ones cannot be constructed and why.

The reader will be expected to understand concepts from Modern Algebra, particularly the concept of a Field. While Bold does spend time explaining what a Field is, his definition is quick and is assumed to be more of a refresher for someone who has already learned about them. Bold also has a section on Complex Numbers where he derives one of the formulas used later in the book. Again - this section is assumed to be a refresher on Complex Numbers. High School Geometry or Algebra students would have significant trouble understanding his explanations and proofs.

Bold provides problems for the reader to work along the way. These are problems that logically lead to the proof of the problem being studied. The problems are good. As a third year college student majoring in mathematics, I found the explanations/solutions to be sometimes hard to follow. He assumes a great deal about the reader's level of proficiency in math and in geometry. As a result, he liberally skips steps in proofs that are assumed to be "obvious."

If you're expecting simple proofs to these problems, you're not going to find them. If they were simple, they wouldn't have taken 2000 years to solve. But they are explained clearly here in terms that anyone with a college degree should be able to understand.

Overall, a superb book. A must have for anyone interested in the famous problems from the history of Geometry.

Excellent course book, 30 Dec 2007
This is the book that was used in the (now defunct) M435 Topology course at the Open University. Thus I came to know and love this book.

I've taken a look at a few topology texts and they're all more or less difficult to get to grips with. This one is the most accessible of those.

If you need to understand topology for any reason, or you're studying it and it's not on your list of course works, then get it.

Clear and easy to follow, 30 Apr 2007
This is an excellent introduction to basic analytic topology and metric spaces. The fundamental concepts are clearly presented and the theory is developed so that it is easy to follow; but the book is also concise and compact (not topologically, I think!!).

I doubt that you could find a better starting place if you want to learn about metric spaces or basic analytic topology.

Clear and rigorous exposition of elementary material., 20 Dec 2006
Complements nicely the metric and topological spaces course lectured in the first year at Cambridge and develops compactness in the proper manner and finishes cleanly with the Arzela-Ascoli theorem. Perhaps provides far too few applications of the contraction mapping theorem and occasionally hides important and well known results in starred exercises which are generally interesting and well worth doing -- equivalence of norms on R^n, one-point compactification and various counterexamples to unreasonable assumptions of continuity to name just a few.

The author also seems to have completely forgotten the p-adic numbers when developing metric spaces which is a huge loss.

The book largely misses off the more fun aspects of topology which are explored in their beauty in, for example, Henle's "A combinatorial introduction to topology".

Overall, I have yet to find a better book for anyone to begin study of topology from but this book is not enough on its own for the reader to truly enjoy the subject.

A comprehenisve book that is easy to read, 01 Feb 2001
This is a great book for students taking credits in metric and topological spaces, for example the G13MTS course at Nottingham. It is easy to read, and quite small, yet covers nearly all the introductary material very well. Also contains problems to solve if you want some extra practice.

Extremely comprehensive self contained text, 06 Dec 1999
Having scouted the shelves at the Nottingham University library, this was the only text I found which gave an introduction into both Metric and Topological Spaces without the prerequisite for deep previous knowledge. The book starts with a recap of some results from real analysis and develops these comprehensively by first looking at properties of Metric spaces and then the more general topological space scenario. It's a very self contained text with good use of examples and counterexamples to assist learning.

Good historical review, 17 Dec 2008
Unlike John Derbyshire's books on the history of mathematics, this book does not contain much mathematics apart from concepts in topology, but it does contain a good history of the developments leading up to the conjecture in 1904, and the attempts since to provide a proof. The book is intended for a non-specialist audience, but it helps if you have a mathematical background, at least to A level, and are interested in the development of ideas in their historical context. There is not much material on the conjecture itself since to include it would take it out of the scope of a popular book, but one gets the flavour of the idea and its usefulness beyond merely an abtruse mathematical concept. The book is well-written and entertaining and recommended if you don't want a mathematical text-book approach to the subject.

What about mathematics ?, 16 Dec 2008
Only few people have an idea on what mathematicians active at the frontiers of research try to achieve.

If you want to get a glimpse: read this book. No mathematical background is needed (if you nevertheless do have some background then the notes contain interesting material). The book describes the development of mathematical theories (in words, not in formulas) starting with Euclid's fifth postulate, via non-euclidian geometries to the Poincaré Conjecture and its solution by Grigory Perelman in 2003. It does so by introducing the people involved, describing their characters and their ways of living and presenting their contribution to the theory being developed.

The book also gives a clear account of the interaction of mathematics with other sciences.

The book is well written.

I recommend the book to every-one who wants to know what mathematicians do and why that is of importance. By the way the book is also interesting for people with a mathematical background who want to know what the Poincaré Conjecture is about.




A bit too historic, 09 Aug 2008
I have read a book called `The Poincare Conjecture'. I understood the book, I even enjoyed the book; yet I am none the wiser on the actual Conjecture from where I set out.

I am not sure why D. O'Shea avoided the hard bits. There is no risk of an unsuspecting member of the public picking this up in an airport bookstore before boarding a long haul expecting a Tom Clancy novel.

This book is too focused on historical topics behind the Conjecture and associated topics in topology that make light reading. If you are seeking to learn more about the specifics of the 'Poincare Conjecture' this is not good enough.

A great deal of this book, say 50%, is centered on the evolution of Euclidian to non-Euclidian Geometry. Only about 2/3's of the way in are we actually dealing with the conjecture.
A massive amount of this book focuses on the golden period of mathematics at Göttingen. A fairer title for this book would have been 'Topics in the history of Geometry'.

In fairness, the mathematics behind Perelman's solution are pretty much inaccessible and even the conjecture itself is difficult to understand properly.

This book will not satisfy anyone who is seriously interested in the conjecture nor will it deepen anyone's understanding who wishes to understand it more.

Not quite what I expected..., 22 Aug 2007
There are some magnificent books about mathematics, and in particular on the history of some mathematical breakthrough, like that of Simon Singh on fermat last theorem, which I read more than once. (Simon Singh has a physics degree, if I recall).
This book of Donald O'Shea is not very well written. First of all, I think the book is not well structured. He doesn't conduct the story simply from a to b, he retains himself in too many subjects a bit off topic, not relevant, or doesn't seem quite pertinent to the main subject, which is the poincaré conjecture, (although some are interesting); what's the relevance of the second world war, or the history of united states mathematics and it's universities. He turns back and forth some times, like forgetting something behind. The prose is unpleasant, except maybe in the lasts chapters. The author spent several chapters in the beginning, talking about the shape of the earth, coulumbus travels, history of maps, defining manifolds of dimension 2, pitagoras and euclid elements, euclidian geometry, the fifth postulate, and suddenly jumps over almost every pertinent concept to understand the poincaré conjecture and the solution by perelman. That is, if he starts the book writing to a public with no knowledge on mathematics, he ends it as writing to a professional mathematician. Everyone that buys a book of this sort, obviously knows what a surface is (or even what is a manifold, or have some knowledge on calculus) don't see the point in explaining that. On the other hand, in the end of the book he says something like: "the complements of two knots could be homeomorphic without the knots being isotopic to each other or their mirror image" with no explanation whatsoever. Let me detail a bit more: for example, in page 131 alone O'Shea introduces several fundamental concepts in topology, see how he does:

about "betti numbers", and "homologies":
-Betti associated numbers with manifolds and poincaré reinterpreted this numbers by introducing equations between submanifolds of a manifold called homologies on a manifold that expressed the relation of bounding within the manifold;

about the "fundamental group":
-Poincaré associated a completely new algebraic object with each manifold which e called the fundamental group.
Sure, as I know from the beginning, that all this terms are associated with topology somehow!

In spite of being a mathematician, Donald O'Shea doesn't seem to think like one, he presents concepts, and tries to define them, in a confusing way. There are some mistakes, but not serious: "..a spherical piece of cloth that would fit perfectly on the top of your head. (...) The cloth would have to have less area inside a circle of fixed radius than there would be on a bedsheet."(page 96) Defines at least 2 times wrongly the number pi, as: "the ratio of the diameter of a circle to its radius"(page 208). These are 2 examples. Distractions of course, but nevertheless, doesn't look nice for a mathematician.
If you want to know the recent story about the poincaré conjecture and some facts about perelman's solution, you just need to read the last 3 chapters. And of course, you won't get any clear idea how perelman did it!
Many facts revealed, for example, in the article "Manifold Destiny" published in The New Yorker, important as they are to understand the circumstances of the solution, and all the complications that emerged around it, are simply ignored!!
The book has one good thing though, has lots of references, articles, books and websites.
For a mathematician who took a whole sabbatical to investigate and write this 200 page story, Donald O'Shea, in my view, did quite a miserable job.

Probably the greatest work in graph theory of all time, 08 Oct 2007
rigorous yet accessible - this book is the entire stimulus for my lov of graph theory, and indeed my research project.

One of the best places to start..., 08 Oct 2006
Nakahara's book is one of the best introductions to geometry and topology that I have read. I constantly use the book as the starting place for just about any topic in geometry and topolgy.

After reading the book you will not be able to jump straight into research work, but it does bridge the gap between more advanced texts and papers.

Everybody should have a copy.

Good graduate intro to Differential Geom, 01 Sep 2005
To complete this book, there should be a section on general curvilinear coordinate transformations, the ultimate foundation of tensor calculus.This is a defficiency this book shares with many differential geometry texts.But maybe this can be forgiven at graduate level, for which this book is a decent pedagogical text- if a little terse at times.

The book begins with a survey of those areas of physics to which diff geom are applied , then develops some topology, and goes on to a comprehensive discussion of the theory of finite dimensional manifolds-including a chapter on complex manifolds.You will learn basic exterior calculus, lie derivatives and covariant derivatives , and so on.A first choice for those who have had a little preparation at undergraduate level.