Mark Ronan

Symmetry and the Monster

In the early 1980's there was an unusual buzz in the normally quiet pure mathematics departments of many universities. It looked like the classification of finite groups, a major milestone in the subject, had been completed. Mark Ronan's 'Symmetry and the Monster' gives the history of this process, and in particular the discovery of the last of the 'sporadic' groups known as the 'Monster'. Of particular interest are the 'Moonshine' connections which have been found between this object and totally different areas of mathematics. The book tells the stories of the main contributors to the subject, from Galois up to the present day, and is aimed at the non-specialist reader.

The one problem I found with the book was that I felt the author was shying too much away from technical terms. It's all very well writing a non-technical account, but Ronan seems to get trapped in his simplified terminology, and my feeling is that this book will be read by those who have had some exposure to the subject and want to find out about the history, rather than by complete beginners. Hence using a different terminology is confusing. For those wanting to look further into the subject, I can't help feeling that a search for sporadic simple group is likely to give better results than one for exceptional symmetry atom

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Hardcover 272 pages
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Published: 2006 Oxford University Press, USA

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ISBN: 0192807226
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Published: 2006 OUP Oxford

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Product Description
Mathematics is driven forward by the quest to solve a small number of major problems--the four most famous challenges being Fermat's Last Theorem, the Riemann Hypothesis, Poincare's Conjecture, and the quest for the "Monster" of Symmetry. Now, in an exciting, fast-paced historical narrative ranging across two centuries, Mark Ronan takes us on an exhilarating tour of this final mathematical quest. Ronan describes how the quest to understand symmetry really began with the tragic young genius Evariste Galois, who died at the age of 20 in a duel. Galois, who spent the night before he died frantically scribbling his unpublished discoveries, used symmetry to understand algebraic equations, and he discovered that there were building blocks or "atoms of symmetry." Most of these building blocks fit into a table, rather like the periodic table of elements, but mathematicians have found 26 exceptions. The biggest of these was dubbed "the Monster"--a giant snowflake in 196,884 dimensions. Ronan, who personally knows the individuals now working on this problem, reveals how the Monster was only dimly seen at first. As more and more mathematicians became involved, the Monster became clearer, and it was found to be not monstrous but a beautiful form that pointed out deep connections between symmetry, string theory, and the very fabric and form of the universe. This story of discovery involves extraordinary characters, and Mark Ronan brings these people to life, vividly recreating the growing excitement of what became the biggest joint project ever in the field of mathematics. Vibrantly written, Symmetry and the Monster is a must-read for all fans of popular science--and especially readers of such books as Fermat's Last Theorem.

Product Description

Mathematics is driven forward by the quest to solve a small number of major problems--the four most famous challenges being Fermat's Last Theorem, the Riemann Hypothesis, Poincare's Conjecture, and the quest for the "Monster" of Symmetry. Now, in an exciting, fast-paced historical narrative ranging across two centuries, Mark Ronan takes us on an exhilarating tour of this final mathematical quest.
Ronan describes how the quest to understand symmetry really began with the tragic young genius Evariste Galois, who died at the age of 20 in a duel. Galois, who spent the night before he died frantically scribbling his unpublished discoveries, used symmetry to understand algebraic equations, and he discovered that there were building blocks or "atoms of symmetry." Most of these building blocks fit into a table, rather like the periodic table of elements, but mathematicians have found 26 exceptions. The biggest of these was dubbed "the Monster"--a giant snowflake in 196,884 dimensions. Ronan, who personally knows the individuals now working on this problem, reveals how the Monster was only dimly seen at first. As more and more mathematicians became involved, the Monster became clearer, and it was found to be not monstrous but a beautiful form that pointed out deep connections between symmetry, string theory, and the very fabric and form of the universe.
This story of discovery involves extraordinary characters, and Mark Ronan brings these people to life, vividly recreating the growing excitement of what became the biggest joint project ever in the field of mathematics. Vibrantly written, Symmetry and the Monster is a must-read for all fans of popular science--and especially readers of such books as Fermat's Last Theorem.

Anecdotes and soft math *****
Full of stories and simplified explanations of very deep material, this is one of the best math books I have read. One needn't be a professional mathematician to enjoy or understand it.

Anecdotes and soft math *****

Full of stories and simplified explanations of very deep material, this is one of the best math books I have read. One needn't be a professional mathematician to enjoy or understand it.

The Monster at the End of the Book ****
While it is simple enough to conceive an object in one, two or three dimensions, adding just one more dimension can be mind-bending. The four dimensional cube - or tesseract - cannot be truly perceived, but we can at least get a glimmer of it when we look at its projection, which appears like a cube within a cube. Five dimensions are even harder to perceive. The Monster, the subject of Mark Ronan's Symmetry and the Monster, has 196,884 dimensions. It seems appropriately named. What is the Monster, however? This takes a while to describe, and it all begins with the brilliant Galois, a mathematical genius who would be dead by 20 after being on the losing side in a duel. Galois would make some major strides in the field of algebra known as group theory. A group is really just a self-contained set of numbers (or other components) with an operation (such as addition) and certain properties (such as closure, the idea that when you do the operation on two members of the set, you get another member of the set; for example, with the whole numbers and addition, adding any two positive integers gets you another positive integer). Groups can be both finite and infinite, and among finite groups, there are so-called simple groups (or what Ronan calls atoms of symmetry). These are not simple as in easy, but simple as they cannot be deconstructed into simpler groups, just as when you factor a number, you cannot factor any further when you reach the prime factors. Most simple groups fit into certain families, but there also 26 exceptional groups (or sporadic groups). Determining that the number was 26 and finding all these groups is what Symmetry and the Monster is all about. The final group would be the biggest, by far: the Monster. Perhaps the best book dealing with the solution of a tough problem is Simon Singh's Fermat's Enigma, dealing with the proof of Fermat's Last Theorem. Ronan's book is not as easy of a read, but then again, he has a tougher row to hoe: while Fermat's Last Theorem is relatively easy to understand (though difficult to prove), the concept of symmetry groups is a bit more esoteric. Operating within this constraint, Ronan does a good job, writing clearly, with both a sense of history and sense of humor. This is not an easy subject to really grasp, but it may be ultimately rewarding to those who stick with it.

The Monster at the End of the Book ****

While it is simple enough to conceive an object in one, two or three dimensions, adding just one more dimension can be mind-bending. The four dimensional cube - or tesseract - cannot be truly perceived, but we can at least get a glimmer of it when we look at its projection, which appears like a cube within a cube. Five dimensions are even harder to perceive. The Monster, the subject of Mark Ronan's Symmetry and the Monster, has 196,884 dimensions. It seems appropriately named.

What is the Monster, however? This takes a while to describe, and it all begins with the brilliant Galois, a mathematical genius who would be dead by 20 after being on the losing side in a duel. Galois would make some major strides in the field of algebra known as group theory. A group is really just a self-contained set of numbers (or other components) with an operation (such as addition) and certain properties (such as closure, the idea that when you do the operation on two members of the set, you get another member of the set; for example, with the whole numbers and addition, adding any two positive integers gets you another positive integer).

Groups can be both finite and infinite, and among finite groups, there are so-called simple groups (or what Ronan calls atoms of symmetry). These are not simple as in easy, but simple as they cannot be deconstructed into simpler groups, just as when you factor a number, you cannot factor any further when you reach the prime factors. Most simple groups fit into certain families, but there also 26 exceptional groups (or sporadic groups). Determining that the number was 26 and finding all these groups is what Symmetry and the Monster is all about. The final group would be the biggest, by far: the Monster.

Perhaps the best book dealing with the solution of a tough problem is Simon Singh's Fermat's Enigma, dealing with the proof of Fermat's Last Theorem. Ronan's book is not as easy of a read, but then again, he has a tougher row to hoe: while Fermat's Last Theorem is relatively easy to understand (though difficult to prove), the concept of symmetry groups is a bit more esoteric. Operating within this constraint, Ronan does a good job, writing clearly, with both a sense of history and sense of humor. This is not an easy subject to really grasp, but it may be ultimately rewarding to those who stick with it.

Symmetry and the Monster ****
Interesting reading. The description of the lives of the mathematicians who contributed to the development of group theory helps create a basis of understanding for the topic. As a student, I wanted a different point of view than that of the text, and this book has done just that.

Symmetry and the Monster ****

Interesting reading. The description of the lives of the mathematicians who contributed to the development of group theory helps create a basis of understanding for the topic. As a student, I wanted a different point of view than that of the text, and this book has done just that.

Good but not enough example to illustrate ideas ***
This is an interesting read in general but the author doesn't include enough examples to illustrate idea (e.g. some graphical examples of different rigid geometric transformaion of solids will be great at the beginning of the book). The author also introduce mathematical concepts without enough explanation. While some of the concepts are simple enough to be understood without clarification, some of the more complicated ones in the later chapters are not. So the readers who are not already familiar with the subjects might find it difficult to follow the author's arguments.

Good but not enough example to illustrate ideas ***

This is an interesting read in general but the author doesn't include enough examples to illustrate idea (e.g. some graphical examples of different rigid geometric transformaion of solids will be great at the beginning of the book). The author also introduce mathematical concepts without enough explanation. While some of the concepts are simple enough to be understood without clarification, some of the more complicated ones in the later chapters are not. So the readers who are not already familiar with the subjects might find it difficult to follow the author's arguments.

Slightly too dumbed-down ****
According to the blurb on the back, the American Mathematical Monthly described this book as "truly a page-turner". I have to say it is not. Mark Ronan's task is to take us through the history of group theory culminating in the recently-completed project to classify the finite simple groups. This has taken decades of work by large numbers of highly-skilled mathematicians, with proofs so long and abstruse that there is a genuine concern that no future generation of mathematicians will be able to comprehend them. How do you communicate this to a lay audience? The key decision for the writer is to gauge his audience. Ronan's view is a readership which knows no group theory. He therefore can't even define a simple group: "a simple group is a group which is not the trivial group and whose only normal subgroups are the trivial group and the group itself" - Wikipedia. The reader, lacking help in engaging with the subject matter, is instead entertained by concise and amusing mini-biographies and anecdotes about the many participants in the quest. Ronan is a little dry as a writer, but in general this works well enough, although he is too indulgent of such monstrous personages as Sophus Lie. The final milestone in the classification project was confirmation of discovery of the mathematical Monster, the largest of the 26 sporadic groups. This was big news even on conventional news outlets, such as the BBC. In conclusion, this book will work for mathematicians who know some group theory and who like the historical context spelled out. I don't think many people not educated in mathematics will make it through to the end. With this in mind, Ronan could have profitably added a chapter at the beginning (or even an appendix) where he took the reader through normal subgroups, quotient groups and on to simple groups. He would then have been able to use correct terminology (his own merely irritates) and the journey would have been a lot more satisfying. Perhaps for the second edition?

Slightly too dumbed-down ****

According to the blurb on the back, the American Mathematical Monthly described this book as "truly a page-turner". I have to say it is not.

Mark Ronan's task is to take us through the history of group theory culminating in the recently-completed project to classify the finite simple groups. This has taken decades of work by large numbers of highly-skilled mathematicians, with proofs so long and abstruse that there is a genuine concern that no future generation of mathematicians will be able to comprehend them.

How do you communicate this to a lay audience? The key decision for the writer is to gauge his audience. Ronan's view is a readership which knows no group theory. He therefore can't even define a simple group: "a simple group is a group which is not the trivial group and whose only normal subgroups are the trivial group and the group itself" - Wikipedia.

The reader, lacking help in engaging with the subject matter, is instead entertained by concise and amusing mini-biographies and anecdotes about the many participants in the quest. Ronan is a little dry as a writer, but in general this works well enough, although he is too indulgent of such monstrous personages as Sophus Lie. The final milestone in the classification project was confirmation of discovery of the mathematical Monster, the largest of the 26 sporadic groups. This was big news even on conventional news outlets, such as the BBC.

In conclusion, this book will work for mathematicians who know some group theory and who like the historical context spelled out. I don't think many people not educated in mathematics will make it through to the end. With this in mind, Ronan could have profitably added a chapter at the beginning (or even an appendix) where he took the reader through normal subgroups, quotient groups and on to simple groups. He would then have been able to use correct terminology (his own merely irritates) and the journey would have been a lot more satisfying. Perhaps for the second edition?

Too simplified? **
Reading this book left me somewhat frustrated, as the need for popularisation of a very difficult subject leads the author to cover the maths too lightly, in my view. The maths in the book is very easy to follow but unfortunately the result is that you get very little insight into what this is really about. However, there are some very nice and elaborate descriptions of historic events and past mathematicians, so basically I think this book will cater for two types of readers: 1) Mathematicians with a good insight into group theory who know what this is all about and want a quick and entertaining recap of the history of the field, and 2) people who don't care about the maths, but enjoy the history of how scientists discover stuff. For the rest of us, who are after popular presentations of real science, I think there are much better alternatives.

Too simplified? **

Reading this book left me somewhat frustrated, as the need for popularisation of a very difficult subject leads the author to cover the maths too lightly, in my view. The maths in the book is very easy to follow but unfortunately the result is that you get very little insight into what this is really about. However, there are some very nice and elaborate descriptions of historic events and past mathematicians, so basically I think this book will cater for two types of readers: 1) Mathematicians with a good insight into group theory who know what this is all about and want a quick and entertaining recap of the history of the field, and 2) people who don't care about the maths, but enjoy the history of how scientists discover stuff. For the rest of us, who are after popular presentations of real science, I think there are much better alternatives.

Aptly titled *****
Popularisations of mathematics are difficult to do well because you need to have a fair amount of the language of maths under your belt before you can follow the arguments. To that end, putting across the ideas in a non-technical manner needs a skill that few possess. Ronan does a sparkling job here. The basic concepts of group theory are glossed over without going into tedious detail (and despite my affection for this particular branch of maths, I consider a lot of the detail extremely tedious), and once the story gets under way, the ideas are brought forward in a flowing, almost breathlessly excited, style which is infectious. The author himself was involved in this stupendous quest of classification, so he knows what he's talking about. One of the aspects of such a popular account is the bringing to life of the people behind the name, many of whom I'd never heard, quite a few of whom I'd already encountered in my travels through an undergrad degree in mathematics. Neither does the author shrink from confronting the political circumstances in which certain of the mathematicians were working, which adds a further dimension of interest to the tale. The first thing one wants to do having read this book is to go and find out the mathematics behind it all. Be warned: it is difficult area to get to grips with. The basics are simple but the detail is diabolical.

Aptly titled *****

Popularisations of mathematics are difficult to do well because you need to have a fair amount of the language of maths under your belt before you can follow the arguments. To that end, putting across the ideas in a non-technical manner needs a skill that few possess.

Ronan does a sparkling job here. The basic concepts of group theory are glossed over without going into tedious detail (and despite my affection for this particular branch of maths, I consider a lot of the detail *extremely* tedious), and once the story gets under way, the ideas are brought forward in a flowing, almost breathlessly excited, style which is infectious.

The author himself was involved in this stupendous quest of classification, so he knows what he's talking about.

One of the aspects of such a popular account is the bringing to life of the people behind the name, many of whom I'd never heard, quite a few of whom I'd already encountered in my travels through an undergrad degree in mathematics. Neither does the author shrink from confronting the political circumstances in which certain of the mathematicians were working, which adds a further dimension of interest to the tale.

The first thing one wants to do having read this book is to go and find out the mathematics behind it all. Be warned: it is difficult area to get to grips with. The basics are simple but the detail is diabolical.

Beauty and the Beast *****
They always say "mathematics is not a spectator sport", and this is true, but in reading this book, I felt I was as close as possible to being an enthralled spectator at a great game that I would have little or no idea how to play myself. I almost literally couldn't put the book down. The author takes great pains to make the subject as simple as possible, but not simpler (to echo the quotation from Einstein which heads the penultimate chapter). Such a compromise cannot be perfect, and as someone who knows a little maths, I found it mildly irritating to have to replace each occurrence of the phrase "atom of symmetry" with "simple group", rather than have the metaphor explained once, and the correct term used thereafter. On the other hand, I was quite happy to keep reading about "cross-sections", rather than have to keep stumbling over the phrase "involution centralizer" and be thereby reminded of how little I know about group theory! If I would take issue with any of the author's choices of vocabulary, it is his use of the term "deconstruction" instead of "decomposition", which is an equally familiar word, with one less letter (but one more syllable), and it is one which doesn't cause the susceptible reader to imagine that Jacques Derrida was somehow involved in the project. (God forbid!) Each reader, at whatever level of knowledge, will have his or her own preference as to the appropriate amount of technical vocabulary to use, and the author has clearly struck a considered balance in this respect. The result is, I think, an easy read even for complete non-mathematicians, but which still contains plenty to fascinate even the professional who is not a specialist in this most specialised of areas. It is, indeed, a specialised field; the classification of finite simple groups is not your average piece of mathematical research. I was already aware of the length of the famous (notorious?) Feit-Thompson "odd order paper", but was not aware that, at 255 pages (occupying an entire issue of a mathematical journal), this was a mere bagatelle compared to some of the prodigiously (monstrously?) long papers and typescripts (some not yet published, and never likely to be) which played an essential role in this heroic project. One shivers when one reads of the fears of those involved that the dismayingly formidable techniques required for this area of mathematics - and apparently for it alone - would not be passed on orally to future generations, and the understanding of them would be lost, like that of hieroglyphics. (A second vast project, the Revision of the Classification, is still underway to try to ensure that this does not happen.) But often the best way to understand something is to look at extreme cases, and I think it is no accident that this is the best popularisation of mathematics that I have read. Something of the soul of mathematics is laid bare here.

Beauty and the Beast *****

They always say "mathematics is not a spectator sport", and this is true, but in reading this book, I felt I was as close as possible to being an enthralled spectator at a great game that I would have little or no idea how to play myself. I almost literally couldn't put the book down.

The author takes great pains to make the subject as simple as possible, but not simpler (to echo the quotation from Einstein which heads the penultimate chapter). Such a compromise cannot be perfect, and as someone who knows a little maths, I found it mildly irritating to have to replace each occurrence of the phrase "atom of symmetry" with "simple group", rather than have the metaphor explained once, and the correct term used thereafter. On the other hand, I was quite happy to keep reading about "cross-sections", rather than have to keep stumbling over the phrase "involution centralizer" and be thereby reminded of how little I know about group theory! If I would take issue with any of the author's choices of vocabulary, it is his use of the term "deconstruction" instead of "decomposition", which is an equally familiar word, with one less letter (but one more syllable), and it is one which doesn't cause the susceptible reader to imagine that Jacques Derrida was somehow involved in the project. (God forbid!)

Each reader, at whatever level of knowledge, will have his or her own preference as to the appropriate amount of technical vocabulary to use, and the author has clearly struck a considered balance in this respect. The result is, I think, an easy read even for complete non-mathematicians, but which still contains plenty to fascinate even the professional who is not a specialist in this most specialised of areas.

It is, indeed, a specialised field; the classification of finite simple groups is not your average piece of mathematical research. I was already aware of the length of the famous (notorious?) Feit-Thompson "odd order paper", but was not aware that, at 255 pages (occupying an entire issue of a mathematical journal), this was a mere bagatelle compared to some of the prodigiously (monstrously?) long papers and typescripts (some not yet published, and never likely to be) which played an essential role in this heroic project. One shivers when one reads of the fears of those involved that the dismayingly formidable techniques required for this area of mathematics - and apparently for it alone - would not be passed on orally to future generations, and the understanding of them would be lost, like that of hieroglyphics. (A second vast project, the Revision of the Classification, is still underway to try to ensure that this does not happen.)

But often the best way to understand something is to look at extreme cases, and I think it is no accident that this is the best popularisation of mathematics that I have read. Something of the soul of mathematics is laid bare here.