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Symmetry and the Monster: One of the greatest quests of mathematics [Paperback]

Mark Ronan
5.0 out of 5 stars  See all reviews (1 customer review)
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Book Description

Aug. 15 2007 0192807234 978-0192807236 1
Imagine a giant snowflake in 196,884 dimensions... This is the story of a mathematical quest that began two hundred years ago in revolutionary France, led to the biggest collaboration ever between mathematicians across the world, and revealed the 'Monster' - not monstrous at all, but a structure of exquisite beauty and complexity. Told here for the first time in accessible prose, it is a story that involves brilliant yet tragic characters, curious number 'coincidences' that led to breakthroughs in the mathematics of symmetry, and strange crystals that reach into many dimensions. And it is a story that is not yet over, for we have yet to understand the deep significance of the Monster - and its tantalizing hints of connections with the physical structure of spacetime. Once we understand the full nature of the Monster, we may well have revealed a whole new and deeper understanding of the nature of our Universe.

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`Review from previous edition 'Succeed[s] in bringing to the fore an aspect of mathematics that some popularizers miss - that math is not a science of monuments, but a living tradition as vibrant as physics or ethics or law, one in which new monuments pop up weekly... ...And readers of [Symmetry and the Monster] will know, at least in part, where it's happening now, and even (maybe) where it's going to happen next.'' iSeed Magazine/i Jun/Jul 2006

a fascinating book that will appeal to anyone with an appetite for exploration and discovery, and which is accessble to all.

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1 of 1 people found the following review helpful
5.0 out of 5 stars The insigth into symmetry April 9 2011
By Dr. Bojan Tunguz TOP 50 REVIEWER
Format:Paperback
One of the greatest achievements of the 20th century mathematics has been the classification of the finite simple groups. Groups are mathematical objects that tells us about symmetries, and like many other mathematical objects they are relatively easy to describe, but can be fiendishly difficult to fully understand. Sometimes understanding comes from a single brilliant insights by an incredibly gifted individual, and these individuals become part of the mathematical lore that can even touch upon the popular imagination. However, most of the time these days the game of mathematics has become complex enough that it can become increasingly difficult for any individual to fully contribute to on its own to the full problem. Professional mathematicians don't mind this at all: they thrive in collaborations and feed off of each other's work and enthusiasm. The collaborative nature of mathematics is at full display when it comes to the classification of finite simple groups, an effort that spanned hundreds of articles in scientific journals between 1955 and 1983. I have always been curious to find out more about this enterprise, and this book does a remarkable job at presenting it to the general reader. It is comprehensive without becoming technically hard to follow. Anyone who has ever taken a college level mathematics course should be able to read it without much difficulty, although some basic understanding of group theory and modern algebra would be great bonus. The book also doesn't dumb down mathematics to the point that it becomes irritating for those who have some mathematical sophistication, so even professional scientists and mathematicians can find it very informative and a rewarding read.

And if you are curious, the Monster from the title refers to the special simple finite group that has been one of the most fascinating mathematical objects discovered so far.
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Amazon.com: 3.6 out of 5 stars  14 reviews
14 of 15 people found the following review helpful
4.0 out of 5 stars Slightly too dumbed-down Oct. 2 2007
By Nigel Seel - Published on Amazon.com
Format:Paperback
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?
9 of 9 people found the following review helpful
4.0 out of 5 stars The Monster at the End of the Book March 4 2008
By mrliteral - Published on Amazon.com
Format:Paperback|Verified Purchase
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.
6 of 6 people found the following review helpful
5.0 out of 5 stars I couldn't put it down Nov. 6 2008
By Peter Haggstrom - Published on Amazon.com
Format:Paperback|Verified Purchase
This book is a ripping yarn - I couldn't put it down. My wife asked when I came to bed at midnight : "Maths porn again darling? " Although I have done some group theory, my knowledge was nowhere near enough to make any meaningful attempt to understand the detail of the Monster project and like many others, it remained an intractable beast that others were battling with. Ronan explains in a high level way the history to the Monster starting with Galois and working his way through the historical development. He peppers the account with all manner of interesting observations about the participants which are revealing in ways that one does not often find in maths books. For instance, there is a revealing comment on page 152 about someone of the stature of John H Conway who confessed he "felt like a fraud" in giving talks early on in his work on Monster. It seems a graduate student asked him the obvious question namely " How do you now that your new group can't be decomposed into something simpler?" Maths is an unforgiving business.

Mark Ronan who has worked with and/or knows most of the heavy hitters in the field has done a wonderful job explaining the history of what is an extraordinary undertaking not only in purely intellectual terms but also in personal terms. The sociological dimensions of this immense task are reflected in all manner of small and large stories. Thus John H Conway bargains with his wife to have blocks of time away from the 4 kids so he can crack some problems and he manages in 12  hours to prove something important about the Leech Lattice. That set him up for life. The proofs in this field can be hundreds of pages long - one by Mason is 800 pages long and has not been published. This itself imposes huge strains on referees. The classification task (which I had read about but had no detailed knowledge of what was involved other than a vague idea it was the equivalent of the 30 Years War) demonstrates what a small group of intensely committed people can do. What they were doing was to provide a set of knowledge that subsequent mathematicians could understand given that the barriers to entry to the detailed knowledge are so high.

At a purely personal level one has to marvel at how some of the people concerned threw their lot in with this "monstrous" task. Every budding PhD students knows that problem selection is important and it does not pay to spin one's wheels forever on some obscure problem.

There are some truly astonishing connections revealed in this book. The connection between the number theoretic j function and the character set of the Monster (see pages 192-193) is remarkable but then there is the even more remarkable connection between light rays and the Leech Lattice (see page 224).

Mark Ronan has done a great service to all those who have served and still served in the battle with the Monster. Most of the main workers in the field are no longer with us so Ronan's book provides the general community with some sense of their achievements.

For those interested in Lie Theory may I suggest John Stillwell's accessible book "Naïve Lie Theory" as a starting point. He strips a way a lot of the overheard that makes Lie Theory so daunting.

Peter Haggstrom
Bondi Beach
Sydney, Australia
7 of 8 people found the following review helpful
4.0 out of 5 stars Good, but read Fearless Symmetry first July 1 2009
By oldtaku - Published on Amazon.com
Format:Paperback|Verified Purchase
This takes you into the strange and fascinating realms of symmetry groups. Ronan does a good job at trying to take even the mathematically uneducated on this journey, but I am going to tell you this: if you don't know a little about symmetry groups going in you are going to be hopelessly lost by 1/3 of the way through the book. Oh, you can follow along and still be entertained by the anecdotes and stories, and somewhat grasp what's going on, but you will have completely lost track of exactly WHAT is being discussed, other than it involves extremely large groups of... things.

I suggest:
1) If you can handle Calculus or Godel, Escher, Bach: An Eternal Golden Braid, you can handle this subject. Group theory can be astoundingly complex (I certainly don't pretend to understand it all), but you can handle the depth presented here and in the book you should read first, which is...
2) Fearless Symmetry: Exposing the Hidden Patterns of Numbers (New Edition) - read this first. It will educate you on the basic math and what's going on (Galois groups and symmetry groups have a very precise meaning in mathematics). Ronan almost completely glosses over the details in the quest for accessibility, but without this it's a hollow pursuit, like a video game you can beat just by mashing the X button.
3) If you can't follow 'Fearless Symmetry' (or it bores you to tears), don't bother with 'Symmetry and the Monster'.
4) If you can follow and enjoy 'Fearless Symmetry, then 'Symmetry and the Monster' takes you on a quick tour of the 'here be dragons' areas of the map - it will be immeasurably more relevant and fascinating and a five star book.
4 of 5 people found the following review helpful
3.0 out of 5 stars Good but not enough example to illustrate ideas Feb. 6 2008
By Lawcool - Published on Amazon.com
Format:Paperback
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.
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