Customer Reviews

Fearless Symmetry: Exposing the Hidden Patterns of Numbers

5 star:		(11)
4 star:		(4)
3 star:		(2)
2 star:		(1)
1 star:		(1)

 

 

We may be entering a golden age of popularized mathematics literature. On the heels of John Derbyshire's recent superb book about Algebra, which alternates historical discussion with mathematical primers that illuminate rather than confuse, comes this excellent book that covers the fascinating topic of mathematical symmetry: especially Evariste Galois' final frenzied creation, Group Theory. From its birth to a productive maturity in Number Theory; where it has found extensive practical usage in acoustics, radar, codes and ciphers (and of course particle physics), Fearless Symmetry unfurls the threads of Galois Theory and follows their path through several branches of mathematics. It doesn't utilize Derbyshire's stark method of alternating chapters between history and mathematics. Rather, it enfolds the historical narrative into a clear presentation of the requisite mathematics.

Simplified abstraction, is probably the best explanation of the author's technique. Group Theory discussion leads to Andrew Wiles and Fermat's Last Theorem, Fibonacci numbers, Pythagorean Triples and the Riemann Hypothesis. In the process, Fearless Symmetry becomes the first popularized exposition of representation theory and reciprocity laws. It also discusses how mathematicians prove theorems and solve problems. The all-important rules of mathematics are also discussed. This is a wide-ranging work that manages to avoid the obfuscation often found in math books. A willingness to solve problems that are simply and clearly posed are all that's required from the reader. The authors even suggest that readers disinclined to solve problems can skip them. That would be a severe loss given the nature of this book. In any case, the problems are not difficult, offer instantaneous feedback as to the reader's understanding of the material and are an extension of the text. Mathematicians may enjoy the book for its elegance in uniting so many disparate topics. As a non-mathematician (a molecular biologist by training), I can attest to the clarity of the discussion. I found the book fascinating, truly informative, endlessly challenging to my own assumptions of the way math is done. If you don't mind some mathematics on the printed page, this book may provide several hours of sheer intelligent pleasure. Strongly recommended.

Mike Birman

This book, like the curate's egg, is good in part. The title for a start is somewhat misleading. The book is actually about Galois groups and how this theory lead to the recent (1995) proof of Fermat's last theorem. All of that is attempted "in a friendly style for a general audience", an ambitious undertaking to be sure. The technical level here is between that of John Derbyshire's "Unknown Quantity" which can be read by almost anyone and Emil Artin's "Galois Theory" which is unreadable unless you're in grad school.

I will list my own experience since it may be helpful to other potential readers. I read the first fifteen chapters with profit and great pleasure. This is about two thirds of the book and clearly merits five stars on its own. I particularly recommend the charming and humorous explanation of absolute Galois groups in chapter 8.

Unfortunately I finally got to chapter 16 (on Frobenius elements) and there my reading pleasure came to a screeching halt. I have tried several times but I just cannot understand this material. It doesn't help either that the authors seem at this point to have given up on their "friendly style for the general reader". They contradict themselves twice in one paraqraph ("Actually this is a lie.....In fact we just lied again") and refer to an appendix which is "probably opaque". Unfortunately the rest of the book appears to depend on chapter 16 so I was not able to read that either. Hence my final rating of three stars only.

The authors aim to reveal the role of Galois groups in modern number theory. "Groups" are mathematical descriptions of the patterns and symmetries of objects or mathematical equations. (Hence the reference to symmetry and patterns in the book's title.) Galois groups are used at the forefront of number theory research: they were used to prove Fermat's last theorem regarding Pythagoras's theorem of right triangles.

The book introduces the reader to the rudiments of groups, modular arithmetic, fields, varieties, the law of quadratic reciprocity, matrices, elliptic curves, and representations. Using these basic concepts, the authors show how the absolute Galois group of Qalg (the field of algebraic numbers) is used in contemporary research into the theory of polynomial equations.

Unfortunately this book is overambitious. It repeats the same mistake that every college professor makes: Typically, a professor belabors the rudiments of a subject; but then, as the course nears its midpoint, he realizes that he's running out of time, and the pace accelerates from a slow trot to a full gallop. Essential concepts are vaguely defined; e.g., Frobenius element (pp. 178-9) and unramified prime (p. 184). Towards the end of the book, concepts are introduced without definition; e.g., Fourier coefficients (p. 236), weight of a modular form (p. 246), conductor (p. 246), deformation (p. 252), complete noetherian local ring (p. 253). One of the phrases that's repeated most often is that some concept "is beyond the scope of this book."

Curious college grads -- even those with some exposure to abstract algebra -- will become hopelessly lost after Chapter 16. Graduate math students will be bored by the early chapters and will be left unsatisfied by the sketchy presentation of the later chapters. Grad' students should try Hellegouarch's "Invitation to the Mathematics of Fermat-Wiles".

In both the physics and mathematics community something very exciting is happening. Highly competent physicists and mathematicians have for the last six or seven years been writing books that give deep insight into the concepts and intuition behind their specialties. A voluminous literature of course exists that is written for the specialist in the field of relevance, and these are written in a high-level, formal style, and no motivation, either historically or technically is given. Those interested in entering the field will have to rely on getting verbal explanations from the researchers themselves, which may be difficult if they are not close to them in a geographical sense. This is also another reminder that there is definitely an oral tradition in mathematics, and experts it seems are reluctant to explain themselves to newcomers. Physicists are particularly sensitive to this state of affairs. They need to not only understand a large amount of material in physics, but they require deep insight into the mathematical tools that must be used to formulate their theories, and this insight must be obtained rather quickly. They do not have time to wait until the mathematical concepts "come to them."

This book gives a great deal of this insight in the field of Galois theory, the theory of equations, and algebraic number theory. But the reader also gets a taste of such esoteric topics such as etale cohomology and the proof of Fermat's Last Theorem. The authors pull all of this off in 267 pages, an amazing feat considering the nature of the subject matter. The book can be best appreciated by the advanced undergraduate student or graduate student of mathematics, but even professional mathematicians in other fields of mathematics will no doubt find the book helpful in introducing them to the subject. High-energy physicists will love the book, even the parts that are really a review of some elementary linear algebra.

The authors know when to stop when discussing a topic, so as to not lead the unprepared reader into a morass of highly technical argumentation. But they wet the reader's appetite enough to motivate them to consult the references for further reading. This book, and others like it thankfully are becoming more prevalent. Mathematicians are realizing that there is nothing wrong in engaging in a little hand waving in order to explain their ideas. This has enormous didactic power, and one can only imagine the ramifications of a large number of these kinds of books appearing in the next few years. With the deep insights they grant to aspiring mathematicians, this reviewer predicts an enormous explosion of new mathematics in the next decades, even greater than the current rate of progress, incredible as it is.

This is one of the finest books that lucidly and simply explain very complex concepts and terms in number theory, group theory and abstract algebra to anyone with reasonable mathematical maturity -- Easy to read, self contained and all topics are developed in simple illustrative manner without requiring dry proofs without sacrificing rigor.


Contrary to what the books front flap and back cover says, I think this books is much easier to understand if you at least have college calculus I and II. Not that anything in the book requires knowledge from those courses, but level of mathematical maturiy developed during those courses will make the book an easy read. However I commend Avner Ash and Robert Gross for ensuring that all terms are defined within the book and 'technically speaking' a high school graduate should be able to follow through the book, if he/she is a careful mathematical thinker.

If in school you never ended up taking abstract algebra or number theory and wished that you had taken those courses, but now consider these courses to be too hard to be studied in self teaching mode - READ this book. It is written for you.

In spite of some of the comments posted already and in spite of what is on the book's back cover - this is a math book - this is a serious math book. I personally don't see that average person getting anything out of this if they hadn't had say Linear Algebra in particular. Calculus is not required but higher alegra is.

The reason I bought this book is that I read Ian Stewert's book on Symmetry and Beauty and found it lacking as it was not very mathematical.
I was not dissapointed in the level of math in this book. If anything, I got overwhelmed by the end.

I call this type of book "drill deep" but not wide. I like that idea.

The author's have a real ambitious goal. It's laid out on pages 11 and 12:
"in this book we explore ..representations...we consider sets, groups, matrices and functions between them. We show you in detail in one particular case that we develop throughout the book that sets us to our goal: mod p linear representations of Galois groups."

THIS IS THE GOAL OF THIS BOOK. They are not kidding this is what the book sets out to do and I belive accomplishes.


The authors are true to this goal in the "drill deep" mode. Example: Chapter 2 is Groups - not everything about Group Theory is presented but enough that is needed for the rest of the book. In a similar manner one chapter is on so called reciprocity laws. Chapter 4 is on Modular Arithmetic a crucial aspect to this book.

One prior reviewer indicated that each chapter is far more difficult than the last; this is sortof the general tenure of the book - but with exceptions if you know that material. Example, Chapter 5, Complex Numbers, for me was a relief sandwiched in between Modular Artimetic and Equations and Varieties. I can attest that for the subject "Complex numbers" - that they treated it at a relativley elementary level and focused on just those aspects needed later on. I am sure that for all subjects like "Quadratric reciprocity" that was the case. However, if you hadn't been exposed to quadratic reciprocity and Legendre symbols it is a tough slog.

For me the high point of the book was Chapter 8, I felt that I understood the difficult concept of the the Absolute group of the field of algebraic numbers by the end of the chapter. It is an infinite group that only elements can really be enumerated - Identity and complex conjugation. It fills in some (but not all) of the points in the number line between the group of rational numbers and the line with no gaps the field of real numbers.

Chapters 13 to 22 my ability to follow went way downhill and I just skimmed to get some highpoints.

I might return to this book in the future. I like the idea of not having to learn every aspect of something like alebraic ring theory , then every aspect of permutation theory etc. but just learning enough to accomplish some higher level of understanding like ultimatley how Fermat's Last Therom was solved.

I would recomend Stwert's book on Symmetry and Beauty first if you feel you want a more general understanding of this subject as opposed to a real math book which this is.

This books starts off elementary beyond belief. This book begins with almost painfully long and thorough expositions of undergraduate mathematics, including linear algebra, group theory, and number theory. Suddenly the book takes a turn for the more complicated and quickly loses it's friendliness. While the material becomes more interesting and enjoyable, more details are left unmentioned, concepts left un-fleshed out, and examples are sparse. The material presented in this book was magnificently selected and the organization of the vast amount was admirable. However, the later chapters lacked in the quality of writing I was growing to appreciate from the two authors. I highly recommend any undergraduate mathematics or similar major to purchase this book. A semester of abstract will be useful, and a semester of number theory an added bonus. I would like to refute the claim that this book is enjoyable for non-math readers. Further, any math grad would be bored to tears by this book.

Fearless Symmetry is an engaging, entertaining, wonderful piece of literary mathematics. It gives a concise view of number theory without the dryness of a textbook. The authors do a wonderful job of explaining and solidifying the abstraction in mathematics, bringing the average non-mathematically minded reader into the realm of modern mathematics.
This book is a 'must read' for any undergraduate student of mathematics who may be experiencing difficulty understanding the basic concepts of modern algebra.

Galois Group is the most difficult but the most important theory in Modern Algebra, yet this book remarkably explains it in a very friendly, easy-to-follow manner for non-mathematicians.

The first few foundation chapters may be too simple for math students (Group, Matrices), the authors quickly accelerate in depth to Quadratic Reciprocity, Legendre symbol, Varieties, Elliptic Curves, Galois Group and representations, etc. The flow of the chapters are tightly linked, one hurdle cleared before leading to the next, the readers are well guided into the 'treasure' of algebra kingdom.

Except Chapter 16 & Part III, which are written not for the faint-hearted, the authors should have done a smoother transition from Part I & II to more difficult concepts leading to FLT proof. I agree the topics are not easy by nature, and the authors have good intention not to overwhelm the readers with details, but the cursory introduction of Frobenius elements, (un)ramified ('good' / 'bad') primes, Character, Modular forms, etc, are too abrupt jump for most readers.

However, Chap 22 on FLT (The 3 pieces of FLT proof) is excellent. I learnt that FLT (for n>2) is equivalent to proving FLT (for odd primes >= 5): by n|4 (which Fermat had proved for n=4 => n = 4Z); or n not |4 (=> n= odd primes, but n=3 had been proved by Gauss).

One previous reviewer predicted an enormous explosion of New Math in the next decades. I wish more popular math writers like Ash & Gross, Derbyshire, Ian Stewart, etc, will contribute to the proliferation of New Math literatures.

New Math needs not be kept in the 'ivory tower', it should be fun to read by every man (woman) on the street -- as David Hilbert had said.

Fearless symmetry is a beautiful piece of work by Avner Ash.
If you are interested in numbers and you have a reasonable knowledge of basic principles of math , you will enjoy this book and you are able to do some work for yourself at the excercises , which is still more enjoyable and will give you a deeper insight in the structure of numbers , without the need of being rigorous in theorems.
A lot of thanks to both the authors.

Hans Niekamp