Fearless Symmetry: Exposing the Hidden Patterns of Numbers (New Edition) (Paperback)

by Avner Ash (Author), Robert Gross (Author) "Before we narrow our focus to mathematical concepts, we start by discussing the general concept of a representation..." (more)
Key Phrases: Fermat's Last Theorem, United States, North Pole (more...)

Editorial Reviews

Review
"It delivers a tour through basic number theory, Galois theory, and the rudiments of arithmetic geometry". -- Jordan Ellenberg, Seed, June/July 2006

"The view from the top is spectacular but from halfway up, it's pretty good too". -- Justin Mullins, New Scientist --This text refers to an out of print or unavailable edition of this title.

Review
The authors are to be admired for taking a very difficult topic and making it . . . more accessible than it was before.
(Timothy Gowers Nature )

The authors . . . outline current research in mathematics and tell why it should hold interest even for people outside scientific and technological fields.
(Science News )

The book . . . does a remarkable job in making the work it describes accessible to an audience without technical training in mathematics, while at the same time remaining faithful to the richness and power of this work. I recommend it to mathematicians and nonmathematicians alike with any interest in this subject.
(William M. McGovern SIAM Review )

Unique. . . . [T]his book is an amazing attempt to provide to a mathematically unsophisticated reader a realistic impression of the immense vitality of this area of mathematics.
(Lindsay N. Childs Mathematical Reviews )

To borrow one of the authors' favorite words, this book is an amazing attempt to provide to a mathematically unsophisticated reader a realistic impression of the immense vitality of this area of mathematics. But I think the book has another useful role. With a very broad brush, it paints a beautiful picture of one of the main themes of the Langlands program.
(Lindsay N. Childs MathSciNet )

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Product Details

• Paperback: 312 pages
• Publisher: Princeton University Press; New edition (August 4, 2008)
• Language: English
• ISBN-10: 0691138710
• ISBN-13: 978-0691138718
• Product Dimensions: 9.1 x 6 x 0.9 inches
• Shipping Weight: 1 pounds (View shipping rates and policies)
• Average Customer Review: 3.9 out of 5 stars See all reviews (17 customer reviews)
• Amazon.com Sales Rank: #446,576 in Books (See Bestsellers in Books)

Customer Reviews

Most Helpful Customer Reviews

64 of 68 people found the following review helpful:

5.0 out of 5 stars Excellent discussion of mathematical symmetry, June 5, 2006

By	Mike Birman (Brooklyn, New York USA) - See all my reviews (TOP 500 REVIEWER)

This review is from: Fearless Symmetry: Exposing the Hidden Patterns of Numbers (Hardcover)

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

38 of 40 people found the following review helpful:

3.0 out of 5 stars Good in part, December 19, 2006

By	Wolfgang Zernik (Doylestown, PA USA) - See all my reviews (REAL NAME)

This review is from: Fearless Symmetry: Exposing the Hidden Patterns of Numbers (Hardcover)

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.

23 of 25 people found the following review helpful:

3.0 out of 5 stars Too much, too fast, March 14, 2007

By	A reader (Shrewsbury, MA USA) - See all my reviews

This review is from: Fearless Symmetry: Exposing the Hidden Patterns of Numbers (Hardcover)

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".