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

Avner Ash (Author), Robert Gross (Author)

Book Description

ISBN-10: 0691138710 | ISBN-13: 978-0691138718 | Publication Date: August 4, 2008 | Edition: New

Mathematicians solve equations, or try to. But sometimes the solutions are not as interesting as the beautiful symmetric patterns that lead to them. Written in a friendly style for a general audience, Fearless Symmetry is the first popular math book to discuss these elegant and mysterious patterns and the ingenious techniques mathematicians use to uncover them.

Hidden symmetries were first discovered nearly two hundred years ago by French mathematician Évariste Galois. They have been used extensively in the oldest and largest branch of mathematics--number theory--for such diverse applications as acoustics, radar, and codes and ciphers. They have also been employed in the study of Fibonacci numbers and to attack well-known problems such as Fermat's Last Theorem, Pythagorean Triples, and the ever-elusive Riemann Hypothesis. Mathematicians are still devising techniques for teasing out these mysterious patterns, and their uses are limited only by the imagination.

The first popular book to address representation theory and reciprocity laws, Fearless Symmetry focuses on how mathematicians solve equations and prove theorems. It discusses rules of math and why they are just as important as those in any games one might play. The book starts with basic properties of integers and permutations and reaches current research in number theory. Along the way, it takes delightful historical and philosophical digressions. Required reading for all math buffs, the book will appeal to anyone curious about popular mathematics and its myriad contributions to everyday life.

Editorial Reviews

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

Review

All too often, abstract mathematics, one of the most beautiful of human intellectual creations, is ground into the dry dust of drills and proofs. Useful, yes; exciting, no. Avner Ash and Robert Gross have done something different--by focusing on the ideas that modern mathematicians actually care about. Fearless Symmetry is a book about detecting hidden patterns, about finding definitions that clarify, about the study of numbers that has entranced some of our great thinkers for thousands of years. It is a book that takes on number theory in a way that a nonmathematician can follow-systematically but without a barrage of technicalities. Ash and Gross are two terrific guides who take the reader, scientist or layman, on a wonderful hike through concepts that matter, culminating in the extraordinary peaks that surround the irresistible, beckoning claim of Fermat's Last Theorem.
(Peter Galison, Harvard University ) --This text refers to an out of print or unavailable edition of this title.

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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 x 6 x 0.9 inches
• Shipping Weight: 1 pounds (View shipping rates and policies)
• Average Customer Review: 4.2 out of 5 stars  See all reviews (19 customer reviews)
• Amazon Best Sellers Rank: #864,233 in Books (See Top 100 in Books)

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Most Helpful Customer Reviews

72 of 77 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
(VINE VOICE)    (TOP 500 REVIEWER)   

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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

51 of 54 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.

33 of 35 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".