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Why Beauty Is Truth: The History of Symmetry [Hardcover]

Ian Stewart
5.0 out of 5 stars  See all reviews (1 customer review)
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Book Description

April 10 2007 046508236X 978-0465082360
At the heart of relativity theory, quantum mechanics, string theory, and much of modern cosmology lies one concept: symmetry. 'In Why Beauty Is Truth', world-famous mathematician Ian Stewart narrates the history of the emergence of this remarkable area of study. Stewart introduces us to such characters as the Renaissance Italian genius, rogue, scholar, and gambler Girolamo Cardano, who stole the modern method of solving cubic equations and published it in the first important book on algebra, and the young revolutionary Evariste Galois, who refashioned the whole of mathematics and founded the field of group theory only to die in a pointless duel over a woman before his work was published. Stewart also explores the strange numerology of real mathematics, in which particular numbers have unique and unpredictable properties related to symmetry. He shows how Wilhelm Killing discovered "Lie groups" with 14, 52, 78, 133, and 248 dimensions-groups whose very existence is a profound puzzle. Finally, Stewart describes the world beyond superstrings: the "octonionic" symmetries that may explain the very existence of the universe.

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From Publishers Weekly

Anyone who thinks math is dull will be delightfully surprised by this history of the concept of symmetry. Stewart, a professor of mathematics at the University of Warwick (Does God Play Dice?), presents a time line of discovery that begins in ancient Babylon and travels forward to today's cutting-edge theoretical physics. He defines basic symmetry as a transformation, "a way to move an object" that leaves the object essentially unchanged in appearance. And while the math behind symmetry is important, the heart of this history lies in its characters, from a hypothetical Babylonian scribe with a serious case of math anxiety, through Évariste Galois (inventor of "group theory"), killed at 21 in a duel, and William Hamilton, whose eureka moment came in "a flash of intuition that caused him to vandalize a bridge," to Albert Einstein and the quantum physicists who used group theory and symmetry to describe the universe. Stewart does use equations, but nothing too scary; a suggested reading list is offered for more rigorous details. Stewart does a fine job of balancing history and mathematical theory in a book as easy to enjoy as it is to understand.Line drawings. (Apr.)
Copyright © Reed Business Information, a division of Reed Elsevier Inc. All rights reserved.

From Booklist

*Starred Review* Werner Heisenberg recognized the numerical harmonies at the heart of the universe: "I am strongly attracted by the simplicity and beauty of the mathematical schemes which nature presents us." An accomplished mathematician, Stewart here delves into these harmonies as he explores the way that the search for symmetry has revolutionized science. Beginning with the early struggles of the Babylonians to solve quadratics, Stewart guides his readers through the often-tangled history of symmetry, illuminating for nonspecialists how a concept easily recognized in geometry acquired new meanings in algebra. Embedded in a narrative that piquantly contrasts the clean elegance of mathematical theory with the messy lives of gambling, cheating, and dueling mathematicians, the principles of symmetry emerge in radiant clarity. Readers contemplate in particular how the daunting algebra of quintics finally opened a conceptual door for Evaniste Galois, the French genius who laid the foundations for group theory, so empowering scientists with a new calculus of symmetry. Readers will marvel at how much this calculus has done to advance research in quantum mechanics, relativity, and cosmology, even inspiring hope that the supersymmetries of string theory will combine all of astrophysics into one elegant paradigm. An exciting foray for any armchair physicist! Bryce Christensen
Copyright © American Library Association. All rights reserved

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2 of 2 people found the following review helpful
5.0 out of 5 stars A Beautiful Book Jan. 29 2011
By Ian Robertson TOP 100 REVIEWER
Format:Paperback
'Why Beauty Is Truth' is a well written, well constructed history of symmetry. It is not, however, for readers without an interest in or reasonable facility for mathematics. Stewart does an excellent job of leading readers step by step and chronologically from the Babylonian's early work through to the mathematical underpinnings of quantum physics and string theory, using straightforward examples, as little jargon as possible, and a liberal sprinkling of helpful diagrams.
Despite the author's excellent skills, the content can be challenging, especially as it progresses from Euclidian geometry into quadratic equations, Lie groups, relativity and multiple dimensions.
Nonetheless, Stewart weaves a compelling story about how mathematics and symmetry have advanced our understanding of our world since the earliest days, and I found myself more and more engaged as the book progressed. Like many of the best stories, the interesting but factual early pages lay the foundation for the excellent middle and end of the book. In fact, I found the second half of the book reminded me of another gifted author, Brian Greene and his first two books, 'The Elegant Universe: Superstrings, Hidden Dimensions, and the Quest for the Ultimate Theory' and 'The Fabric of the Cosmos: Space, Time, and the Texture of Reality', though of course Greene writes from a combined mathematician's/physicist's perspective, while Stewart writes from a purely mathematical one. (Stewart is professor of mathematics at the University of Warwick; Greene a professor of mathematics and physics at Columbia University).
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Amazon.com: 4.0 out of 5 stars  27 reviews
81 of 84 people found the following review helpful
5.0 out of 5 stars A Walking Tour of Group Theory in Math and Physics May 21 2007
By Edward F. Strasser - Published on Amazon.com
Format:Hardcover
"Beauty" in Stewart's title refers to symmetry in mathematics and physics, and to the mathematical structures called groups, which express this symmetry. "Truth" refers to the fact that the fundamental laws of the universe are described by such symmetries.

Before Stewart goes into this, he builds up for about 100 pages, giving the historical background of the ideas leavened with some biographical sketches. Then he gives two simple examples which form a basis for going into the later topics. I can't match Stewart's simplicity in a brief review, but I hope I can give you an idea of the nature of the examples.

Symmetry here has a somewhat more general meaning than in ordinary language. Ordinarily we say that something is symmetric if it looks the same as its mirror reflection. It is often said that a starfish has "radial symmetry" because, if it is rotated by 72 degrees (1/5 of a circle), it still looks the same, right down to the legs pointing in the same directions. Stewart considers the rotations and reflections of an equilateral triangle and defines a sort of "multiplication" of these turnings. The turnings together with the "multiplication" have a structure known as a "group". (It is called "multiplication" because it follows the same rules as multiplication of numbers. Any set of things which follow these rules is a group.)

There is also purely mathematical symmetry. For example, suppose you have a formula containing 3 numbers. If you rearrange those numbers in any order and the value of the formula is still the same, that rearrangement is called a symmetry. Instead of preserving the shape of an object, it preserves the value of an expression. Stewart shows that there is a deep connection between this group and the triangle group: both have the same multiplication table.

From there, Stewart goes on to applications of groups, symmetry and connections, mostly in physics. Here, he can't go into as much detail because the mathematics is too advanced. Like others who write on Physics for a general audience, he gives an impression of what the physics is like. This is why I called it a "walking tour". Unlike many others, however, he makes it clear he's not telling the whole story. For example, when talking about the spin of a particle, authors often have a drawing of a ball with a curved arrow indicating a spinning motion. "The particles did not spin in space, like the Earth or a spinning top. They "spin" -- whatever that means -- in more exotic dimensions." Before I read this, I wasted a lot of time trying to figure out explanations while visualizing a spinning ball. Now I just understand that spin is an abstract property and I have a better feel for the character of the science. I think that many readers will have a clearer notion of Einstein's (and Riemann's) curved space than can be gotten from the misleading "rubber sheet geometry" analogy that is so popular with science writers.

As he gets into the physics, Stewart brings up a new type of mathematical object, the Lie groups. I have seen these a few times before with no understanding at all. I assumed that they involved some abstruse math that would require more work than I was willing to put out. But Stewart defines two of these groups, called O(2) and SO(2), and they turn out to be very like the triangle group. No one had been able to explain this for general readers before because no one was prepared to spend over a hundred pages working up to it.

There is a lot of good material in this book and none of it requires any knowledge of high school math, although a little algebra will enhance some people's appreciation.

At this point I have to mention that I have a Ph.D. in math, although not in areas related to group theory. Much of the material in this book is new to me. Over the past few decades I have spent considerable energy learning how the general public sees math and science and thinking of how to explain ideas in non-technical ways, so I am confidant when I say: Why Beauty Is Truth is an excellent book to give general readers a view of how the beauty of symmetry, expressed in the language of groups, has helped to shape modern physics.

Addendum: (This is strictly for people who want to think seriously about the math.) The "multiplication" I mentioned in the triangle example means one turning followed by another. Once you get to the definition in the book, you might like to do some calculations to verify that the turnings really do form a group, that the "multiplication" table is correct, and that the triangle group and the permutation group have the same table. (This kind of equivalence is very important in mathematics.) I don't recommend this for all readers, but for some it will give a real insight into how mathematicians work. I do recommend it very strongly for young readers who might like to major in math.
34 of 35 people found the following review helpful
5.0 out of 5 stars A well-written book for the non-specialist July 16 2007
By Bruce R. Gilson - Published on Amazon.com
Format:Hardcover
Some of the reviews of this book seem to feel it doesn't present enough group theory. I think they are looking for a more technical book than Stewart meant to write, and so they are downgrading the book for reasons that are not fair to the book.

I reviewed a book by Mario Livio called "The Equation that Couldn't Be Solved," and gave it 5 stars. After reading this book, I almost want to go back and lower my rating of Livio's book, but of course, I shouldn't do that just because a better book has come out since. Livio's book concentrates on a shorter timespan than this, but both feature the same things -- mathematicians' attempts to solve equations of higher and higher degrees, from quadratics to cubics to quartics, and failure to find a solution to the quintic, only to find (due to the work of Abel and Galois) that it couldn't be done; and Galois' invention of group theory to make his proof, followed by other mathematicians' revelation that group theory is just what the doctor ordered to explain symmetry.

Stewart's book goes further back in time than Livio's, and also devotes more space to the modern uses of symmetry in physics. So it puts everything in more context. And, simply put, Stewart is a captivating writer. I enjoyed Livio's book, but I could hardly put down Stewart's. This book gets a high 5-star rating from me.

But it IS a book for the non-specialist. It isn't a course in group theory, or the Galois theory of equations; it is an attempt to give a non-mathematician some idea of these subjects. It should not be rated on a set of criteria that ignore what Stewart was trying to do. The negative comments really are unjustified; but yes, I'll warn you away from this if you expect it to teach you all the group theory you'll need to do particle physics, or crystallography, or any of the subjects that depend on group theoretic concepts of symmetry these days.
19 of 19 people found the following review helpful
5.0 out of 5 stars Keats may have been right all along May 5 2007
By viktor_57 - Published on Amazon.com
Format:Hardcover
I have always enjoyed Professor Ian Stewart's works for general audiences, including "Letters to a Young Mathematician" and "Flatterland", among others. In "Why Beauty Is Truth: A History of Symmetry", Stewart continues to explain seemingly esoteric and difficult mathematical topics with a clarity and humanity that illuminate not only the topics themselves, but also the people who developed them and the importance of their work to us in the present day.

In his latest book, "Why Beauty Is Truth", Stewart recounts the history of a concept most of us understand intuitively, symmetry, by describing the lives of people who made important contributions to the mathematics of this seemingly simply concept which turned out to have extraordinary implications. From the development of ancient number systems and algebra to the discovery of Lie groups, Stewart explains the mathematics and concepts in an intuitive way, sprinkling in equations when necessary, but mostly relying on his ability to imagine how a non-mathematician might best understand even the most abstract concepts, whether by example, metaphor, or even some fictional drama.

Stewart is the rare mathematician who seems equally at home with the technical aspects of his subject and its history, including the biographies of those who made important contributions. Stewart is also a fine writer and enthusiastic popularizer, showing how the development of symmetry from the beginnings of counting has led to some of the most important developments in physics, including general relativity and string theory. Math and physics enthusiasts will undoubtedly enjoy "Why Beauty Is Truth", as will the curious lay reader who enjoys new discoveries and lively, engaging and intelligent writing.
17 of 18 people found the following review helpful
4.0 out of 5 stars Very biographical May 16 2007
By J. Jenkins - Published on Amazon.com
Format:Hardcover|Verified Purchase
I actually do agree with the terse reviewer below who mentions there is not enough on group theory. I think the reader or buyer should be forewarned that there is a lot of historical and biographical information in here, going back to the ancient greeks: Pythagoras, etc., traveling through history until string theory (on which subject the writer seems to be quite enthusiastic albeit fence-sitting). Throughout that excursion there is a lot of biographical information, some of it quite uninteresting and irrelevant (in my opinion!). Certainly some people might be interested to read the lives of the mathematicians but I was hoping for a book that dallied with the more philosophical implications of beauty and truth, mathematics and reality, such as for example Paul Davies does so well. As such I don't think he really explains why it might be correct to say that "beauty is truth." Nonetheless the book is really really well written and approachable, and Ian Stewart does a fantastic job of explaining complicated math concepts. Towards the end it feels like he is hurrying through some of the most interesting topics, such as how group theory applies to the standard model of quantum mechanics, which seems to be the most surprising or fascinating application of the concept of symmetry to reality. A few physicists get bunched together in the last couple of chapters where they might each have merited a chapter on their own.
30 of 34 people found the following review helpful
2.0 out of 5 stars Spreads itself too thinly Nov. 18 2007
By Israel Ramirez - Published on Amazon.com
Format:Hardcover
This book covers an enormous range of topics beginning with Mesopotamia number systems and ending with string theory. It simultaneously describes mathematical theories, the history of how these ideas evolved over time, and details about the lives of the mathematicians. Several of the brief biographies are very well done; the treatments of Gauss, Omar Khayyám, and Galois are outstanding. Others are sketchy, hardly more than a list of parent's occupations, siblings, spouse, and children. As a result of the broad coverage, each mathematical concept gets very brief treatment. I often felt that I wasn't given enough information to understand a concept. Lie groups, in particular, turn out to be very important for contemporary physics but the description is so brief and jargon encrusted that the physical applications were unintelligible to me.
The author is not certain about his intended audience. He apologizes to the reader for the complexity of the solution to the cubic equation, even though this is a straightforward extension of high school algebra. Yet later on he assumes that the reader will easily grasp that a Fano plane is a finite projective geometry. The book was simultaneously too nontechnical and too technical for me (a computer technologist and a former scientist).
It is not clear what the purpose of the book is. Many of the topics covered have no obvious connection to symmetry except in the sense that everything is related to symmetry. The historical evolution of representations of numbers is interesting, for example, but doesn't help understanding the multidimensional algebras that somehow relate to symmetry.
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