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Why Beauty Is Truth: The History of Symmetry Hardcover – Apr 10 2007
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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.
*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
Top Customer Reviews
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.Read more ›
Most Helpful Customer Reviews on Amazon.com (beta)
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.
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.
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.
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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