Why Beauty Is Truth: A History of Symmetry | Ian Stewart | A Walking Tour of Group Theory in Math...

Why Beauty Is Truth: A History of Symmetry | Ian Stewart | Any collection strong in science needs this discussion.

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Why Beauty Is Trut...

	Why Beauty Is Truth: A History of Symmetry Ian Stewart Perseus Books Group, 2007 - 290 pages average customer review:based on 17 reviews view larger image
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highly recommended

A Walking Tour of Group Theory in Math and Physics

"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.

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Any collection strong in science needs this discussion.

Ian Stewart's WHY BEAUTY IS TRUTH: A HISTORY OF SYMMETRY discusses the history and nature of the concept of symmetry, from early times to the latest physics involvement. This analysis comes from a famous mathematician who charts the discoveries and investigative processes of fellow scientists and geniuses as they developed the concept of symmetry and drew important connections between its science and nature. Any collection strong in science needs this discussion.

Very biographical

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.

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group theory

good book, but there is little original in his presentation that is not available from other recent sources. I agree with other viewers that what is needed is a good book written as well as this on the subject of group theory in relation to particle physics, nuts and bolts of applied symmetry. Nothing on the market that I know of. Any suggestions?

Dissapointed

This book had a wonderful review in Scientific American.

I am a Chemist with a fair amount of math. The major reason I was dissapointed is basically I did not learn anything mathematical. There were some fascinating biographies of physicists and mathematicians. I am not saying I did not learn anything because I know it all already. When there was a subject introduced that it did not know, it was introduced using analogies that really stretched what was going on - like building a multistory building.

It was a good read for the personalites involved, but really not a place to learn anything.

I would suggest the classic "Chemical Applications of Group Theory" for those really wanting to learn something.

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reviews: 1, 2, page 3, 4

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