This is a hack job by someone who is clearly interested in the subject matter, but handles it very unevenly. The book begins with the question of why symmetry is significant from an aesthetic perspective, but then gives a mathematical explanation of symmetry (primarily the contributions of group theory), mixed with some choppy, historically dubious biographies of the involved mathematicians that go far off track, and then makes a poor attempt to tie the book together in the end. ( )

Both times now that I've read a book by Mario Livio, I've thought that it was a great topic, but that someone else would have done a better job of writing about it. ( )

In Chapter One, Mario Livio promises to open our eyes to the magic of symmetry through the language of mathematics. To do so, he first acquaints us with group theory of modern algebra. A group is any collection of elements (they need not be numbers) that have the properties of (1) closure, (2) associativity, (3) an identity element, and (4) an inverse operation. The fact that this simple definition leads to a theory that unifies all symmetries amazes even mathematicians.

Livio give us a little of the history of algebra, beginning with the ancient Greeks and Hindus, who solved the general quadratic. The story of the solution of the general cubic is a fascinating one involving allegations of cheating and libel among 16th century Italian mathematicians. Moreover, the solution required the invention of imaginary numbers. Once the cubic was solved, the solution or the quadratic quickly followed, but the quintic remained a mystery.

Even Euler and Gauss were stumped by the quintic, and they began to think the problem was insoluble. In fact, the work of two very young mathematicians, Niels Henrik Abel and Evariste Galois, proved that there could be no general solution to the general form of the quintic equation. The solution proved to be a surprise in that it depended on the relations among the coefficients of the variables. Only those quintics with a proper symmetry among the coefficients can be solved by purely algebraic operations. Livio does not actually show why the previous statement is true, probably because it requires real math. Nevertheless, the conclusion is pretty startling even to a math tyro like me.

The book gets a bit bogged down in its biographical sections, devoting more time to Galois’s life than I found interesting. Nevertheless, it is worth reading.

(JAB) ( )

I read picked this book because I have since my early algebra days been interested in the quintic (e.g. x^5 + 2x^4 + … + 1). It presented a very good explanation of the history that led up to its ultimate proof that it’s impossible to solve in the general case using standard arithmetic operators and extraction of roots.

Although it covered that well, it kind of went off on many tangents to fields that sort of had to do with symmetry. Perhaps I should have got a book focused more — but all around it was interesting. ( )

The story of group theory (Abel, Galois, et al) -- another good pop-math book.