The Equation that Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry

The astrophysicist and popular science writer Mario Livio is clearly not afraid of long titles and subtitles. His previous books include The Golden Ratio: The Story of Phi, the World's Most Astonishing Number and The Accelerating Universe: Infinite Expansion, the Cosmological Constant, and the Beauty of the Cosmos and both books take a single topic and analyze it from many different angles. Livio's new book, The Equation that Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry continues in this niche, both in its format and in its unwieldy title. The book is perhaps best viewed as a series of disjoint chapters, all of which are related to the overall ideas of group theory, but presented without a coherent narrative.

The first chapter is entitled "Symmetry" and covers much of the same ground as Hermann Weyl's classic book of the same name, albeit with many of the references updated from the 1950's to include Andy Warhol, Woody Allen, and poker, among others. This chapter and the next discuss examples of symmetry in nature and art and why it is aesthetically as well as biologically important to us. In the second chapter, Livio also introduces the notion of a group and some nice examples of symmetry groups. He then uses this terminology to help discuss many of the symmetry phenomena which he has already described.

After these two introductory chapters, the tone of the book changes entirely as Livio moves into discussing some aspects of the history of mathematics for three chapters. The first of these three chapters starts as close to the beginning of mathematics as one gets, in the Sumerian communities of the fourth millennium BC. Livio then discusses the emergence of equations as a mathematical tool, and in just over a dozen pages takes the reader through the introduction of algebra and up to attempts to solve the cubic equation.

The story of Fiore and Tartaglia, two mathematicians who lived in the sixteenth century and who had public 'duels' to see who was better at solving cubic equations, is a great story, and one that never fails to captivate students to whom I have told it. I won't spoil the ending for those of you who have never heard these tales, but this chapter alone is worth the price of admission to Livio's book.

After discussing cubic equations (and quartic equations, which make a surprise cameo appearance in the above tale), the natural question to consider is the quintic equation, and how to find a solution to a general fifth degree polynomial equation. The title of Livio's book gives away the punchline which is almost certainly familiar to any mathematician — that there is no way to solve a general quintic equation and that many such equations have no easily expressible solution — but it is a very interesting tale of how this punchline was reached by different mathematicians.

The fourth and fifth chapters of Livio's book tell the stories of Niels Abel and Evariste Galois, two of the mathematicians who tackled the problem of solving the quintic but whose work on this problem led to much deeper ideas and to the creation of entirely new fields of mathematics. The two stories have a decent amount in common — both men lived in the early part of the nineteenth century, both men died young, both led tragic lives which would make good made-for-television movies, and both made great contributions to mathematics. Livio tells the stories of Abel, the "poverty-stricken mathematician", and Galois, the "romantic mathematician" in a very interesting way, even for a reader who already knows the stories, and includes many nice photographs and reproductions of historical documents such as the mathematicians' manuscripts.

Of course, the one detail that many people know about Galois's life is that he died in a duel — one with pistols rather than cubic equations. Depending on whom you ask you may hear that he was killed over a woman or that he was killed for political reasons or for any number of other reasons. Livio details all of these 'conspiracy theories' and devotes an entire section of the chapter on Galois to a Patricia Cornwall-esque attempt to unravel the mysteries surrounding Galois's death. His conclusions, not to mention his presentation of the 'evidence' are very compelling and offer a take on the story that I had not previously seen.

After spending over a hundred pages looking like a history book, the book then does another 180 and once again looks like a math book in a chapter entitled simply "Groups." Luckily for the reader, Livio is just as good at writing in this manner. He begins by expanding his earlier notions of symmetry groups by looking at the permutations of the letters in the word GALOIS, and before you know it has discussed Rubik's cubes, the draft lottery for the Vietnam War, Levi-Strauss's Elementary Structures of Kinship, and non-euclidean geometry, all in the name of developing some of the key ideas in group theory. Most impressively, Livio gives a very readable two page summary of Galois's proof of the insolvability of the quintic. Livio does not give a rigorous treatment of group theory by any stretch of the imagination — and there are ways that I think he could have approached the material in a better way, such as including the concept and language of group actions — but he does a good job at giving a flavor of the kinds of results and ideas in the field that will hopefully inspire the non-mathematician reader to go sign up for the nearest course in abstract algebra.

The seventh chapter changes flavors once again, and focuses primarily on ways in which principles of symmetry show up in physics, ranging from Newton's study of the dynamics of celestial bodies to more modern ideas such as supersymmetry and Lie groups. The eighth chapter brings us full circle, as Livio once again discusses snapshots of symmetry in fields such as cognitive science and biology. Livio also once again digresses into some high-powered mathematics, finishing the main body of the book with a discussion of the so-called Monster group, and discussing the classification of finite simple groups.

The book concludes with a chapter entitled "Requiem for a Romantic Genius" in which Livio tackles the very notions of creativity and genius, and makes a compelling argument that Galois should be considered one of the most creative and influential mathematicians throughout history, an argument that this reviewer would certainly agree with.

As I have mentioned several times, Livio is a very engaging writer. The Equation that Couldn't Be Solved is a very well written book about very interesting subject matter. Livio supplements his words with many great illustrations, and also throws in pop culture references ranging from Shakespeare to the John Cusack film Serendipity.

Personally, I found the lack of a cohesive narrative and the scattershot nature of the different chapters to be a bit disconcerting, and I would have preferred a book that seemed a bit more intentionally put together. On the other hand, I imagine that many readers will enjoy reading the different views on the material, and the fact that much of the book is not very technical might trick a few civilians (by which I mean non-mathematicians) into learning some of the ideas of group theory and seeing the beauty of abstract mathematics.

In the end, the misgivings I have are far outweighed by the quality of the book. As a mathematician, I did not learn much in the way of mathematics from reading this book, but I did learn quite a bit of history and I enjoyed reading the exposition of the math and physics. I also think that The Equation that Could Not Be Solved would be an excellent book for a student to pick up to get drawn into the world of abstract algebra, and I have already been recommending the book to friends in the other science departments on campus. This is the kind of book, and Livio is the kind of author, that will convince the kind of scientifically-minded people who read magazines like Seed and Discover (the latter of which put Livio's book on their list of the Best Science Books of 2005) that mathematics in general, and group theory in particular, is an exciting and relevant field.

Darren Glass is an Assistant Professor at Gettysburg College. His mathematical interests include algebraic geometry, number theory, and (wait for it) Galois Theory. He can be reached at dglass@gettysburg.edu .