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

I've earlier reviewed Livio's book on the Golden Ratio and if you'll read my review of that book, you'll see that I found it somewhat disappointing, though liking parts of it enough to give an overall 4-star rating. This book is much better. Basically, it addresses two topics: the attempts made over the ages to use formulas (such as the one we learned in algebra in school for the quadratic equation) to solve higher degree equations, which failed when algebraists got to the fifth degree, and the mathematics that describes symmetry, called group theory. These two topics would seem to be unrelated, but in fact, when Evariste Galois proved that the formula could not be found for fifth degree equations, he did it by inventing group theory! This book explores this connection, while also giving a lot of biographical information about both Galois and Niels Abel, who duplicated Galois' result about fifth degree equations. I found that this book reads very well, and I highly recommend it.

This book represents an ambitious effort of Mario Livio to explain and describe the mathematical origen of group theory, the language of symmetry, in its historical context. The subject is developed to show the importance of the theory for some of the most important scientific achievements of the 20th century (the general theory of relativity). The author's account of the lives of the 2 young mathematicians who did the seminal work in this field presents a very gripping story and evokes the tragic sense that much was lost with their untimely deaths. The research involved in writing this book is truly extraordinary, and the ability of the author to interrelate the resulting information in a meaningful way is impressive.

At some level mathematics becomes developes a beauty and a mystique not unlike a symphony or a poem or a painting. And like these, the most beautiful math is often done by very young men. The story of the solving of the quintic equation is a tale worth telling. The mathematics of algebra began thousands of years ago. The Greek Pythagoras in the sixth century BC made a great advance with the solving of the quadratic equation. Over the next couple of thousand years further work got us to the solution of the quartic equation during the Renaissance. Then came the quintic, the subject of this book. Two young mathematicians solved the problem Norwegian Niels Henrik Abel, and Frenchman Evariste Galois. The work they did was brilliant, and they both died very young. This book is a tale of what they did, and the implications of that work. The results of their work lead to the concepts of symmetry. This book, written almost like a detective story then take symmetry into the musid of Bach, the basic forces of nature, and the selection of sexual partners. It's a delightful read.

In this book Mario Livio tells, in his typically clear and gripping style, the story of the quintic (5th degree) equation, which resisted solution till the early 19th century, in spite of enormous efforts by many gifted and well-known mathematicians. It was finally solved independently by two young and brilliant mathematicians, Niels Henrik Abel from Norway and Evariste Galois from France, both of whom died tragic deaths at a very young age, Abel at the age of 27, and Galois at the age of 20. The approach taken by Galois, who had unusual mathematical abilities, is truly brilliant and revolutionary, and opened the door to a whole new branch in mathematics, the group theory, the "official language" of symmetry. The social views of Galois were equally revolutionary, who died at the tender young age 20 in a duel resulting from a love fight, with his final words to his brother, "Don't cry for me, I need all my courage to die at 20." Ironically, the circumstances leading to the tragic and spectacular death of Galois were themselves shrouded in mystery. No doubt there are many conspiracy theories, but most of them do not explain many known facts about the incidence. Mario Livio has put an enormous amount of effort in solving this mystery, which coherently explains most of the known facts, which he has presented in this book. The book combines the fascinating life stories of the great two mathematicians told in the style of a modern thriller, with amazing insights into the purpose and meaning of their discoveries expressed in a language easily accessible by the general public. The book is replete with fascinating details on art, history, science and mathematics coming, as it does, from a uniquely gifted expert in all these subjects. The book makes for a truly fascinating and inspiring reading.

I picked up this book not knowing anything about symmetry and, frankly, not being too interested in it. What I discovered was a brilliant, cerebral yet entertaining examination of both the mathematical foundations of this concept and its artistic, cultural, and social significance. Perfectly mixing mathematical analyses with fascinating biographical, historic and artistic information (as well as the occasional amusing anecdote), Livio's incredibly well-researched book is as illuminating as a great work of philosophy and as thrilling as a Sherlock Holmes mystery. Those with absolutely no knowledge of mathematics (like me) should not be deterred, because the author inventively elucidates any difficult concepts, leaving nothing unexplained yet never digressing unnecessarily from the central narrative. Above all, the haunting character of Evariste Galois will remain with readers for a long time after they have completed reading this masterful account.