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The Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry Hardcover – September 13, 2005

4.3 out of 5 stars 39 customer reviews

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Editorial Reviews

From Publishers Weekly

The idea of symmetry has been heavily deployed in recent science popularizations to introduce advanced subjects in math and physics. This approach usually backfires—mathematical symmetry is much too difficult for most laypeople to understand. But this engaging treatise soft-pedals it in a crowd-pleasing way. The title's formula is the "quintic" equation (involving x raised to the fifth power), the analysis of which gave rise to "group theory," the mathematical apparatus scientists use to explore symmetry. Inevitably, the author's attempts to explain group theory and its applications in particle physics and string theory to a general audience fall sadly short, so readers will just have to take his word for the Mozartean beauty of it all. Fortunately, astrophysicist Livio (The Golden Ratio) keeps the hard stuff to a minimum, concentrating instead on interesting digressions into human interest (e.g., the founder of group theory, Evariste Galois, was a revolutionary firebrand who died in 1832 at age 20 in a duel over "an infamous coquette"), pop psychology (women have more orgasms when their partners have symmetrical faces), strategies for finding a soul mate and some easy math puzzles readers might actually solve. The result is a somewhat shapeless but intriguing excursion. Photos.
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From Scientific American

The so-called quintic equation resisted solution for three centuries, until two brilliant young mathematicians independently discovered that it could not be solved by any of the usual methods — and thereby opened the door to a new branch of mathematics known as group theory. This book is the story of these two early 19th-century mathematicians— a Norwegian, Niels Henrik Abel, and a Frenchman, Evariste Galois, both of whom died tragically, Galois in a duel at the age of 20. Livio, an astrophysicist now at the Space Telescope Science Institute and author of The Golden Ratio, interweaves their story with fascinating examples of how mathematics illuminates a wide swath of our world.

Editors of Scientific American


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Product Details

  • Hardcover: 368 pages
  • Publisher: Simon & Schuster; First Edition edition (September 13, 2005)
  • Language: English
  • ISBN-10: 0743258207
  • ISBN-13: 978-0743258203
  • Product Dimensions: 6.4 x 0.9 x 9.4 inches
  • Shipping Weight: 1.3 pounds
  • Average Customer Review: 4.3 out of 5 stars  See all reviews (39 customer reviews)
  • Amazon Best Sellers Rank: #431,164 in Books (See Top 100 in Books)

Customer Reviews

Top Customer Reviews

By Steve Koss VINE VOICE on February 17, 2006
Format: Hardcover
Mario Livio's title suggests an exploration of unsolvable equations, in particular the drama enshrouding the mathematical conundrum of solving general, fifth degree polynomial equations, known as quintics. His subtitle, "How Mathematical Genius Discovered the Language of Symmetry," indicates that his work will also explore the role of symmetry in ultimately resolving the question of whether such polynomials could be solved by a formulas using nothing more than addition, subtraction, multiplication, division, and nth roots. These two subjects portend an interesting discussion on the solvability of equations and the peculiar mathematical race in Renaissance Europe to "discover" the magical formulas for solving cubics and quartics.

One could reasonably expect that the groundbreaking work of Tartaglia, Cardano. Ferraro, Galois, Abel, Kronecker, Hermite, and Klein would be encompassed in this survey, and indeed they are. However, purchasers of this book are given no indication that they will spend well over half their reading time on rehashes of Abel's tragic life story and the mythology of Evariste Galois's foolish death, Emmy Noether's challenges as a woman mathematician in Germany, a history of group theory, Einstein's theory of relativity, the place of string theory in modern cosmology, the survival benefits of symmetry in evolution, Daniel Gorenstein's 30-year proof that "every finite simple group is either a member of one of the eighteen families or is one of the twenty-six sporadic groups," a trite and unnecessary diversion on human creativity, and finally, an even more outlandish (and utterly inconclusive) "comparison" of Galois's brain with that of Albert Einstein.
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Format: Paperback
I became interested in this book for several reasons. The first is that I find Livio to be an entertaining writer. I read his book on phi and its relationship to beauty and found it interesting and enlightening. I have reviewed that book on amazon earlier. I met Livio in Princeton a little over a month ago when he gave a lecture on symmetry at the Princeton Plasma Physics Laboratory in one of a series of lectures intended for high school students. It was a fascinating presentation and he briefly discussed the book, mentioning how his research into the death of Galois led him to a new theory about how he died in the duel and who killed him. I found this very intriguing and I wanted to read about it.

As a college undergraduate I majored in mathematics and modern algebra was my favorite subject. The course I took on Galois theory was the most fascinating to me and I marveled over the fact that a teenage boy had developed a branch of group theory that answered questions that had stumped the greatest mathematicians for centuries.

So I bought the book and read it with very high expectations. I preface my remarks this way because I was somewhat disappointed in the book and my disappointment leads to my criticism here. But I don't want the critcism to detract from the fact that it is a well written and researched book and written in a style that like his other books makes it accessible to the general public and even the highly motivated high school students.

First of all the title leads you to believe that it is completely about the solving of the problem for which polynomials can be solved by radicals (i.e. equations that only involve basic arithmetical operations a roots, e.g. square cube roots etc,)and which ones cannotbe so solved.
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Format: Hardcover
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.
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Format: Hardcover
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.
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