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(More customer reviews)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. The persevering reader can only conclude that anything and everything that remotely touches upon the quintic and Galois's work was given a chapter of its own, a mathematical version of "everything but the kitchen sink." The end result is an unfortunate mishmash, a sort of treetop skimming of modern mathematics, post-Newtonian physics, and cognitive theory.
Sadly, Mr. Livio misses a number of opportunities to enlighten his readers on the theory of polynomials, the nature of their roots, and the curious symmetries one encounters. For example, he makes no effort to discuss the nature of polynomial roots beyond a short Appendix, and he passes on the chance to detail the marvelous symmetry of imaginary roots in equations such as x^6 = 1. While he outlines the general thrust of Galois's approach to the unsolvability of quintics, Livio also mentions that Hermite found a method to solve the general quintic using elliptic functions, but we are not told how such a solution is discovered. What about sixth degree polynomials and beyond? Mr. Livio doesn't tell us - he's too busy worrying over the fairness of the first draft lottery in 1970. There is also the small matter of the author's style of explication. At times, such as his introduction to symmetry, he writes for a general, non-mathematical audience. Later, he tosses out references to elliptic functions without explanation and culminates his group theory discussion with sentences like, "We can use the family tree of these subgroups to create a sequence of composition factors (order of the parent group divided by that of the maximal normal subgroup)."
What THE EQUATION THAT COULDN'T BE SOLVED really needed was a good editor to bring these widespread ramblings into focus. A bit of truth in advertising might have been appropriate as well, but a book entitled "The Role of Group Theory in Modern Mathematics and Science" (primarily what this book is about, along with the author's peculiar obsession with Evariste Galois's death by duel) wouldn't tap well into the market developed by Keith Devlin, John Allen Paulos, Ian Stewart, Eli Maor, Simon Singh, and other popularizers of mathematics for mass market audiences. In the end, this book falls short of its companions for its sheer lack of focus and somewhat misleading cover presentation. At times, the book is interesting; at others, regrettably, it's simply too much of a superficial slog through too many loosely connected disciplines.
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