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The Equation That Couldn't Be Solved
Joe Kauzlarich writes "There's an ever-growing number of fun niche books seeping onto the mathematics bookshelves, that, while not essential, are almost always guaranteed to leave the reader with a fuller taste of the subject at hand and an appetite to learn more. Mario Livio's The Equation That Couldn't Be Solved is a modest semi-classic of pop-math literature, focusing on the central concepts of group theory, the subject that turned mathematics on its head a century and a half ago and has ever since been one of the delights of studying higher mathematics." Read on for the rest of Joe's review.
The Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry
author
Mario Livio
pages
335
publisher
Simon & Schuster
rating
8/10
reviewer
Joe Kauzlarich
ISBN
0-7432-5820-7
summary
Popular math/science
If you've studied group theory, you've probably heard it called 'the language of symmetry' or referred to by some such vague, colorful non-description, while your professor and textbook direct you to just memorize the handful of basic axioms, definitions, and theorems that reveal little to the unknowing eye in the way of having much to do with symmetry. Livio concentrates on the more colorful aspects of symmetry, spending little time with black and white textbook theory. For this reason, the book makes ideal extra-curricular entertainment for those enrolled in a first-semester course on abstract algebra.
It seems that Mario Livio's technique in writing books is to choose an ostensibly simple topic and explore it from a broad array of angles. In his second and most popular work, The Golden Ratio, he chose to write about the number Phi. The book reads like the front page of Slashdot, skipping quickly from topic to topic, though sticking to the general theme, insuring that the reader must never get bored. The treatment he once gave to Phi, he now gives to symmetry. Livio explores the concept of symmetry as it manifests itself in biology, art, physics and (especially, of course) mathematics. Then he broaches the most important topic of the book, group theory, and ventures upon the two stunning tales of its conception, as the book's two central figures independently discover that a certain equation cannot be solved by means of regular algebra (which, at the time, referred to the sort of formulaic manipulation done by today's undergrad algebra and calculus students; now, the word 'algebra,' in professional circles, includes group theory and much more).
At last, less-experienced readers will find a warm entry-way into one of the most fascinating and advanced branches of mathematics, one which has, through time, permeated most other branches. Experienced readers will revisit a familiar topic in its historical and mathematical-cultural context, as well as gain an 'intuitive' picture of group theoretical symmetry, an aspect often omitted from first semester advanced algebra courses. All readers can be comforted that mathematical notation is hardly anywhere to be found in the book. Experts need not fear wasting money to relearn what they already know and beginners can pick up the math through its brief mostly-English-language descriptions and should feel more comfortable diving into a course on the subject.
What is this Equation That Couldn't Be Solved? The equation in question is the quintic equation-- a polynomial of degree five (i.e. ax^5+bx^4+...+ex+f=0). You've probably studied the quadratic equation-- ax^2+bx+c=0-- as well as the quadratic formula, used to solve this equation-- x= (b(+/-)sqrt(b^2-4ac))/2a. The quintic equation cannot be solved by means of a formula and it took hundreds of years and two very young men to discover this. And as happens in so many famous instances throughout the history of science, the answer to a seemingly innocent little problem becomes the key to a revolution in thought.
A 22-year-old Norwegian named Niels Henrick Abel (1802-1829) and a 20-year-old Frenchman named Evariste Galois (1811-1832), discovered the impossibility of solving the quintic almost simultaneously in the 1820's. Both died within years of their discovery and both went unnoticed and uncelebrated until after their death. The tragedies that preceded their deaths-- Abel died essentially out of poverty; Galois, poor and already half-mad, in a pistol duel-- have served as a valuable lesson to the mathematical community ever since: spot genius early and foster it. Who knows what would have become of these men had they lived through the prime of their talents, just as the great Gauss and his contemporaries were developing the foundations for what would become Modern mathematics? It was Abel and, particularly, Galois, who defined the language of symmetry. Both saw The Equation in a light that had never been seen before.
Mario Livio is a historian as much as he is a scientist and the detail and color he gives to the lives of these tragic figures is unforgettable. Not only was his research thorough, but he even visited the regions he describes, and his results on the mysteries surrounding the death of Galois offer conclusiveness and definitiveness that seem hardly to have been matched in this particular line of research. Additionally, Livio digs up fresh mathematical anecdotes throughout the book, being careful not to repeat those stories or 'factoids' that are repeated ad nauseum across the genre.
Group theory has become an essential requisite of such diverse areas of scientific research as was unimaginable at the time of its inception. The fundamental particles of nature are arranged in groups, making the subject a cornerstone of particle physics and all physical 'theories of everything.' Group theory is the simplest sort of 'mathematical abstraction' (actually, it is a step past set theory) in that numbers and equations play no part in its basic definitions. Once you learn it well, then rings and fields follow. Then comes the fascinating study of topology, and then there is little that can stop you from learning anything you want mathematically (okay, that's a stretch). Cryptography is a modern applied field which requires a good working knowledge of group theory. I'm sure there are many other examples of applied group theory if you can't be convinced of the beauty of the subject in and for itself. Physics enthusiasts will enjoy the later chapter on group theory in modern particle physics, which is meant to show how integral the subject is to understanding and communicating the very laws of our universe.
While this is surely a bias on my part, I wasn't impressed with the amount of actual math described in the book. The very basics of group theory, as I mentioned, are elaborated upon-- the definition of a group, permutation groups, symmetry groups-- but Livio makes few attempts to make clear what group theorists study (mathematically-speaking) beyond these simple sorts of ideas. To his credit, he does explain Galois's proof quite clearly, considering the amount of time a student spends getting to it in textbooks. The book, as I've said, is foremost a look at symmetry, secondarily historical, and lastly, a math text. It is light reading, but-- take my word for it-- extremely entertaining and worth the few bucks. If you aren't much of a math geek, this book provides a great chance for you to get a glimpse at abstract algebra, which, IMHO, is one of the most fascinating branches of mathematics and, oddly, seems normally to be kept well-hidden from the eyes of non-math or non-physics majors."
You can purchase The Equation That Couldn't Be Solved from bn.com. Slashdot welcomes readers' book reviews -- to see your own review here, read the book review guidelines, then visit the submission page.
If you've studied group theory, you've probably heard it called 'the language of symmetry' or referred to by some such vague, colorful non-description, while your professor and textbook direct you to just memorize the handful of basic axioms, definitions, and theorems that reveal little to the unknowing eye in the way of having much to do with symmetry. Livio concentrates on the more colorful aspects of symmetry, spending little time with black and white textbook theory. For this reason, the book makes ideal extra-curricular entertainment for those enrolled in a first-semester course on abstract algebra.
It seems that Mario Livio's technique in writing books is to choose an ostensibly simple topic and explore it from a broad array of angles. In his second and most popular work, The Golden Ratio, he chose to write about the number Phi. The book reads like the front page of Slashdot, skipping quickly from topic to topic, though sticking to the general theme, insuring that the reader must never get bored. The treatment he once gave to Phi, he now gives to symmetry. Livio explores the concept of symmetry as it manifests itself in biology, art, physics and (especially, of course) mathematics. Then he broaches the most important topic of the book, group theory, and ventures upon the two stunning tales of its conception, as the book's two central figures independently discover that a certain equation cannot be solved by means of regular algebra (which, at the time, referred to the sort of formulaic manipulation done by today's undergrad algebra and calculus students; now, the word 'algebra,' in professional circles, includes group theory and much more).
At last, less-experienced readers will find a warm entry-way into one of the most fascinating and advanced branches of mathematics, one which has, through time, permeated most other branches. Experienced readers will revisit a familiar topic in its historical and mathematical-cultural context, as well as gain an 'intuitive' picture of group theoretical symmetry, an aspect often omitted from first semester advanced algebra courses. All readers can be comforted that mathematical notation is hardly anywhere to be found in the book. Experts need not fear wasting money to relearn what they already know and beginners can pick up the math through its brief mostly-English-language descriptions and should feel more comfortable diving into a course on the subject.
What is this Equation That Couldn't Be Solved? The equation in question is the quintic equation-- a polynomial of degree five (i.e. ax^5+bx^4+...+ex+f=0). You've probably studied the quadratic equation-- ax^2+bx+c=0-- as well as the quadratic formula, used to solve this equation-- x= (b(+/-)sqrt(b^2-4ac))/2a. The quintic equation cannot be solved by means of a formula and it took hundreds of years and two very young men to discover this. And as happens in so many famous instances throughout the history of science, the answer to a seemingly innocent little problem becomes the key to a revolution in thought.
A 22-year-old Norwegian named Niels Henrick Abel (1802-1829) and a 20-year-old Frenchman named Evariste Galois (1811-1832), discovered the impossibility of solving the quintic almost simultaneously in the 1820's. Both died within years of their discovery and both went unnoticed and uncelebrated until after their death. The tragedies that preceded their deaths-- Abel died essentially out of poverty; Galois, poor and already half-mad, in a pistol duel-- have served as a valuable lesson to the mathematical community ever since: spot genius early and foster it. Who knows what would have become of these men had they lived through the prime of their talents, just as the great Gauss and his contemporaries were developing the foundations for what would become Modern mathematics? It was Abel and, particularly, Galois, who defined the language of symmetry. Both saw The Equation in a light that had never been seen before.
Mario Livio is a historian as much as he is a scientist and the detail and color he gives to the lives of these tragic figures is unforgettable. Not only was his research thorough, but he even visited the regions he describes, and his results on the mysteries surrounding the death of Galois offer conclusiveness and definitiveness that seem hardly to have been matched in this particular line of research. Additionally, Livio digs up fresh mathematical anecdotes throughout the book, being careful not to repeat those stories or 'factoids' that are repeated ad nauseum across the genre.
Group theory has become an essential requisite of such diverse areas of scientific research as was unimaginable at the time of its inception. The fundamental particles of nature are arranged in groups, making the subject a cornerstone of particle physics and all physical 'theories of everything.' Group theory is the simplest sort of 'mathematical abstraction' (actually, it is a step past set theory) in that numbers and equations play no part in its basic definitions. Once you learn it well, then rings and fields follow. Then comes the fascinating study of topology, and then there is little that can stop you from learning anything you want mathematically (okay, that's a stretch). Cryptography is a modern applied field which requires a good working knowledge of group theory. I'm sure there are many other examples of applied group theory if you can't be convinced of the beauty of the subject in and for itself. Physics enthusiasts will enjoy the later chapter on group theory in modern particle physics, which is meant to show how integral the subject is to understanding and communicating the very laws of our universe.
While this is surely a bias on my part, I wasn't impressed with the amount of actual math described in the book. The very basics of group theory, as I mentioned, are elaborated upon-- the definition of a group, permutation groups, symmetry groups-- but Livio makes few attempts to make clear what group theorists study (mathematically-speaking) beyond these simple sorts of ideas. To his credit, he does explain Galois's proof quite clearly, considering the amount of time a student spends getting to it in textbooks. The book, as I've said, is foremost a look at symmetry, secondarily historical, and lastly, a math text. It is light reading, but-- take my word for it-- extremely entertaining and worth the few bucks. If you aren't much of a math geek, this book provides a great chance for you to get a glimpse at abstract algebra, which, IMHO, is one of the most fascinating branches of mathematics and, oddly, seems normally to be kept well-hidden from the eyes of non-math or non-physics majors."
You can purchase The Equation That Couldn't Be Solved from bn.com. Slashdot welcomes readers' book reviews -- to see your own review here, read the book review guidelines, then visit the submission page.
"...write down the roots of the quintic in terms of square roots, cube roots, and the Bring radical, which is therefore an algebraic solution in terms of algebraic functions of a single variable..."
see: http://en.wikipedia.org/wiki/Quintic_equation
It's really that simple.
He who knows best knows how little he knows. - Thomas Jefferson
It was common that those who would participate in a pistol duel to stay up all night--writing a will, writing down their knowledge for posterity, praying, etc. Without having researched the issue, I can say that his opponent was likely up all night, as well.
Yet it was quite uncommon that the result of their writing down helped shape Mathematics for a couple of centuries...
Silly them. I imagine the man who could sleep the night would have a significant advantage. Of course, if he could sleep he probably felt pretty confident in his duelling ability anyway...
"kept well-hidden"? Sorry, but that at least is utter rubish. No part of mathematics is kept well-hidden by anyone really; it's just that
1. the general public isn't really interested in mathematics (unlike physics, for example; most non-mathematicians I've met seem to have an instinctive averse reaction when you even say "mathematics")
2. mathematics, in general, cannot be dumbed down simplified for laypeople the same way that other natural sciences can. Someone can have a general idea of what a black hole is even when they don't understand the physical theories behind it, but how do you explain to a layperson what a Hilbert space is?
Coupled together, these things mean that the general public isn't really aware of what mathematicians even study or why it's important to them, but it's not the fault of mathematics (or mathematicians).
quidquid latine dictum sit altum videtur.
Just so people don't get the wrong idea, it's not just quintic polynomials which can't be solved with one formula: it's all polynomials of degree five and higher. Also, "can't be solved" is something of a misnomer: there exist five solutions to a degree five polynomial, and they can be expressed either as infinite series or in terms of some non-standard functions. It's just that they can't be solved in terms of addition, multiplication, and exponentiation (i.e., using +, *, and radicals).
http://pomonahistorical.org/12times.htm
Britney Gallivan has solved the Paper Folding Problem. This well known challenge was to fold paper in half more than seven or eight times, using paper of any size or shape.
The task was commonalty known to be impossible. Over the years the problem has been discussed by many people, including mathematicians and has been demonstrated to be impossible on TV.
..the equations of love?
"It is only through the mysterious equations of love that any logic can be found."
No, no. Sure, *some* fifth-order polynomials are factorable to a set of reduced-order polynominals, but not all. What's being said here is that you can't take an arbitrary fifth-order polynomial, in the form ax^5 + bx^4 + cx^3 + dx^2 + ex + f, and have a formula to provide a solution. So there can be no 'quintic formula' along the same lines as the 'quadratic' formula, making polynomials of fifth-order or higher much harder to solve.
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I Hit the Karma Cap, and All I Got Was This Lousy