Book details of 'The Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry'
| Title | The Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry |
| Author(s) | Mario Livio |
| ISBN | 0743258207 |
| Language | English |
| Published | September 2005 |
| Publisher | Simon & Schuster |
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What do the music of J. S. Bach, the basic forces of nature, Rubik's Cube, and the selection of mates have in common? They are all characterized by certain symmetries. Symmetry is the concept that bridges the gap between science and art, between the world of theoretical physics and the everyday world we see around us. Yet the "language" of symmetry--group theory in mathematics--emerged from a most unlikely source: an equation that couldn't be solved. Over the millennia, mathematicians solved progressively more difficult algebraic equations until they came to what is known as the quintic equation. For several centuries it resisted solution, until two mathematical prodigies independently discovered that it could not be solved by the usual methods, thereby opening the door to group theory. These young geniuses, a Norwegian named Niels Henrik Abel and a Frenchman named Evariste Galois, both died tragically. Galois, in fact, spent the night before his fatal duel (at the age of twenty) scribbling another brief summary of his proof, at one point writing in the margin of his notebook "I have no time." The story of the equation that couldn't be solved is a story of brilliant mathematicians and a fascinating account of how mathematics illuminates a wide variety of disciplines. In this lively, engaging book, Mario Livio shows in an easily accessible way how group theory explains the symmetry and order of both the natural and the human-made worlds. Download Description
"What do the music of J. S. Bach, the basic forces of nature, Rubik's Cube, and the selection of mates have in common? They are all characterized by certain symmetries. Symmetry is the concept that bridges the gap between science and art, between the world of theoretical physics and the everyday world we see around us. Yet the ""language"" of symmetry--group theory in mathematics--emerged from a most unlikely source: an equation that couldn't be solved. Over the millennia, mathematicians solved progressively more difficult algebraic equations until they came to what is known as the quintic equation. For several centuries it resisted solution, until two mathematical prodigies independently discovered that it could not be solved by the usual methods, thereby opening the door to group theory. These young geniuses, a Norwegian named Niels Henrik Abel and a Frenchman named Evariste Galois, both died tragically. Galois, in fact, spent the night before his fatal duel (at the age of twenty) scribbling another brief summary of his proof, at one point writing in the margin of his notebook ""I have no time."" The story of the equation that couldn't be solved is a story of brilliant mathematicians and a fascinating account of how mathematics illuminates a wide variety of disciplines. In this lively, engaging book, Mario Livio shows in an easily accessible way how group theory explains the symmetry and order of both the natural and the human-made worlds. "
--This text refers to the Digital
edition. From the Publisher
"Essential reading for anyone who wants to understand how the apparently remote concerns of pure mathematics can lead too deep and practical insights into the natural world." --Ian Stewart, author of DOES GOD PLAY DICE? "A highly readable and illuminating book." --Sir Michael Atiyah, Abel Prize in Mathematics Laureate 2004
About the Author
Mario Livio is a Senior Astronomer and the former Head of the Science Division at the Space Telescope Science Institute (STScI), the institute that conducts the scientific program of the Hubble Space Telescope. He received his Ph.D. in theoretical astrophysics from Tel Aviv University in Israel and was a professor in the Physics Department at the Technion-Israel Institute of Technology from 1981 to 1991, when he joined STScI. Dr. Livio has published more than 400 scientific papers and has received numerous awards for research and for excellence in teaching. He is the author of The Golden Ratio, a highly acclaimed book about mathematics and art for which he received the International Pythagoras Prize and the Peano Prize, and The Accelerating Universe.
Excerpt. © Reprinted by permission. All rights reserved.
Chapter One: Symmetry An inkblot on a piece of paper is not particularly attractive to the eye, but if you fold the paper before the ink dries, you may get something that looks like figure 1 that is much more intriguing. In fact, the interpretation of similar inkblots forms the basis for the famous Rorschach test developed in the 1920s by the Swiss psychiatrist Hermann Rorschach. The declared purpose of the test is to somehow elicit the hidden fears, wild fantasies, and deeper thoughts of the viewers interpreting the ambiguous shapes. The actual value of the test as an "x-ray of the mind" is vehemently debated in psychological circles. As Emory University psychologist Scott Lilienfeld once put it, "Whose mind, that of the client or the examiner?" Nevertheless, there is no denial of the fact that images such as that in figure 1 convey some sort of attractive and fascinating impression. Why? Is it because the human body, most animals, and so many human artifacts possess a similar bilateral symmetry? And why do all those zoological features and creations of the human imagination exhibit such a symmetry in the first place? Most people perceive harmonious compositions such as Botticelli's Birth of Venus as symmetrical. Art historian Ernst H. Gombrich even notes that the "liberties which Botticelli took with nature in order to achieve a graceful outline add to the beauty and harmony of the design." Yet mathematicians will tell you that the arrangements of colors and forms in that painting are not symmetric at all in the mathematical sense. Conversely, most nonmathematical viewers do not perceive the pattern in figure 3 as symmetrical, even though it actually is symmetrical according to the formal mathematical definition. So what is symmetry really? What role, if any, does it play in perception? How is it related to our aesthetic sensibility? In the scientific realm, why has symmetry become such a pivotal concept in our ideas about the cosmos around us and in the fundamental theories attempting to explain it? Since symmetry spans such a wide range of disciplines, what "language" and what "grammar" do we use to describe and characterize symmetries and their attributes, and how was that universal language invented? On a lighter note, can symmetry provide an answer to the all-important question posed in the title of one of the songs of rock star Rod Stewart -- "Do Ya Think I'm Sexy?" I will try to provide at least partial answers to all of these questions and many more. Along the way, I hope that the story as a whole will depict both the humanistic side of mathematics and, even more importantly, the human side of mathematicians. As we shall see, symmetry is the paramount tool for bridging the gap between science and art, between psychology and mathematics. It permeates objects and concepts ranging from Persian carpets to the molecules of life, from the Sistine Chapel to the sought-after "Theory of Everything." Yet group theory, the mathematical language that describes the essence of symmetries and explores their properties, did not emerge from the study of symmetries at all. Rather, this astonishingly unifying idea of modern thought emanated from a most unlikely source -- an equation that couldn't be solved. The dramatic and tortuous history of this equation is an essential part of this intellectual saga. At the same time, this tale will shed light on the loneliness of genius and on the tenacity of the human intellect in the face of seemingly insurmountable challenges. I have put a tremendous effort into trying to solve the two-centuries-old mystery of the death of the protagonist of this story -- the brilliant mathematician Évariste Galois. I believe that I have come closer to the truth than was ever possible before. The witty playwright George Bernard Shaw once said, "The reasonable man adapts himself to the world; the unreasonable one persists in trying to adapt the world to himself. Therefore all progress depends on the unreasonable man." In this book we shall encounter many unreasonable men and women. The creative process, by its very nature, seeks uncharted intellectual and emotional terrain. Brief forays into mathematical abstraction will offer a peek into the very nature of creativity. I begin with a concise exploration of the wonderland of symmetries. Immunity to Changes The word symmetry has ancient roots, coming from the Greek sym and metria, which translate into "the same measure." When the Greeks labeled a work of art or an architectural design symmetric, they meant that one could identify some small piece of the work, such that the dimensions of all the other parts contained that piece a precise number of times (the parts were "commensurable"). This early definition corresponds more to our modern notion of proportion than to symmetry. Nevertheless, the great philosophers Plato (428/427-348/347 BC) and Aristotle (384-322 BC) were quick to associate symmetry with beauty. In Aristotle's words, "The chief forms of beauty are orderly arrangement [in Greek taxis], proportion [symmetria], and definiteness [horismenon], which are revealed in particular by mathematics." Following in the Greeks' footsteps, the identification of symmetry with "due proportion" was subsequently propagated by the influential Roman architect Vitruvius (ca. 70-25 BC), and it persisted all the way through the Renaissance. In his De Architectura Libri Decem (Ten Books on Architecture), literally the architectural bible in Europe for centuries, Vitruvius writes: The design of a temple depends on symmetry, the principles of which must be carefully observed by the architect. They are due to proportion. Proportion is a correspondence among the measures of the members of an entire work, and of the whole to a certain part selected as standard. From this result the principles of symmetry. The modern meaning of symmetry (first introduced in the late eighteenth century) in the precise mathematical sense is really "immunity to a possible change." Or, as mathematician Hermann Weyl (1885-1955) once put it, "A thing is symmetrical if there is something you can do to it so that after you have finished doing it it looks the same as before." Examine for example the verses Is it odd how asymmetrical Is "symmetry"? "Symmetry" is asymmetrical. How odd it is. This stanza remains unchanged if read word by word from the end to the beginning -- it is symmetrical with respect to backward reading. If you envision the words as being arranged like beads along a string, you could regard this reverse reading as a sort of (not literal) mirror reflection of the stanza. This stanza does not change when mirror-reflected in the above sense -- it is symmetrical with respect to such mirror reflection. Alternatively, if you prefer to think in terms of reading the poem out loud, then the backward reading corresponds to a time reversal, somewhat like rewinding a videotape (again, not literally, because the individual sounds are not reversed). Phrases with this property are called palindromes. The invention of palindromes is generally attributed to Sotades the Obscene of Maronea, who lived in the third century BC in Greek-dominated Egypt. Palindromes have been extremely popular with many word-play wizards such as the Englishman J. A. Lindon, and with the superb recreational-mathematics author Martin Gardner. One of Lindon's amusing word-unit palindromes reads: "Girl, bathing on Bikini, eyeing boy, finds boy eyeing bikini on bathing girl." Other palindromes are symmetric with respect to back-to-front reading letter by letter -- "Able was I ere I saw Elba" (attributed jokingly to Napoleon), or the title of a famous NOVA program: "A Man, a Plan, a Canal, Panama." Surprisingly, palindromes appear not just in witty word games but also in the structure of the male-defining Y chromosome. The Y's full genome sequencing was completed only in 2003. This was the crowning achievement of a heroic effort, and it revealed that the powers of preservation of this sex chromosome have been grossly underestimated. Other human chromosome pairs fight damaging mutations by swapping genes. Because the Y lacks a partner, genome biologists had previously estimated that its genetic cargo was about to dwindle away in perhaps as little as five million years. To their amazement, however, the researchers on the sequencing team discovered that the chromosome fights withering with palindromes. About six million of its fifty million DNA letters form palindromic sequences--sequences that read the same forward and backward on the two strands of the double helix. These copies not only provide backups in case of bad mutations, but also allow the chromosome, to some extent, to have sex with itself -- arms can swap position and genes are shuffled. As team leader David Page of MIT has put it, "The Y chromosome is a hall of mirrors." Of course, the most familiar example of mirror-reflection symmetry is that of the bilateral symmetry that characterizes the animal kingdom. From butterflies to whales, and from birds to humans, if you reflect the left half in a mirror you obtain something that is almost identical to the right half. I will, for the moment, ignore the small if tantalizing external differences that do exist, and also the fact that neither the internal anatomy nor the functions of the brain possess bilateral symmetry. To many, the word symmetry is actually assumed to mean bilateral symmetry. Even in Webster's Third New International Dictionary, one of the definitions reads: "Correspondence in size, shape, and relative position of parts that are on opposite sides of a dividing line or median plane." The precise mathematical description of reflection symmetry uses the same concepts. Take a drawing of a bilaterally symmetric butterfly and mark a straight line down the middle of the figure. If you flip the drawing over keeping the central line in place, perfect overlapping will occur. The butterfly remains unchanged -- invariant -- under reflection about its central line. Bilateral...
