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134 of 140 people found the following review helpful:
5.0 out of 5 stars
"The World's Most Astonishing Number", April 7, 2003
Are some numbers more important than others? Certainly numbers like the primes, pi and "e" have properties that make them interesting to mathematicians and physical scientists alike. Then there are numbers like 7, 13 and 666 that have other connotations for theologians, numerologists and the like. And yet, some numbers have not gotten their due in recent years. Phi--a number variously referred to as the golden ratio, golden section, and divine proportion among others--is one. But Mario Livio has written a book in an attempt to remedy this situation. Phi received its original definition from Euclid as an "extreme and mean ratio" when a straight line is cut so that the ratio of the entire line to the longer division of the segment is the same as the ratio of the longer division of the segment to the shorter. And yet, much like the better known geometrical example of pi, phi turns out to have many more applications beyond its simplest geometrical definition. Though measurable, phi is an irrational number with relationships to the Fibonacci sequence, fractals, the physical structure of things from plant growth and spiral shell development to the appearance of large-scale objects like galaxies, and more. And beyond this, phi has been used as a basis applications in numerology and aesthetics. Livio does a very good job of covering all this ground and more. He is especially good at giving us a historical overview of the development of our understanding of this important number as well as explaining the mathematics in a way that is complete but easy to understand. He is also very good at presenting the various mystical ways phi has been interpreted over the centuries, giving each a rigorous challenge--rejecting many but open-minded to the possibilities that any good Platonist would be. In fact, if there is a weakness in this book, it is that Livio spends a lot of time covering these more esoteric applications of phi. And yet, these applications are part of the history of the number and cannot be ignored whatever a reader might feel about the value of these applications. Phi may not quite live up to the hype as "the world's most astonishing number" but certainly any reader with an interest in mathematics will not want to miss this book.
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106 of 112 people found the following review helpful:
4.0 out of 5 stars
Interesting book, interesting number, February 8, 2003
In mathematics, there are a few irrational numbers that stand out from the infinitely crowded field. First in the bunch would probably be the square root of two, which was kind of the "first" irrational number. Then there is the everpresent pi, and then the less well-known but completely important "e". If there was a fourth place prize, however, it would probably go to the golden number, phi, or roughly 1.618. In this book, Livio gives a brief history of mathematics and phi's place in it. Intimately related to the Fibonacci numbers, a sequence of numbers in which any given number is the sum of the previous two (after the first couple); these numbers (1,1,2,3,5,8,13...) have shown up in some unlikely places such as sunflowers and nautilus shells. Livio shows us the significance of phi in both the mathematical and physical world. Livio also makes a good case that phi may be the most overrated of all numbers. Although it has a wonderfully golden name, it actually doesn't live up to its reputation; Livio shows that phi's presence in art and architecture is more fictional than real and that there is nothing about phi that automatically confers aesthetic beauty. A good portion of the book is dedicated to debunking these golden myths. Overall, this is a good book. Livio's writing is appealing to both mathematician and non-mathematician alike. He does have a tendency to meander from his topic, which can be distracting (even if entertaining), although he eventually does get back on track. For those who like reading about math and the significance of certain numbers (I have also read books on pi, e, i, 0 and infinity), this is a worthwhile read.
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58 of 60 people found the following review helpful:
5.0 out of 5 stars
All About 1.618033988749894848204586834365638117720309179..., January 14, 2003
Of all the irrational numbers, the best known is pi, which shows up all over the place. However, if you read _The Golden Ratio: The Story of Phi, The World's Most Astonishing Number_ (Broadway Books) by Mario Livio, you will gain an appreciation for the ubiquity of another irrational with all sorts of amazing properties. You can try this one on your calculator: Phi equals 1.6180339887... (As an irrational, its string of numbers goes infinitely beyond the decimal point, and you can be sure computers have calculated it to millions of places). Take the inverse of that number; that is, divide it into one. You will get 0.6180339887...; in other words, the inverse looks just like phi itself, but with a zero instead of one left of the decimal. Or try this: start with a 1, followed by a 1. The next number will be the two previous ones added together, which is 2; the next number, in turn, is again the two previous ones added together, which is 3. The series goes 1, 1, 2, 3, 5, 8, 13, 21, 34, 55... This is the famous Fibonacci sequence, and is investigated widely within this book as it is intimately connected to phi. Take any number in the series and divide it by the number before it, and you will get a number close to phi; the higher the number in the series, the closer the result comes. (13 / 8 = 1.615 ; 55 / 34 = 1.6176....). These sorts of number tricks abound in Livio's book, and the mathematics is not daunting. It is also a history of phi, which turns out to be a representative slice of the history of mathematics. Euclid knew the number, but Leonardo Fibonacci in the twelfth century developed the series with its ratio. It shows up in breeding rabbits; spirals in pine cones, sunflowers, galaxies, and hurricanes; tilings and fractals; and many more surprising places. Livio has enormous fun giving and explaining all these examples. Showing up as it does all over the place, perhaps phi is just being seen because that is what is being looked for. Livio, whose day job is being Head of the Science Division at the Hubble Space Telescope Science Institute, is refreshingly dismissive of attempts to try to see a Golden Ratio in everything, which people have tried to do for centuries. It isn't in the pyramids, nor in the Parthenon, nor in Leonardo's paintings. Without forcing the issue, however, it is easy to see that the Golden Ratio, logarithmic spirals, and Fibonacci numbers are all over the place; there is even a _Fibonacci Quarterly_ mathematical journal. This leads to larger final issues, which Einstein expressed as the question, "How is it possible that mathematics, a product of human thought that is independent of experience, fits so excellently the objects of physical reality?" Do mathematical concepts have a universal and timeless existence "out there" and are just waiting for us to discover them? Or is mathematics a human invention that resides only within the human brain? It can't be surprising that this classic conundrum is not definitively solved here. Livio's ideas about it, however, well expressed and tied to this remarkable numerical constant, are well worth thinking about.
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