Rating: 
Summary: A Noble Attempt that fails
Review: Much ado has been made of Mario Livio's latest pop science book. I admire his aim, but I can't praise his writing or organization. This book caroms from one tangent to another, the prose is muddy and academic, and worse, it fails to clearly define what phi is all about. Those seeking mathematics writ large and small--and writ well--should seek out "Just Six Numbers: The Deep Forces that Shape the Universe" by Martin J. Rees
Rating: 
Summary: All About 1.618033988749894848204586834365638117720309179...
Review: 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.
Rating:
Summary: Excellent overview of the significant number
Review: OK, I don't come across PHI as often as I come across e or pi. This book was nevertheless a fascinating read, even though (as one of the reviewers pointed out) Livio tries to talik about math and applications which can seem somewhat tedious at times.
The books deserves five stars for the following things:
1. It's very well written (I read it in an afternoon)
2. It's informative and not too 'scientific' - you can understand it even if you're not good with numbers at all.
3. It's realistic - most books about this topic tend to be more fantasy than reality. Mario Livio has made sure that he debunks myths and relies only on well checked facts. - If you like popular science books don't miss this one, it's well worth your time.
Rating: 
Summary: Misleading Endorsement
Review: Probably not unlike many others who purchased this book, the on-the-cover endorsement by Dan Brown, author of "The DaVinci Code", caught my eye. Yet about ten per cent of the way through masochistically forcing myself to finish this thankfully short book, I began to ask myself: "Did Dan Brown even read this?"
Mr. Brown states: "... you will never again look at a pyramid, pinecone, or Picasso in the same light." The author, Mario Livio, however, proceeds to argue AGAINST the possibility that the golden ratio was used consciously or subconsciously by ancient architects, rennaissance or modern artists, musicians, or anyone else. Each chapter follows the same pattern:
1. "Did ____ use the golden ratio in his celebrated work ____?"
2. "Let's consider the possibility over the next several pages..."
3. "Alas, upon careful consideration, I must conclude that ____ did not use the golden ratio."
4. "Let's look at another example, and repeat steps 1 through 3."
In fairness, Mr. Livio does a reasonably good job of presenting cases of phi showing up in nature, reasoning that the presence of this number leads to efficient designs in everything from the arrangement of flower petals to the proportions of a chambered nautilus. So, I guess the pinecone part wasn't misleading. ;)
Rating:
Summary: Predictably Prejudiced
Review: The big problem with this book is it seems to exist only to debunk. It tends to debunk in a Matrin Gardner-ish manner, by stating only some of the facts about a subject, then simply saying that they aren't true, or stating that no one takes this subject seriously any more, or some such statement implying that any form of serious study of the subject is beneath the author's serious consideration, and, therefore, also beneath yours.
His discussion of Joseph Schillinger's monumental System Of Musical Composition is flawed by this same wet pointing approach. There's much more to Schillinger's work than the use of the Fibonacci series or the Golden Mean in musical composition or arranging, but Livio seems to think he's covered the System by a paragraph or two on its use of the Golden Mean.
I suppose I should be thankful that Schillinger is even mentioned any more these days, despite the fact that he pioneered and predicted the extensive use of electronics in music today, among other things. His work is still in extensive use by many composers, but Livio would have us believe he's a totally forgettable crackpot.
Rating: 
Summary: Good but lengthy
Review: The book could really be summed up in few pages. Moreover could get the taste of phi from the internet itself, one need not buy this book. Definately Phi is an irrational numbers of all irrational numbers but what is the rational behind making it the golden ratio and fall in love with it. To me Pi is also a fasinating number.
Rating:
Summary: Uninspired philosophy of mathematics
Review: The book seems to have two purposes. First, it seeks to debunk the notion that the 'golden mean' is intuitively pleasing to all humans. Secondly, the book argues that we can best understand the cosmic meaning of life via numbers. From these two theses, the author establishes the 'genius' mathematician as mediator and priest between mankind and the cosmos. Ok, enough mumbo-jumbo. If the above interests you, read the last chapter first. It should have been used as the introduction and will clarify the purpose of arguments presented throughout the book. Getting back to the book's narrative, the early sections seek to debunk various claims that Egyptian and Mesopotamian civilization use the Fibonacci ratio. Reports of its use in China are ignored. Later, there is a section debunking the notion that great art 'uses' the Golden Mean. Scattered throughout are developments in number theory, starting with Pythagoras and continuing with the standard European mathematical genius roll call. The last chapters reveal the relationship between the Golden Mean and complexity/chaos theory/fuzzy logic/quantum theory/etc. Livio's Fibonacci sequence, Penrose tilings, and quasi-crystals stories will probably engage the recreational mathematicians among us, and provide a handy 'all in one place' summary of such matters. Others will find the philosophic overtones tangential and/or distracting. Any successful philosophy of math needs to address the issues of cardinality and ordinality at the level of intuition. The two topics are dismissed by page 15, but they underlie the whole issue of 'intuitive acceptance of number theory'. This topic is discussed throughout the book as Livio seeks to explain why so many people see the Golden Mean in ancient works of architecture, modern art and stock market charts. Curiously, Livio argues they all got it wrong and only genius mathematicians have the gift of 'intiutive acceptance of number theory'. Not an argument which many will find satisfying, but for the passionately recreational mathematician, it may be seductive.
Rating:
Summary: Good, but limited
Review: The idea of the Golden Ratio is a very interesting one, and certainly excellent fodder for a book. The number crops up everywhere, from geometry to biology. When I purchased this book, hot on the heels of the excellent "The Code Book: The Evolution of Secrecy" by Simon Singh, I was primed for an adventure in mathematics. The topic was fascinating and the book very well-reviewed. However, I believe that, in this case, Livio missed the literary mark. The weakness of the book is that I found the style somewhat dry and turgid. Whereas many other books on similar themes (for example, Gleick's outstanding "Chaos") make strong use of narrative and style in order to move the reader along, Livio seems to jump from topic to topic, not giving enough meat in places and failing to sufficiently join the dots in others. Furthermore, he leaves many questions unanswered, while dwelling for too long on debunking Golden "myths". The end result was unfulfilling - giving me enough of a taste to wet my appetite, but not enough substance to satisfy. On the plus side, the book is highly suitable for non-Mathematicians (while not simplifying things to the extent that those trained in the Sciences will reject it), seems factually accurate, and does nicely illustrate the many (and extraordinary) places that Phi turns up in science. I'm certainly glad I read the book, and would recommend it (reservedly) - the lukewarm praise is due to how much more it could have - and should have - been.
Rating:
Summary: good history...but a tinge of arrogance
Review: This book does provide an excellent account on the history of Phi. That much is not in dispute. However, what is in dispute is the way in which Mr. Livio goes about "exploding" the "myths" of Phi. Rather than taking such a rigid approach to the mystical history of Phi, Livio should have taken greater care to state that, in many of his examples, we really don't know one way or another. For example, to say that the Egyptians had no knowledge of phi, based on the great pyramid being off by however miniscule of a percent, is at best nitpicking and at worst ridiculous. Such a discrepancy could be attributed to mere erosion. This is just one example, but read his numerous criticisms and the ask yourself whether he his truly proving his case or just blowing smoke. The biggest error of all in his reasoning is that Phi has not been proven to truly relate to aesthetic preference. He should have done more homework, and included Feckner's study on shape preference. It clearly presents evidence to the contrary. In general, this book's greatest flaw is the author's poor attempt to make a name for himself by slandering the work and theory of numerous figures without backing it up with any real evidence. Perhaps we're just supposed to take his word for it. I, for one, cannot, for blind cynics are just as bad as the blindly credulous.
Rating:
Summary: Dense
Review: This book is an exploration into the history and concepts of Phi, the Golden Ratio. Livio begins his story by tracing the earliest known uses of numbers and counting systems. He progresses through Pythagoras and the discovery of irrational numbers. The history of Phi takes us to many mathematicians and their work, including Plato, Euclid, Abu'l-Wafa, Fibonacci, Pacioli, and Kepler, from the foundations of plane geometry through computer-generated fractals. Livio describes the special properties of Phi in triangles and pentagrams, the mysterious Fibonacci series phenomena, and the beauty of equiangular spirals. Concepts and examples are illustrated throughout the book with formulae, diagrams, drawings, and black-and-white reproductions of paintings. At the end of the book are appendices with geometrical and mathematical proofs, an extensive list of suggestions for further reading organized by chapter, and an index.
Livio cites dozens of examples of how Phi comes into play in geometry and mathematics, and he also points out where the numbers or shapes relating to the Golden Ratio appear in nature, such as in snail shells or pineapple scales. He then examines claims that the Golden Ratio was used explicitly in art, music, and poetry, and argues that such claims, are the for the most part, widely overstated. Indeed, only a few artists, musicians, or poets have explicitly attempted to work with the Golden Ratio (Le Corbusier, for one), and much of their work comes across as somewhat contrived rather than natural. On the use of the Golden Ratio for aesthetics, Livio concludes, "In spite of the Golden Ratio's importance for many areas of mathematics, the sciences, and natural phenomena, we should, in my humble opinion, give up its application as a fixed standard for aesthetics, either in the human form or as a touchstone for the fine arts.
The material in this book is quite fascinating for those with a mathematical bent. Even readers who have heard of and worked with the Golden Ratio before are bound to come across new facts, details or phenomena involving the Golden Ratio that they were not aware of before. Livio's presentation draws many, many facets of the Golden Ratio together into one cohesive story, while analyzing some claims for uses of the Golden Ratio in the arts that perhaps go too far. Livio assumes that readers will have a decent background in algebra and a firm foundation in geometry. The material in this book is dense and requires thoughtful reading, so it is not a quick read, but it is quite informative and interesting.
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