The Golden Ratio: The Story of PHI, the World's Most Astonishing Number

Arrived on time, can’t wait to start reading it. It’s like new. The sender packaged it with care and sent me a book mark along with a $100 free wine card. Thank YOU!

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.

"The Golden Ratio" is truly master piece by Mario Livio. I was impressed by Livio's first book: "The Accelerating Universe", not just by its style and contents, but also by the clear and engaging manner in which Livio had presented the material. In this book Livio has exceeded even the high standards of his earlier book in producing what I found to be a very fascinating and educative book on a seemingly ordinary number. Phi, the so called 'golden ratio', originated from a geometrical concept: if we divide a line into two segments such that the ratio of the length of whole line to that of the bigger segment is the same as the ratio of the length of the bigger segment to that of the smaller one, then this ratio is 'phi'. On the face of it, this looks like a mundane, dull and insignificant ratio. Then Livio leads us through innumerable examples --- examples as varied as the breeding patterns of rabbits to optics of light rays --- where the same golden ratio appears again and again. With Livio's magic touch, the seemingly dull number widens to a fascinating world of its own. We find the same ratio hidden in the delightful petal arrangements in a red rose, in Salvador Dali's famous painting "Sacrament of the last supper", in the spiral shells of mollusks, in the spiral patterns of face-on Galaxies, ..... the list goes on. The book also contains philosophical discussions on such topics as "Is God a Mathematician?", where Livio tries to pierce through the meaning of it all. This book is likely to remain a classic and true source book on the golden ratio for a long time. The book is full of information, and cleverly written. It makes for a very interesting reading: and in the process you will not only learn all about the fascinating world of the golden ratio but also about paintings, flowers, astronomy, and a lot more, in an effortless and enjoyable manner.....

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 no

Following in the steps of his earlier, just as fascinating account linking cosmology and the arts ("The Accelerating Universe"), Mario Livio continues to prove he is one of the most original, exciting and literate writers of popular science today. "The Golden Ratio" is a witty and learned journey generally following the trail of the number Phi, but stopping along the way to take in subjects as diverse as philosophy, history, art, religion, the sciences, architecture, etc. Writing about science in a way that is both knowledgeable and understandable for the common reader is an infamous hurdle, but Livio leaps over it with the greatest of ease, giving clear explanations of every potentially difficult matter and providing the scientific proofs in the appendices, for those more mathematically inclined. Overall, though, it is the great humanity of Livio's worldview that shines through the book and makes it, at least for me, one of the most memorable reads of the year.