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

I had thought the Golden Ratio was simply the ideal aesthetic ratio between the length and the height of a painting or that of objects within a painting. According to Author Mario Livio, however, it has very little to do with the arts but a great deal to do with nature and the laws of physics, as well as some amazing abstract mathematical characteristics (discovered over the last several centuries). I believe the sub-title of the book is correct: it IS the world's most astonishing number. In other words, though it does not in the author's view have much to do with the Mona Lisa, the Parthenon, or the Pyramids, it does have some fascinating connections to nature, as well as numbers in the abstract, and their characteristics.Well, what is the Golden Ratio anyway? Basically, phi or the Golden Ratio is such that if you break a line AB into 2 parts by adding point C to make AC and CB, such that AC is greater than CB and AC/AB = AB/AC. I t sounds pretty boring, but it gets a lot better, since it is also the convergence of something called the Fibonacci Sequence, a set of numbers beginning with 0 such that any 2 consecutive numbers added together equals the next number in the sequence (0,1,1,2, 3, 5, 8, 13, etc.). The Fibonacci Sequence can also be proved to be the same as the continued fraction of all 1's and also the convergence of the continuous nested square roots of 1's. (You can look on the net to see what these expressions look like, both somehow very satisfying aesthetically). I was amazed that these connections could have been made at all with phi, and that the Fibonacci Sequence is the most irrational of all possible numbers; that is, it converges the most slowly to its final irrational value. Call me weird, but that just blew me away!I was most amazed that minds could think of these abstract things, and that the math connections to phi worked out so beautifully. Phi's abstract qualities are, in my opinion, every bit as impressive as its connections to nature itself (galaxies, sunflowers, hurricanes, and more). How did they think this stuff up, and why does it fit together so well? Some of the more bizarre are as follows:The inverse of phi has the same numbers to the right of the decimal point as phi itself.The square root of phi also has the same numbers to the decimal point as phi.The sum of 10 consecutive Fibonacci numbers is = to the 7th number times 11.The unit digit of a given Fibonacci number occurs exactly every 60 numbers.All Fibonacci primes have prime subscripts (with the exception of 3).The product of the first and third Fibonacci numbers in a set of 3 consecutive Fibonacci numbers is within 1 of the 2nd number squared.Who would even think of looking into such things, and why does it work out so well?There were also a couple of tangential points that were really neat to me. How about the First Digit Phenomenon (Benford's Law), that says if you have a random set of numbers, the probability of the first digit being a 1 is greater that it being a 2 is greater that it being a 3, and so on. How is that even possible in the real world? I'll have to think about that one a little more. And how about proof for the irrationality of the square root of 2? This elegant little proof was worth the price of the book, at least for me. It is a derivation of something called reductio ad absurdum: you prove something is true by starting with the opposite assumption and taking it to its logical conclusion to prove it can't be true.Finally, I was struck by a broader question raised by the Mario Livio: how is it that math can so concisely define the laws of nature (gravity, motion, etc.)? I don't think that thought once crossed my mind throughout my high school and college careers in engineering! The book says that Kepler's Third Law, for example, states that the square of a planet's period divided by the cube of its semi-major axis is constant for all planets. How does that work out so well in such a brief, elegant formula, and how in the world did Kepler think of it? Are we talking Coincidence or Creator?I was a little let down by this book as far as art is concerned; Livio simply doesn't believe it is a factor (except for a little 20th century art in the cubist genre perhaps). But I was surprisingly excited by some of the abstract characteristics of the Golden Ratio, and the minds that somehow put it all together. It was as exciting to me as seeing rare, beautiful, exotic creatures on a TV nature show.The Golden Ratio is a strange, beautiful, and rare bird indeed!

http://www.amazon.com/kindleforandroid/Indeed this is also a most astonishing book telling the story of a number. In bibliographical terms this book is mathematics, or slightly more narrowly history of mathematics. Nota bene! The number is not the well-known Pi (3,14..), related with circle, but equally well-known, not as a number but as ratio of long and short edge of an ideal four corner surface. Everybody knows and has a conception of what is a Golden Ratio without ever thinking it as a number. At least I bumped to the number reading this book, first time in my over 75 years of life. The magic number is PHI (1,618), the ratio of the long side to short side of any Golden cut surface. What is so special about that innocent-looking number?Read this book and you will be astonished. As if the whole Universe would be planned on the basis of this magic number: all 'natural' dimensions fron snow-flakes to the form of galaxies, masterpieces of painting, sculpture and music have this Golden ratio as the basic measure of their inner proportions. We seek it instinctively everywhere and are disappointed, if we do not find it. As an example I am extremely irritated of the brute deviation from this ideal of the format of paper journals; in addition to being unpleasant looking, they also are clumsily flabby for holding in hands.Despite of presenting the innocent looking simple number Phi (with alternative formulas behind it) this book plunges right away to the deepest mysteries of mathematics referring to dozens, hundreds of authorities. And yet, you will have no difficulty reading and understanding the text. High level mathematics is usually thought as pages full of formulas and dissertations of 20 pages, which very few persons understand. This book is not that way, although it goes far beyond mathematics requiring technical knowledge and skills. Just that is the fascination of the book. Embracing structures from flowers to houses and galaxies you get a fantastic feeling of better understanding what you see. Also another very rare feature is included. The host of personalities contributing to this discovery of hidden secrets of our world view are presented on everyday grass root level. Such well-known as Pythagoras, Newton, Gauss, Kepler, Einstein along with many less known but very important geniuses. Believe or not you have the feeling of meeting and chatting with them personally. A real magician this Mario Livio, five stars without any hesitation. Grateful to my friend Viljo, class mate beyond 60 years, who introduced this author to me. Reading already a second Amazon book by Livio, about The Impossible Equation.

Good reading despite the obvious and noteable omissions. As it was authored by a true giant of this field, this was personally disappointing as it negated a comprehensive and truly authoritative learning experience. Several critical themes were underdeveloped and essentially unexplored, for example, the author's rationale for his choice of pronunciation of 'Phi' ("Fee"). It may have been a safe option (possibly even prudent) to side-step this passionate area of debate, but this is the very reason we turn to the professionals to hear their expert opinions and consider their justifications garnered through their analysis and experience. A trivial example of a glaring omission is, although only nominally in nature, is the an extension of the irrepressible association of 5 and Fibonacci with Phi to his rejected prononciation of "Fi" and the first two letters in the spelling of 'five' (Fi-ve) and Fi-bonacci. Insignificant some of the oversights may be, but detracting to comprehensiveness. Nonetheless, still was a good read.

Excelente libro para personas curiosas que les gustan los libros. Como el título lo dice, el libro se enfoca en dar una reseña histórica sobre Phi desde sus inicios en la civilización egipcia, su relación con la sucesión de Fibonacci, hasta en las pinturas del renacimiento.

Being interested in design was familiar with the Golden Mean. This book explains not just Phi, the Fibonacci series and the golden mean but goes deeper into other mathematical series I had never encountered. Wow. Interesting stuff. I am not a mathematician by any stretch of imagination but I can appreciate the beauty of numbers the fit into pattern and even overlapping patterns. I appreciated the big picture story even though I could not always follow all the detail. Author also warns about interpreted use of Golden Mean as it fits nature, art and architecture and shows how results often are dependent on what one decides to include in the measurements.

You need to have a good grasp of arithmetic and geometry in order to get the most out of this fascinating book. It's early chapters cover the maths. that you will need to understand the Golden Ratio. Thereafter, the auhor shows how the Golden Ratio has impacted on human thinking about plant and animal structures, architecture, the visual arts, music and the world of finance. Importantly, the book debunks the myths surrounding the Golden Ratio. It is certainly a worthwhile purchase for those with a serious interest in the pervasive influence of the Golden ,Ratio on human thinking throughout many centuries to the present time.

Terrific book - very objectively written in a smooth flowing, easy-to-read style. Will definitely buy his next two books.