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.

This book, as its name suggests, is about an interesting number, the golden ratio (which I prefer to call "tau," but the author usually refers to as "phi," though explaining the reason for both symbols). For those who do not know what this number is, it can be defined in many ways, but the simplest is as the number which, when it is squared, is increased by 1. The fact that all the other definitions gives the same number is the reason for its great interest among recreational mathematics fans. The biggest problem with this book is that it tries to do two different things. One of the "two books" that I see in this one is about the _mathematical_ properties of the golden ratio. And this part of the book covers a lot of ground, and as a result I like it very much, as one of the few recreational math books I've seen recently that is easy to read yet still teaches me something I didn't know before I read it. The other part, however, is simply a refutation of claims made by many people that this or that artist consciously employed the golden ratio in his work. And it's interesting at first, but becomes tedious as he marshals more and more evidence refuting these claims. If the book confined itself to a discussion of the mathematical properties of the golden section (which is intimately related to such things as the Fibonacci sequence, Penrose tilings, and quasi-crystals), it would have merited 5 stars from me. But the attempt to refute all the artistic claims causes it to bog down for me, and causes me to cut one star off. One thing that totally puzzles me is his terminological decision to use "phi" rather than "tau." Since "phi" comes from a tribute to Phidias (a famous Greek architect/sculptor) and one of the points of the book is that neither Phidias nor his contemporary Greeks actually used the number in their designs, his statement that he uses "phi" to conform with most recreational math books is strange. I would, as I have said, gone with "tau," which was the earlier-introduced symbol and has the merit of coming from the initial of the Greek word for "section" (in keeping with the term "golden section" for this number).

I own several books on dynamic symmetry, which includes the Golden Ratio. Unfortunately all of the other books (3 of them, some rare and out of print) are heavily mathematic in their texts. I would say at least 90% math and about 10% English. So I was stuck trying to decipher the text via the sparce English. This book is not only fun to read (for a non-mathmetician type), it is fairly easy to understand (not a quick read, and at some points it takes some effort), and I would guesstimate about 80% English, 10% essential illustrations, and 10% math. And, since I find all this dynamic symmetry and Phi stuff fascinating--and do not want to go back to school to learn advanced algebra and trig--this book is a godsend! One more thing. This is written by a scientist that can write for non-scientists. He has done his research very thoroughly and covers the subject comprehensively.

Until now I during my life have been reading most books concerning history and mathematic. And I find that this book is the result of much research, as it contains much data concerning many people's individual works during thousands of year. And it's good by looking at the golden ratios connection to respectively architecture, nature, paintings, music, fractals, Wall Street and so on. That is, if there at al is any connection. But maybe, because born in Denmark I would have liked a couple of more lines concerning the connection between the Danish count Tycho Brahe and Johannes Kepler, that is, why Kepler at al became Brahe's assistant in Prague. And besides also with Brahe's connection to the count in Scotland who invented the logarithm, which Kepler used. The book is excellent with the many drawing and especially by the including of 10 mathematical appendixes back in the book.

Actually, Mario Livio's "What Makes Us Curious" is becoming my prequel to "The Golden Ratio" (next on my summer reading list). Long story, very short: My curiosity (much more than hobbyist-level math expertise) resulted in discovery of rPi (impromptu nickname; rPi = radial Pi, relating to this right triangle's first discovery within a circle). This ratio (1.1283791670955125738961589031215..) of the hypotenuse to long side of a circle-squaring right triangle effectively defines a circle and its square (a side of the square). Perhaps, "the world's most astonishing number" (Phi) will soon be eclipsed by "the world's most impossible number" (squaring the circle is impossible, according to Ferdinand von Lindemann, et al.). Update: August, 2017 Stop the presses! 2(sqrt(1/Pi)) is eclipsed by a new "golden rectangle" and its related Phi (iPhi) = 1.913058380271100794740307828.. (approx.) Re: The Golden Ratio by Mario Livio, 2002, p. 85. "The Golden Rectangle is the only rectangle with the property that cutting a square from it produces a similar rectangle." Impressive, however ... The new golden rectangle is the only rectangle with the property that its diagonal has length equal to the diameter of a circle and the long side (of the right triangle) has length equal to a side of that circle's square. The magic? The circle-squaring right triangle with hypotenuse-to-long-side ratio 2(sqrt(1/Pi)) and the new iPhi ratio. (Figure 26, p.85, of Livio's book shows a similar geometric pattern). Note: 'i' of "iPhi" alludes to the "impossible" squaring of the circle.

This is a pretty good book. I supposed mathematicians will love it. I must confess reading it made my mind spin enormous amounts of math. Unfortunately, I was disappointed to see the author end up the same way all writers do when they tackle things they can’t truly explain. So we end up with the same idea that this all man made , evolution , blah blah. But read it for yourselves and see that the author wants to say it . Intelligent design is real. The language of math is not some product of a monkey who became smart due to climate conditions , meteorites , etc etc. Pretty sure 2 plus 2 equal 4. Don’t believe me just look at your fingers.

Golden ratios and the like attract significant comment, much is hyperbole. Livio is particularly good at debunking these excesses, while leaving the good substance. The end section in which he discusses whether math was a something that was discovered or a construct. He sides with the idea that Math is something we construct. Bottom line, Math is Darwinian in that what works survives and is retained - a welcome rebuttable of the mumbo jumbo started by Aristotle and Plato that math reveals the rules by which all was created.

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.

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.

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.

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.

This book is fascinating. As an artist I was intetested in understanding the golden ratio as the claims it's in so much work from Greek architecture to Da Vinci confused me. This book explains it all so well and also sparked an interest in maths again. I never much liked it at school but now I'm on my way to being a maths geek.