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

It is little wonder that such numbers as the Golden Ratio were considered magical. The never ending, never repeating number that cannot ever be expressed as a fraction has an uncanny tendency to show up in the oddest places, not only galactic structure and nautilus shells, but in plant parts and composition of paintings and music. Unfortunately magical numerology can lead to far-fetched relationships, as to the so-called number of the beast (666), and to academicism in art. Just because the Golden Ratio results in a pleasing relationship in a composition we are not tied to always measure art on how well it fits that ratio!

Livio has illuminated the history of the Golden Ratio in such a way that much of the associated themes can be understood by the reasonably educated laymen. While some of the book can be tough sledding for most of us non-mathematicians, the gist is available to all with some effort.

Read this book to learn about the history of interpretation and misinterpretation of mathematical concepts.


Astonishing, Math Book Made Interesting   February 19, 2007
J. head (littlteton, nh USA)
8 out of 8 found this review helpful

There are many books explaining the places where the value PHI and the Golden Ratio show up. One discovers that maybe the ancient Greeks based their architecture upon it, and the value was one of the mysteries of the Pythagoreans. Some authors explain this golden ratio to the dimensions of the pyramids. Some books contain the view that the architecture of all life contains an undercurrent of the value of PHI. I have read many books about this subject and this is the book to read. It runs the whole gamut, history, architecture, art-history, and mathematics. The author, Mario Livio, cites many others books and studies, often critically showing that not all artists that knew of the golden ratio used it in their art and that any monument's dimensions when forced through enough mathematical gyrations will portray whatever is desired. The illustrations within the book are superb, right to the point and easy to understand. The author explains a difficult concept in an easy to read manner, even though it involves Fibonacci sequences and fractals. The author is Head of the Space Telescope Science Institute , he knows the subject of which he writes and more importantly can relate this knowledge to the reader. Kudos to the author, well done.


beauty, mathematics, mysticism: all in a number   December 15, 2002
Michael R. Chernick (Malvern, PA)
16 out of 31 found this review helpful

This is a book of 253 pages and 10 appendices about a number called the golden ratio. I confess that I have not read it thoroughly. But I believe that I have seen enough to give it 5 stars and review it intelligently. It is a book for mathematicians and non-mathematicians alike. The first question I asked was how can an entire book be devoted to one number. Well Beckman wrote a book about the number pi and certainly that was interesting. There is a lot to say about the geometry of pi and many mathematical and statistical properties it has. Some including the Buffon needle problem are related by Livio in this book. He contrasts pi to the golden ratio (phi) which also has geometric and mystical properties. The quantity pi is a transcendental number meaning it is not the solution of any algebraic equation. On the other hand phi is algebraic as it is the solution to a quadratic equation.

Other strange properties of phi are:
1. If you subtract 1 from it you get its reciprocal
2. Add 1 to it and you get its square

To see the marvelous algebraic and geometric properties of phi you need only scan through the 10 appendices. Scan through the book and the pictures show you the many artistic properties related to phi.

Although algebraic phi is an irrational number. By applying the quadratic formula to its solution (see Appendix 5 in the book) you will see that its solution involves the square root of 5. Pythagoras and his followers in ancient Greece were said to have discovered irrational numbers (a natural consequence when you study right triangles) and hid this knowledge from the populous.

Phi is defined by Euclid as the "extreme and mean ratio". As Livio quotes Euclid " A straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the lesser". This leads to an equality of proportions that yields phi=1.6180339887 rounded to ten decimal places.

If you have the time read the book thoroughly. Write a review that adds to what has been said if you like. Or skim through the pages and appreciate the artist properties of phi along with its algebraic and geometric properties. Read about fractals and myths. Enjoy this wonderful book!


Why did I become so old before this book was written?   September 7, 2005
Cecil Fox (Little Rock, AR United States)
4 out of 5 found this review helpful

I feel that my life has mostly slipped by without knowing more of the Golden Ratio. I knew that windows in Oude Delft in Holland are Golden Rectangles, and I had some inkling of the shape of proportion in Hogarth, but Livio has sent me scurrying to get the marvelous new edition of Euclid (Amazon) and to find my compass and protractor. But learning on my own shows the shallowness of my understanding and the brilliance of this ultimate exposition of the liberal arts.


A strange, beautiful, and rare bird!   June 4, 2007
Reading Fan (Baltimore)
1 out of 2 found this review helpful

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!