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Showing 1-10 of 85 reviews(Verified Purchases). See all 165 reviews
on September 9, 2017
Needless to say everyone needs to know the subject. I read this book and brushed up on my old knowledge. This made me enjoy the inner beauty of buildings and monuments from perspective of its engineers on my travels abroad. This book is full of indispensable knowledge. Some fundamental high school knowledge is needed for better comprehension of materials. It is well worth the time and the money
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on June 24, 2013
As required for a school project, I chose to touch upon the subject of the mysterious, yet ever present ratio of Phi. I had already known quite a bit about the ratio and its presence in architecture around the world through the vast reaches of human history, but this book showed me a world more. Some theories that I had previously believed were discussed in an unbiased, scientific way and showed that however close they were to being representative of phi, were actually not.

Likewise, it shows how seemingly unrelated objects share this golden ratio in their structures, throughout nature and even further out into the reaches of space. If you want a clear definition of the ratio of Phi and want an unbiased analysis of the world's most geometrically beautiful structures and patterns, this book is a must.
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on August 28, 2017
Have several of Mario Livio's books and he never disappoints. I enjoyed it and would recommend it.
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on June 11, 2017
A wonderful book.
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on May 21, 2017
Excellent book
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on September 10, 2016
authors goal is to ascertain, in the chapter on art, when the golden mean is applied. he certainly left Charles bouleau in a tenable position, which brings up another point. 14 & 15 century Italian artists had strict contracts that dictated subject, composition, colour, & delivery. so if it was applied the patron was dictating.
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on April 1, 2017
I like this book so much that I ordered copies for two of my friends.
Anyone interested in numbers will enjoy this book.
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on March 20, 2017
Math History generally clear
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on April 19, 2013
got this for hubby. he loves it. fascinating book. good price and good quality. would recommend it for someone interested in learning about this.
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on June 3, 2007
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!
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