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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Not a novel
This is a book of facts written by a mathematician and is filled with facts and well thought out arguments. I loved reading it. I found it captivating, scientific, logical, well explained for the non-analytical brain. Well organized and planned out. Touches on the subjects of science, math, art, poetry, music, geometry, God, and a conclusive chapter. I enjoyed the straight-forwardness and the dismantling of lies told by others who simply want to imagine the divine where it is not.If you are looking for fictional science or a novel like reading books without conclusive supportive evidence then this is not the book for you.If you want the facts from a logical perspective without much force feeding you will probably enjoy this book.I appreciate the names and book references along with dates in this book because it aids in the learning experience garnered from the time spent reading. It is very explanatory and kept my attention throughout, and included many visual aids.I bought this book second hand from a thrift book store and the last five or so pages were after the appendix and upside down. Other than that the print is a little small but my eyes are not as young as they used to be.I really enjoyed the book, it was exactly what i was looking for- facts and not fluff. Science not imagination. Logical and organized. It read just as good as a captivating novel. My thanks to the author, its not common in my experience to come across someone with logic who can also tell a very good story of facts.
Top reviews from other countries
Peter Mills
Enlightening for those prepared to put in the effort
Reviewed in the United Kingdom on July 26, 2015
