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Throughout history, thinkers from mathematicians to theologians have pondered the mysterious relationship between numbers and the nature of reality. In this fascinating book, the author tells the tale of a number at the heart of that mystery: phi, or 1.6180339887 ... This curious mathematical relationship, widely known as "The Golden Ratio," was discovered by Euclid more than two thousand years ago because of its crucial role in the construction of the pentagram, to which magical properties had been attributed. Since then it has shown a propensity to appear in the most astonishing variety of places, from mollusk shells, sunflower florets, and rose petals to the shape of the galaxy. Psychological studies have investigated whether the Golden Ratio is the most aesthetically pleasing proportion extant, and it has been asserted that the creators of the Pyramids and the Parthenon employed it. It is believed to feature in works of art from Leonardo da Vinci's Mona Lisa to Salvador Dali's The Sacrament of the Last Supper, and poets and composers have used it in their works. It has even been found to be connected to the behavior of the stock market! This book is a captivating journey through art and architecture, botany and biology, physics and mathematics. It tells the human story of numerous phi-fixated individuals, including the followers of Pythagoras who believed that this proportion revealed the hand of God; astronomer Johannes Kepler, who saw phi as the greatest treasure of geometry; such Renaissance thinkers as mathematician Leonardo Fibonacci of Pisa; and such masters of the modern world as Goethe, Cezanne, Bartok, and physicist Roger Penrose. Wherever his quest for the meaning of phi takes him, the author reveals the world as a place where order, beauty, and eternal mystery will always coexist.… (more)
User reviews
It pops up in the construction of a pentagon and the pentagram (the sides of the 'star' are divided in this ratio) and, supposedly, in the Parthenon of Athens. There is a tendency for leaves and twigs to branch off at angles determined by phi, the proportions of the human body as well as many animals are believed to be determined by Φ, and on and on.
The author goes through a long series of alleged uses of Φ, some of which are debunked (the Parthenon and the Pyramids) and some seem to be legitimate. As is to be expected, there is considerable information on the Fibonacci numbers, where the new term in the series is the sum of the two proceeding, this series converges to phi and has many appearances in nature, like the left and right hand spirals of a sunflower's seeds are always consecutive Fibonacci numbers. The math in the book is minimal, proofs are relegated to a series of appendices, and the bulk of things actually revolve around the concept of Φ being 'aesthetically pleasing', a long standing idea. Livio examines art that is claimed to use Φ in its dimensions somehow and tends to fail to find it.
The best way to sum up this book is that I drew out a logarithmic spiral for a friend and mentioned that any line drawn through the center of the spiral intersects the spiral over and over at the exact same angle. His response was, 'So what?', and yours probably is as well, but he was interested to know that a diving hawk follows this exact spiral so it can easily keep its eyes on its prey on the ground. I guess that's enough.
For the interested reader there are a bunch of proofs of various theorems in the back, which is a nice touch.