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Available
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Publication
New York : Broadway Books, 2002.
Description
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
LibraryThing member melydia
If you divide a line so that the ratio of the smaller to the larger is equal to the ratio of the larger to the whole, you have the golden ratio, phi. There has been an abundance of literature on the presence of phi in a number of unexpected locations, and this book addresses many of these
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appearances intelligently. It is organized more or less historically, starting with the Pythagoreans' obsession with phi (due to its presence in the pentagon and other neat little number tricks) and continuing through the present. The author avoids doctoring numbers to fit phi into famous works of art and architecture, and indeed debunks several such cases. While some of the direct appearances of phi are pretty nifty (such as leaf growth patterns on plant stems), much of the book covers subjects that are only related to phi by a few generations, usually through the pentagon or the Fibonacci numbers. I do not fault the author for this; tangents are to be expected in books about such a narrow subject as a single number.The final chapter, "Is God a Mathematician," includes leading theories in response to that question (yes, no, and sort of) and Livio's personal opinion. I understand the desire to address such a topic, since mathematics is pretty amazing and phi is no small example of this, but this chapter seemed sort of forced, like the author was at a loss on how to wrap up the book. The explanation of the dual nature of light was sort of random, and the rather unsubtle promotion of Stephen Wolfram's then-unpublished book (which was not well received by the math community) was sort of irritating. I imagine that Livio's desire was to instill a lingering thirst for knowledge in his reader, to encourage further study, but it felt more like an advertisement for a newfangled religion that will change the way you look at the world. Despite the final few pages, I found this book to be informative and quite readable, which is always high praise for a book about math. Perhaps if Livio had left out his personal opinion I would have finished it feeling more satisfied. Show Less
LibraryThing member DirtPriest
The Golden Ratio, or phi, or even better, Φ, is an irrational number (equal to 1.618...) that pops up in many strange instances and has some odd properties. In its basic form, it is a simple ratio of a line segment divided in such a way that the proportion of the whole to the longer length is
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.
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equal to the longer length to the shorter.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.
Show Less
LibraryThing member hcubic
1.618033989... is a magic number. Its magic may not be as obvious as the most famous irrational, pi, nor as familiar as e (both of which are also transcendental), but its connection to the Fibonacci series (1, 1, 2, 3, 5, 8, ..., in which each element is the sum of the two previous) is a both
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intimate and surprising, and its role in the spiral of mollusc shells, inscribed pentagons, pineapple segments, fir cones, and the arrangement of seeds in a sunflower provides remarkable evidence that when nature speaks, she does so in the language of mathematics. Astrophysicist Mario Livio, who also wrote "The Equation That Couldn't Be Solved" (about group theory), takes a tempered approach to his subject. Claims have frequently been made that phi was the design principle of the Egyptian pyramids, the Parthenon, the works of Leonardo da Vinci, and in many other artistic creations. The evidence for most of them is weak - often based on a proportion that, when measured in a certain way, comes close to phi or its reciprocal. Livio is rightfully skeptical of most of these claims, but he also gives the Golden Ratio credit for works in which it is clearly implicated. This book provides a wonderful connection from science, art, music, and architecture to geometry and mathematics. Show Less
LibraryThing member themulhern
To write a whole book about the number zero requires a good deal of imagination and random association. That does not seem to be the case for the golden ratio. I've only gotten as far as Chapter 5, but there has been a good deal of interesting material presented. Chapter 2 introduces the
For the interested reader there are a bunch of proofs of various theorems in the back, which is a nice touch.
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Pythagoreans and many of their number theoretic discoveries. In this chapter the author gives a Babylonian formula for generating Pythagorean triples and argues that the existence of this formula and the fact that it generates a Pythagorean triple for every pair p, q where p > q proves that there are an infinite number of Pythagorean triples. I think that there are an infinite number, but the existence of the formula does not prove it. After all, the formula f(n) = (3 * n, 4 * n, 5 * n) yields an infinite number of Pythagorean triples, but they happen to be all in the same ratio, and therefore, essentially the same. Chapter 3 discusses assertions about the presence of the golden ratio in various ancient artifacts and buildings with scepticism. I liked the debunking, those pictures with a golden rectangle superimposed on the Parthenon always seemed questionable to me. Chapter 4 introduces lots of interesting properties of the ratio that were investigated before the dark ages. For the interested reader there are a bunch of proofs of various theorems in the back, which is a nice touch.
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LibraryThing member fpagan
Most of what you ever wanted to know about the irrational number 1.61803 39887 49894 .... (The value through 2,000 decimal places is on p 81.) But glossed over is the fact that it, unlike e and pi and all but a countably infinite subset of the other real numbers, is merely algebraic and not
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transcendental. (It is one-half of 1+sqrt(5), a solution of x^2-x-1=0.) Show Less
LibraryThing member jrgoetziii
This was a poor book. Aside from the many references to Livio's "friend"s, Livio accuses others of number-juggling while himself forgetting to explain how phi obtained the name "The Golden Ratio." He claims that there is no aesthetic appeal to the number, which may or may not be true; but in the
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process, explains only how it obtained the name "The Divine Proportion." Even so, he delves significantly off-topic, which was actually one of the better parts of the book. Which...says enough about the rest of it. Show Less
LibraryThing member EvilJohn
An intersting look at a fascinating number.
LibraryThing member palaverofbirds
For the love of Fibonacci numbers, roses, geometry and fractals. What do pineapples and spiral galaxies have in common? PHI! This is nerd candy.
LibraryThing member DanielAlgara
This was a great read, even for a math idiot like myself.
LibraryThing member jefware
The Golden Ratio dwells lovingly on each pearl in the necklace that is Phi.
Subjects
Awards
ALA Outstanding Books for the College Bound (Science & Technology — 2004)
Language
Original publication date
2002
Physical description
viii, 294 p.; 25 cm
ISBN
0767908155 / 9780767908153