The Golden Ratio: The Story of Phi, the World's Most Astonishing Number (Hardcover)

~ (Author) "The famous British physicist Lord Kelvin (William Thomson; 1824-1907), after whom the degrees in the absolute temperature scale are named, once said in a lecture:..." (more)
Key Phrases: golden ratio, divine proportion, baby pairs, Great Pyramid, Golden Rectangle, Luca Pacioli (more...)

Editorial Reviews

From Publishers Weekly

Most readers will have at least dim memories from geometry class of the irrational number pi. Theoretical astrophysicist Livio gives pi's overlooked cousin phi its due with this lively account, the first on the subject written for the layperson. Phi is the golden ratio of antiquity (1.6180339887), a never-ending number so lauded for its harmonious qualities that in the 16th century it was dubbed the divine proportion. It is related to phenomena as diverse as the petal arrangements of roses, the breeding patterns of rabbits and the shape of our galaxy. Phi is also claimed to have been crucial in the design of the Great Pyramids, the composition of the Mona Lisa and the construction of Stradivarius violins. Livio (The Accelerating Universe) carefully investigates these and other claims and does not hesitate to debunk myths perpetuated by overzealous enthusiasts he calls "Golden Numberists." This is an engaging history of mathematics as well, addressing such perennial questions as the geometric basis of aesthetic pleasure and the nature of mathematical objects. Useful diagrams and handsome illustrations of works under discussion are amply provided. Livio is gifted with an accessible, entertaining style: one typical chapter bounds within five pages from an extended discourse on prime numbers to a clever Oscar Wilde quote about beauty to an amusing anecdote about Samuel Beckett and finally to an eminently clear explanation of G"del's incompleteness theorem. With a guide to the history of ideas as impassioned as Livio, even the math-phobic can experience the shock and pleasure of scientific discovery. This thoroughly enjoyable work vividly demonstrates to the general reader that, as Galileo put it, the universe is, indeed, written in the language of mathematics.
Copyright 2002 Reed Business Information, Inc.

From School Library Journal

Adult/High School-Take something as simple as a line segment and mark it at just the right place. Looking at it with a mathematician's eye, an interesting relationship appears: the ratio between the whole line and the larger of the pieces it was broken into is the same as the ratio of the larger piece and smaller piece. Better known as "the golden ratio" or phi, 1.618- is a number that has fascinated humans for several hundred years, and people have claimed evidence of phi in all manner of things. Livio takes readers on a treasure hunt for phi from ancient times through the present. On the way, he debunks a number of popular myths (e.g., the notion that Mondrian used it in his abstract paintings) and does a wonderful job explaining the Fibonacci sequence and its relationship to phi. Small, black-and-white photos and reproductions demonstrate items mentioned in the text. While it may seem that the author wanders in his expositions, his excursions into history and number games add fun and depth for those who wish to follow. To get the most out of The Golden Ratio, it is best to have an understanding of algebra and basic trigonometry, although the book is great for general readers who don't mind working a little to gain a lot of understanding.
Sheila Shoup, Fairfax County Public Library, VA
Copyright 2003 Reed Business Information, Inc.

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Product Details

• Hardcover: 320 pages
• Publisher: Broadway; 1 edition (October 29, 2002)
• Language: English
• ISBN-10: 0767908155
• ISBN-13: 978-0767908153
• Product Dimensions: 9.3 x 6.2 x 1.1 inches
• Shipping Weight: 1.3 pounds
• Average Customer Review: 3.8 out of 5 stars  See all reviews (87 customer reviews)
• Amazon.com Sales Rank: #189,457 in Books (See Bestsellers in Books)

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Customer Reviews

Most Helpful Customer Reviews

125 of 130 people found the following review helpful:

5.0 out of 5 stars "The World's Most Astonishing Number", April 7, 2003

By	Timothy Haugh (New York, NY United States) - See all my reviews (TOP 500 REVIEWER)    (REAL NAME)

Are some numbers more important than others? Certainly numbers like the primes, pi and "e" have properties that make them interesting to mathematicians and physical scientists alike. Then there are numbers like 7, 13 and 666 that have other connotations for theologians, numerologists and the like. And yet, some numbers have not gotten their due in recent years. Phi--a number variously referred to as the golden ratio, golden section, and divine proportion among others--is one. But Mario Livio has written a book in an attempt to remedy this situation.

Phi received its original definition from Euclid as an "extreme and mean ratio" when a straight line is cut so that the ratio of the entire line to the longer division of the segment is the same as the ratio of the longer division of the segment to the shorter. And yet, much like the better known geometrical example of pi, phi turns out to have many more applications beyond its simplest geometrical definition. Though measurable, phi is an irrational number with relationships to the Fibonacci sequence, fractals, the physical structure of things from plant growth and spiral shell development to the appearance of large-scale objects like galaxies, and more. And beyond this, phi has been used as a basis applications in numerology and aesthetics.

Livio does a very good job of covering all this ground and more. He is especially good at giving us a historical overview of the development of our understanding of this important number as well as explaining the mathematics in a way that is complete but easy to understand. He is also very good at presenting the various mystical ways phi has been interpreted over the centuries, giving each a rigorous challenge--rejecting many but open-minded to the possibilities that any good Platonist would be.

In fact, if there is a weakness in this book, it is that Livio spends a lot of time covering these more esoteric applications of phi. And yet, these applications are part of the history of the number and cannot be ignored whatever a reader might feel about the value of these applications. Phi may not quite live up to the hype as "the world's most astonishing number" but certainly any reader with an interest in mathematics will not want to miss this book.

97 of 102 people found the following review helpful:

4.0 out of 5 stars Interesting book, interesting number, February 8, 2003

By	mrliteral - See all my reviews (TOP 500 REVIEWER)

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In mathematics, there are a few irrational numbers that stand out from the infinitely crowded field. First in the bunch would probably be the square root of two, which was kind of the "first" irrational number. Then there is the everpresent pi, and then the less well-known but completely important "e". If there was a fourth place prize, however, it would probably go to the golden number, phi, or roughly 1.618.

In this book, Livio gives a brief history of mathematics and phi's place in it. Intimately related to the Fibonacci numbers, a sequence of numbers in which any given number is the sum of the previous two (after the first couple); these numbers (1,1,2,3,5,8,13...) have shown up in some unlikely places such as sunflowers and nautilus shells. Livio shows us the significance of phi in both the mathematical and physical world.

Livio also makes a good case that phi may be the most overrated of all numbers. Although it has a wonderfully golden name, it actually doesn't live up to its reputation; Livio shows that phi's presence in art and architecture is more fictional than real and that there is nothing about phi that automatically confers aesthetic beauty. A good portion of the book is dedicated to debunking these golden myths.

Overall, this is a good book. Livio's writing is appealing to both mathematician and non-mathematician alike. He does have a tendency to meander from his topic, which can be distracting (even if entertaining), although he eventually does get back on track. For those who like reading about math and the significance of certain numbers (I have also read books on pi, e, i, 0 and infinity), this is a worthwhile read.

46 of 48 people found the following review helpful:

5.0 out of 5 stars All About 1.618033988749894848204586834365638117720309179..., January 14, 2003

By	R. Hardy "Rob Hardy" (Columbus, Mississippi USA) - See all my reviews (TOP 50 REVIEWER)    (REAL NAME)

Of all the irrational numbers, the best known is pi, which shows up all over the place. However, if you read _The Golden Ratio: The Story of Phi, The World's Most Astonishing Number_ (Broadway Books) by Mario Livio, you will gain an appreciation for the ubiquity of another irrational with all sorts of amazing properties. You can try this one on your calculator: Phi equals 1.6180339887... (As an irrational, its string of numbers goes infinitely beyond the decimal point, and you can be sure computers have calculated it to millions of places). Take the inverse of that number; that is, divide it into one. You will get 0.6180339887...; in other words, the inverse looks just like phi itself, but with a zero instead of one left of the decimal. Or try this: start with a 1, followed by a 1. The next number will be the two previous ones added together, which is 2; the next number, in turn, is again the two previous ones added together, which is 3. The series goes 1, 1, 2, 3, 5, 8, 13, 21, 34, 55... This is the famous Fibonacci sequence, and is investigated widely within this book as it is intimately connected to phi. Take any number in the series and divide it by the number before it, and you will get a number close to phi; the higher the number in the series, the closer the result comes. (13 / 8 = 1.615 ; 55 / 34 = 1.6176....).

These sorts of number tricks abound in Livio's book, and the mathematics is not daunting. It is also a history of phi, which turns out to be a representative slice of the history of mathematics. Euclid knew the number, but Leonardo Fibonacci in the twelfth century developed the series with its ratio. It shows up in breeding rabbits; spirals in pine cones, sunflowers, galaxies, and hurricanes; tilings and fractals; and many more surprising places. Livio has enormous fun giving and explaining all these examples. Showing up as it does all over the place, perhaps phi is just being seen because that is what is being looked for. Livio, whose day job is being Head of the Science Division at the Hubble Space Telescope Science Institute, is refreshingly dismissive of attempts to try to see a Golden Ratio in everything, which people have tried to do for centuries. It isn't in the pyramids, nor in the Parthenon, nor in Leonardo's paintings.

Without forcing the issue, however, it is easy to see that the Golden Ratio, logarithmic spirals, and Fibonacci numbers are all over the place; there is even a _Fibonacci Quarterly_ mathematical journal. This leads to larger final issues, which Einstein expressed as the question, "How is it possible that mathematics, a product of human thought that is independent of experience, fits so excellently the objects of physical reality?" Do mathematical concepts have a universal and timeless existence "out there" and are just waiting for us to discover them? Or is mathematics a human invention that resides only within the human brain? It can't be surprising that this classic conundrum is not definitively solved here. Livio's ideas about it, however, well expressed and tied to this remarkable numerical constant, are well worth thinking about.