The Golden Ratio: The Story of PHI, the World's Most Astonishing Number

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.


All About 1.618033988749894848204586834365638117720309179...   January 14, 2003
R. Hardy (Columbus, Mississippi USA)
41 out of 43 found this review helpful

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.


Remarkable Catalyst for Thought   March 17, 2003
Francis J. Mcinerney (Commonwealth)
35 out of 36 found this review helpful

Author Mario Livio has added another wonderful book that bridges the gap between writings meant for the academic and for the layperson. "The Golden Ratio", was not quite as accessible as some other books that I have read, but it is certainly worth the extra effort even if many of the proofs found at the book's end remain a mystery.

The book spent a great deal of time disproving the myths of application that have been attributed to the remarkable relationships of the number PHI. Like the more familiar PI it is a decimal that has yet to be proven to ever repeat itself, and it demonstrates its presence both widely and in fascinating manners. Whether or not Mozart used the number or Leonardo Da'Vinci did, or even if a building long considered to be predicated upon the number truly is or is not, does not detract from the wonders of this numbers appearance.

The writer will take you through the commonality in the structure of a Nautilus shell, the arrangement of leaves around the stem of a plant, and even how by tossing a coin can prove truth or fraud in the accounting practices of business. He shares an example of having two groups, one tosses a coin 200 times and records each result. The second group does nothing with the coin, they just manufacture the results. The two groups can be detected, the why is a wonderful find.

Certain shapes whether they are the outline of a room or the dimensions of a painting will generally be found the most pleasing by the majority of people. And lest you think these rules are confined to the shell on the beach or an image on your wall, they extend to those galaxies of which we are a part, why planets move in the orbits they do, and what would happen if the slightest changes were made.

As I said, this particular work requires more effort than others I have read and commented upon recently, and the enjoyment or aversion you have to math will play a role in how much pleasure this read will bring. However, even if the formulas leave you cold, the spirals upon rectangles that form, "The Eye Of God", are guaranteed to fascinate.


Don't be scared off; it's for non-math folks like you and me   December 19, 2002
J Scott Morrison (Middlebury VT, USA)
11 out of 11 found this review helpful

I don't want to repeat what others have written, but I must tell you this is a book that is so reader-friendly that you hardly realize you're being gently led into greater mathematical understanding painlessly. Along the way you learn all kinds of fascinating things about this almost mystical number, phi (which, strangely, was known as tau until about 1900).

Livio's writing is a pleasure. He obviously did not simply pile up a lot of facts until they were book-length. The narrative takes an orderly path, moving from concept to concept, tying things together rather like James Burke did in his column, "Connections" in Scientific American, and in PBS documentaries. You start out in Egypt and wind up, say, talking about the music of Bartok or Messaien. I love how this guy's mind works.

So, jump in. The water's phine.


Dense   November 4, 2004
Erika Mitchell (E. Calais, VT USA)
7 out of 8 found this review helpful

This book is an exploration into the history and concepts of Phi, the Golden Ratio. Livio begins his story by tracing the earliest known uses of numbers and counting systems. He progresses through Pythagoras and the discovery of irrational numbers. The history of Phi takes us to many mathematicians and their work, including Plato, Euclid, Abu'l-Wafa, Fibonacci, Pacioli, and Kepler, from the foundations of plane geometry through computer-generated fractals. Livio describes the special properties of Phi in triangles and pentagrams, the mysterious Fibonacci series phenomena, and the beauty of equiangular spirals. Concepts and examples are illustrated throughout the book with formulae, diagrams, drawings, and black-and-white reproductions of paintings. At the end of the book are appendices with geometrical and mathematical proofs, an extensive list of suggestions for further reading organized by chapter, and an index.

Livio cites dozens of examples of how Phi comes into play in geometry and mathematics, and he also points out where the numbers or shapes relating to the Golden Ratio appear in nature, such as in snail shells or pineapple scales. He then examines claims that the Golden Ratio was used explicitly in art, music, and poetry, and argues that such claims, are the for the most part, widely overstated. Indeed, only a few artists, musicians, or poets have explicitly attempted to work with the Golden Ratio (Le Corbusier, for one), and much of their work comes across as somewhat contrived rather than natural. On the use of the Golden Ratio for aesthetics, Livio concludes, "In spite of the Golden Ratio's importance for many areas of mathematics, the sciences, and natural phenomena, we should, in my humble opinion, give up its application as a fixed standard for aesthetics, either in the human form or as a touchstone for the fine arts.

The material in this book is quite fascinating for those with a mathematical bent. Even readers who have heard of and worked with the Golden Ratio before are bound to come across new facts, details or phenomena involving the Golden Ratio that they were not aware of before. Livio's presentation draws many, many facets of the Golden Ratio together into one cohesive story, while analyzing some claims for uses of the Golden Ratio in the arts that perhaps go too far. Livio assumes that readers will have a decent background in algebra and a firm foundation in geometry. The material in this book is dense and requires thoughtful reading, so it is not a quick read, but it is quite informative and interesting.