Customer Reviews

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

5 star:		(38)
4 star:		(21)
3 star:		(19)
2 star:		(7)
1 star:		(6)

 

 

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.

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.

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.

Several years ago I prepared a review for amazon on this book. Since that time there have been many others to contribute. There are those like me who found it fascinating and gave it five stars, others that gave it a 4 or a 3 because they quibbled with the author over some mathematical issues and finally agroup that really hated it and found it boring and gave it only 1 or 2 stars. Some of those in the third group claim to be mathematicians but thought the book had too detailed. I don't see how a true mathematician could not love this book. Here is what I wrote that I still believe.

The book is 253 pages and 10 appendices about a number called the golden ratio. I give it 5 stars. It is a book for mathematicians and non-mathematicians alike. The first question I asked was how can an entire book be devoted to one number. Well Beckman wrote a book about the number pi and certainly that was interesting. There is a lot to say about the geometry of pi and many mathematical and statistical properties it has. Some including the Buffon needle problem are related by Livio in this book. He contrasts pi to the golden ratio (phi) which also has geometric and mystical properties. The quantity pi is a transcendental number meaning it is not the solution of any algebraic equation. On the other hand phi is algebraic as it is the solution to a quadratic equation.
Other strange properties of phi are:
1. If you subtract 1 from it you get its reciprocal
2. Add 1 to it and you get its square

To see the marvelous algebraic and geometric properties of phi you need only scan through the 10 appendices. Scan through the book and the pictures show you the many artistic properties related to phi.

Although algebraic phi is an irrational number. By applying the quadratic formula to its solution (see Appendix 5 in the book) you will see that its solution involves the square root of 5. Pythagoras and his followers in ancient Greece were said to have discovered irrational numbers (a natural consequence when you study right triangles) and hid this knowledge from the populous.

Phi is defined by Euclid as the "extreme and mean ratio". As Livio quotes Euclid " A straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the lesser". This leads to an equality of proportions that yields phi=1.6180339887 rounded to ten decimal places.

Livio also discusses the relationship between the ratio and our concept of beauty (i.e. the quality of the perfect face). It is also interesting that in his new book on the impossibility of solving the 5th degree polynomial by radicals Livio relates the Galois theory of groups to concepts of symmetry. There he also attributes our perception of besuty to symmetry.

If you have the time read the book thoroughly. Write a review that adds to what has been said if you like. Or skim through the pages and appreciate the artist properties of phi along with its algebraic and geometric properties. Read about fractals and myths. Enjoy this wonderful book!

This book, as its name suggests, is about an interesting number, the golden ratio (which I prefer to call "tau," but the author usually refers to as "phi," though explaining the reason for both symbols). For those who do not know what this number is, it can be defined in many ways, but the simplest is as the number which, when it is squared, is increased by 1. The fact that all the other definitions gives the same number is the reason for its great interest among recreational mathematics fans.

The biggest problem with this book is that it tries to do two different things. One of the "two books" that I see in this one is about the _mathematical_ properties of the golden ratio. And this part of the book covers a lot of ground, and as a result I like it very much, as one of the few recreational math books I've seen recently that is easy to read yet still teaches me something I didn't know before I read it. The other part, however, is simply a refutation of claims made by many people that this or that artist consciously employed the golden ratio in his work. And it's interesting at first, but becomes tedious as he marshals more and more evidence refuting these claims. If the book confined itself to a discussion of the mathematical properties of the golden section (which is intimately related to such things as the Fibonacci sequence, Penrose tilings, and quasi-crystals), it would have merited 5 stars from me. But the attempt to refute all the artistic claims causes it to bog down for me, and causes me to cut one star off.

One thing that totally puzzles me is his terminological decision to use "phi" rather than "tau." Since "phi" comes from a tribute to Phidias (a famous Greek architect/sculptor) and one of the points of the book is that neither Phidias nor his contemporary Greeks actually used the number in their designs, his statement that he uses "phi" to conform with most recreational math books is strange. I would, as I have said, gone with "tau," which was the earlier-introduced symbol and has the merit of coming from the initial of the Greek word for "section" (in keeping with the term "golden section" for this number).

"The Golden Ratio" is truly master piece by Mario Livio. I was impressed by Livio's first book: "The Accelerating Universe", not just by its style and contents, but also by the clear and engaging manner in which Livio had presented the material. In this book Livio has exceeded even the high standards of his earlier book in producing what I found to be a very fascinating and educative book on a seemingly ordinary number.

Phi, the so called 'golden ratio', originated from a geometrical concept: if we divide a line into two segments such that the ratio of the length of whole line to that of the bigger segment is the same as the ratio of the length of the bigger segment to that of the smaller one, then this ratio is 'phi'. On the face of it, this looks like a mundane, dull and insignificant ratio. Then Livio leads us through innumerable examples --- examples as varied as the breeding patterns of rabbits to optics of light rays --- where the same golden ratio appears again and again. With Livio's magic touch, the seemingly dull number widens to a fascinating world of its own. We find the same ratio hidden in the delightful petal arrangements in a red rose, in Salvador Dali's famous painting "Sacrament of the last supper", in the spiral shells of mollusks, in the spiral patterns of face-on Galaxies, ..... the list goes on. The book also contains philosophical discussions on such topics as "Is God a Mathematician?", where Livio tries to pierce through the meaning of it all.

This book is likely to remain a classic and true source book on the golden ratio for a long time. The book is full of information, and cleverly written. It makes for a very interesting reading: and in the process you will not only learn all about the fascinating world of the golden ratio but also about paintings, flowers, astronomy, and a lot more, in an effortless and enjoyable manner.....

Following in the steps of his earlier, just as fascinating account linking cosmology and the arts ("The Accelerating Universe"), Mario Livio continues to prove he is one of the most original, exciting and literate writers of popular science today. "The Golden Ratio" is a witty and learned journey generally following the trail of the number Phi, but stopping along the way to take in subjects as diverse as philosophy, history, art, religion, the sciences, architecture, etc.

Writing about science in a way that is both knowledgeable and understandable for the common reader is an infamous hurdle, but Livio leaps over it with the greatest of ease, giving clear explanations of every potentially difficult matter and providing the scientific proofs in the appendices, for those more mathematically inclined. Overall, though, it is the great humanity of Livio's worldview that shines through the book and makes it, at least for me, one of the most memorable reads of the year.

Well, when it comes to information about the number phi, I would say this book is the only game in town. It has flaws and is over-complete however and that is the reason I gave 4 and not 5 stars.
It starts off too slow in a history of some key-figures in the number phi. We all know about Kepler and Durer, and I would say the book would have been better off just dealing with examples that deal, or might deal, with phi.
A strong point of the book is that it covers some interesting examples of well-known cases that were supposed to have the secret of this number built-in in their construction, such as the works of Beethoven and the great pyramid, but after scrutinizing, failed the test. To some readers these examples were not relevant. I do not agree; since you hear them so often quoted, in e.g. Da Vince code (were the teacher said wrongfully that phi was deliberately used in them) it was a good idea to include them here to dismiss these errands once and for all. This way, the book gives a good total picture of what phi is, and is not about. Phi was simply not used in the pyramids nor by Vitruvius, although all too often we read otherwise. I myself have read De Architectura and Vitruvius only talked about nice proportions that were important in architecture, for which he used fractions that didn't even come close enough to phi to suppose humans have a built-in intuition for this proportion.
Of course phi is a number that really does appear in nature. As most of us know by now, phi can be found in rose leaves, fractals and more. Interested as I was on where phi can be found and how, I was not disappointed in this book. The writer explains all these examples well and although this section only comprises the middle of the book, it alone is worth the buy.
I do agree however that some subjects are a bit beyond the scope of the book, especially the last part. Examples are whether maths is discovered or invented. Interesting as these questions are, they really belong to the philosophy of maths and I would recommend the readers who are interested in these subjects to check out books written by S. Shapiro.
My conclusion would be: if you want to know about phi, this book is still the best around

As a non-mathematician I appreciate any help I can get in understanding the more esoteric parts of math. The Golden Ratio is just such a concept. Fortunately, Mario Livio has shown much light on this remarkable corner of geometry in his book "The Golden Ratio."

It is little wonder that such numbers as the Golden Ratio were considered magical. The never ending, never repeating number that cannot ever be expressed as a fraction has an uncanny tendency to show up in the oddest places, not only galactic structure and nautilus shells, but in plant parts and composition of paintings and music. Unfortunately magical numerology can lead to far-fetched relationships, as to the so-called number of the beast (666), and to academicism in art. Just because the Golden Ratio results in a pleasing relationship in a composition we are not tied to always measure art on how well it fits that ratio!

Livio has illuminated the history of the Golden Ratio in such a way that much of the associated themes can be understood by the reasonably educated laymen. While some of the book can be tough sledding for most of us non-mathematicians, the gist is available to all with some effort.

Read this book to learn about the history of interpretation and misinterpretation of mathematical concepts.

This book does provide an excellent account on the history of Phi. That much is not in dispute. However, what is in dispute is the way in which Mr. Livio goes about "exploding" the "myths" of Phi. Rather than taking such a rigid approach to the mystical history of Phi, Livio should have taken greater care to state that, in many of his examples, we really don't know one way or another. For example, to say that the Egyptians had no knowledge of phi, based on the great pyramid being off by however miniscule of a percent, is at best nitpicking and at worst ridiculous. Such a discrepancy could be attributed to mere erosion. This is just one example, but read his numerous criticisms and the ask yourself whether he his truly proving his case or just blowing smoke. The biggest error of all in his reasoning is that Phi has not been proven to truly relate to aesthetic preference. He should have done more homework, and included Feckner's study on shape preference. It clearly presents evidence to the contrary. In general, this book's greatest flaw is the author's poor attempt to make a name for himself by slandering the work and theory of numerous figures without backing it up with any real evidence. Perhaps we're just supposed to take his word for it. I, for one, cannot, for blind cynics are just as bad as the blindly credulous.