4 stars for the depth of research and willingness to de-bunk a lot of the mythic nonsense spread around about the golden ratio.

Conveys a complete and accurate understanding of the number and its significance both artistically and historically.

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

that's alot of hype for the irrational number 1.6180339887... . the author barrages the reader with hyperbole. 'mysterious', 'astonishing', 'amazing', 'wonderful', 'beautiful', 'fascinating', 'curious', 'crucial', 'unimagined', 'divine', etc. etc. not just astonishing, but "the World's Most Astonishing Number".

horse feathers. the people who are 'fascinated' by this are the same who freak out when they see 11:11 on a digital clock; the same who have 'lucky' numbers; the same who fear Friday the 13th.

in fact there are more 'crucial' AND more 'astonishing' numbers. how about 0 or 1 or 2 or 10 or infinity? i guarantee you that if we changed our everyday number base from 10 to , say, 13 that the wheels would fall off of this old bus. now THAT is 'crucial'. and 'divine'? please! what could be more 'divine' than 1? maybe 2 :-) . 'astonishing'? 1 is 'astonishing'. it factors into EVERYTHING! it's everywhere and in everything. 0 doesn't factor into anything. these are more 'astonishing' than phi.

Dali knew how to capitalize off of frenzied hype, so he threw together the "Sacrament of The Last Supper" and when phi's superstitious cultists found out that it featured the 'divine' proportion they took care of turning that ugly, mediocre effort into a 'divine' icon.

the author is supposedly a PhD? whatever.

I bought this book with a thirst to know about this number phi. I did learn about the number phi. However a large part of the book was devoted to instances where various people thought the number phi was present but the author spent considerable time developing the opinion or fact that phi was not influencing this or that particular instance. I got REALLY tired of that.

For me, the first chapter and a half or so and the last two chapters were the meat of the matter for my interest. The book was worth it for the last chapter.

I think that the author would have been better to write a book titled "Why Is Mathematics So Effective?" That seemed to be the central question that really drove the author.

I don't regret reading it. I just feel it wasn't really the book I signed up for.

One of the best books I've read. It is an in depth study of the Golden Ratio...the history, purpose, relationship to other concepts. I am intrigued by math, art, and science and found this book very, amusing. You will need a basic understanding of high school math to fully appreciate some of it. Oh, by the way, the author shoots down most other author's claims that the golden ratio has been used in classic architecture and art. Superb job Mario Livio!

I happened to notice that he says Babylonians found the general solution for the quadratic. General solution of the quadratic was given by Bhaskara. The author has not read Fibbonaci's book. Fibonacci himself said in the preface that he learnt new math from India. Fibonacci numbers were found by Hemachandra. there were many other errors...I would not recommend to my students

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 confess that I have not read it thoroughly. But I believe that I have seen enough to give it 5 stars and review it intelligently. 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.

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!

Highly readable and fascinating book by the well-respected Mario Livio. See http://en.wikipedia.org/wiki/Mario_Livio The book does not require a math background to understand or appreciate. Traces the origins and applications of the golden ratio through time, nature and art. Explores (and generally refutes) myths and misconceptions about the golden ratio. Highly recommended. Fascinating reading. Dan Brown (author of "the DaVinci Code") reportedly loved the book.

I like Livio's cautionary tone throughout the book, ie he's not a starry-eyed Golden Numberist, who believes that the Ratio is the only measure of beauty and proportionality in the universe. That's probably his strongest point.

For one, I'm not so interested in the application of the Golden Ratio or Fibonacci sequence (also somewhat discussed in the book) to art in general. What's so great about Salvador Dali using the Ratio in "Sacrament of the Last Supper", or its apparent extensive use by Bela Bartok in his compositions? In fact, I found a tad irritating Livio's habit of filling page after page discussing the supposed use of the Ratio by some painter or musician, just to conclude that probably it was all just coincidental. Also annoying, at the beginning of the book, is his characterization as "famous" applied to almost every person cited. As if I didn't know that Einstein, Buffon, Pythagoras and Lord Kelvin are all famous!

On the other hand, I enjoyed the book rather terse treatment of the Ratio and the Sequence in pure plane geometry and Platonic solids, and even more interesting is the brief discussion of their serendipitous presence in nature --chambered nautilus and tree's leaf arrangements (phyllotaxis). Brief, sadly.

I also confess to having gotten a little bored and glassy-eyed, and pretty often had to lie down, half-asleep, on my bed while reading this book. Livio often goes off on a tangent, as I said before, following false leads, dead-ends, and even the occasional windmill. After all, I guess that's the price the reader has to pay when the author's trying to meet the publisher's magical, and very much de rigueur, rational proportion of about 250 pages per book.

In conclusion, and on a positive note, this book at the least (re)sparked my interest in the history of number theory, and other special numbers and figures (like 1, zero, pi, e, and i). So, I can't say I wasted my money when I bought it.

I had thought the Golden Ratio was simply the ideal aesthetic ratio between the length and the height of a painting or that of objects within a painting. According to Author Mario Livio, however, it has very little to do with the arts but a great deal to do with nature and the laws of physics, as well as some amazing abstract mathematical characteristics (discovered over the last several centuries). I believe the sub-title of the book is correct: it IS the world's most astonishing number. In other words, though it does not in the author's view have much to do with the Mona Lisa, the Parthenon, or the Pyramids, it does have some fascinating connections to nature, as well as numbers in the abstract, and their characteristics.

Well, what is the Golden Ratio anyway? Basically, phi or the Golden Ratio is such that if you break a line AB into 2 parts by adding point C to make AC and CB, such that AC is greater than CB and AC/AB = AB/AC. I t sounds pretty boring, but it gets a lot better, since it is also the convergence of something called the Fibonacci Sequence, a set of numbers beginning with 0 such that any 2 consecutive numbers added together equals the next number in the sequence (0,1,1,2, 3, 5, 8, 13, etc.). The Fibonacci Sequence can also be proved to be the same as the continued fraction of all 1's and also the convergence of the continuous nested square roots of 1's. (You can look on the net to see what these expressions look like, both somehow very satisfying aesthetically). I was amazed that these connections could have been made at all with phi, and that the Fibonacci Sequence is the most irrational of all possible numbers; that is, it converges the most slowly to its final irrational value. Call me weird, but that just blew me away!

I was most amazed that minds could think of these abstract things, and that the math connections to phi worked out so beautifully. Phi's abstract qualities are, in my opinion, every bit as impressive as its connections to nature itself (galaxies, sunflowers, hurricanes, and more). How did they think this stuff up, and why does it fit together so well? Some of the more bizarre are as follows:

The inverse of phi has the same numbers to the right of the decimal point as phi itself.

The square root of phi also has the same numbers to the decimal point as phi.

The sum of 10 consecutive Fibonacci numbers is = to the 7th number times 11.

The unit digit of a given Fibonacci number occurs exactly every 60 numbers.

All Fibonacci primes have prime subscripts (with the exception of 3).

The product of the first and third Fibonacci numbers in a set of 3 consecutive Fibonacci numbers is within 1 of the 2nd number squared.

Who would even think of looking into such things, and why does it work out so well?

There were also a couple of tangential points that were really neat to me. How about the First Digit Phenomenon (Benford's Law), that says if you have a random set of numbers, the probability of the first digit being a 1 is greater that it being a 2 is greater that it being a 3, and so on. How is that even possible in the real world? I'll have to think about that one a little more. And how about proof for the irrationality of the square root of 2? This elegant little proof was worth the price of the book, at least for me. It is a derivation of something called reductio ad absurdum: you prove something is true by starting with the opposite assumption and taking it to its logical conclusion to prove it can't be true.

Finally, I was struck by a broader question raised by the Mario Livio: how is it that math can so concisely define the laws of nature (gravity, motion, etc.)? I don't think that thought once crossed my mind throughout my high school and college careers in engineering! The book says that Kepler's Third Law, for example, states that the square of a planet's period divided by the cube of its semi-major axis is constant for all planets. How does that work out so well in such a brief, elegant formula, and how in the world did Kepler think of it? Are we talking Coincidence or Creator?

I was a little let down by this book as far as art is concerned; Livio simply doesn't believe it is a factor (except for a little 20th century art in the cubist genre perhaps). But I was surprisingly excited by some of the abstract characteristics of the Golden Ratio, and the minds that somehow put it all together. It was as exciting to me as seeing rare, beautiful, exotic creatures on a TV nature show.

The Golden Ratio is a strange, beautiful, and rare bird indeed!

The Golden Ratio is a great book if you like math and are willing to work your way through something that seems like a text book. I was fascinated to learn that this ratio is reflected in nature in the spiral forms of galaxies and shells, and to some extent in the patterns and scales of modern music and art. The history of the number and of math itself was also interesting, but the star of the book was the math of the Golden Ratio (also called phi) and its relationship to the Fibonacci Series, and the neat things that come out of that relationship.

What is the golden ratio anyway? It is briefly the ratio between a line connecting A and B. If point C is put somewhere between A and B such that CA is larger than CB, and AB/CA = CA/CB, you have phi. Sounds boring but it gets more interesting when the ratio is declared to be the convergence of the Fibonacci series, which is a series of numbers beginning with 0 and 1, such that each number is the sum of the previous two numbers (0, 1, 1, 2, 3, 5, 8, 13, etc.). The further you go into the Fibonacci Series and divide a number in the series by the next number in the series, the closer you get to the ultimate value of phi. We learn that the square any given Fibonacci number is equal to one less than the product of adjacent numbers in the series. An example is the number 5 which is squared to become 25, and whose adjacent numbers of 3 and 8 multiply out to 24, which is, indeed, one less that the square. The Fibonacci Series can be expressed as a `continued fraction' of all ones, which is fascinating to look at, but is also found to be the most `irrational' of all numbers, taking the longest time to converge to its ultimate value. (An irrational number is one that cannot be expressed as a fraction.) That really blew me away - this is the most irrational of all possible numbers! Also, if you square phi (1.61803---) you get a number with the same digits in the same sequence after the decimal point as you get in phi. Likewise if you get the inverse (divide phi into 1), you also get the same digits in the same sequence after the decimal point. Amazing! There are many other unique and surprising relationships that come out of the Fibonacci Series that were a treat for me, and that I am going to revisit in the days to come.

I was less enchanted with the philosophy at the end of the book that discussed whether order is something in our minds or something that really exists. How is it that we have matches between physics and math? I believe the coincidences should stir us to think about ultimate things. Yes, I do believe in a Creator, and, yes, I believe He created the order. I thought the book was on firmer ground when it just stuck with the math.

You've probably gathered that you have to be a math geek to some degree to enjoy this book. I'm sure that is true. But if you are, read this book; it is the `rational' thing to do!

The Golden Ratio is a great book if you like math and are willing to work your way through something that seems like a text book. I was fascinated to learn that this ratio is reflected in nature in the spiral forms of galaxies and shells, and to some extent in the patterns and scales of modern music and art. The history of the number and of math itself was also interesting, but the star of the book was the math of the Golden Ratio (also called phi) and its relationship to the Fibonacci Series, and the neat things that come out of that relationship.

What is the golden ratio anyway? It is briefly the ratio between a line connecting A and B. If point C is put somewhere between A and B such that CA is larger than CB, and AB/CA = CA/CB, you have phi. Sounds boring but it gets more interesting when the ratio is declared to be the convergence of the Fibonacci series, which is a series of numbers beginning with 0 and 1, such that each number is the sum of the previous two numbers (0, 1, 1, 2, 3, 5, 8, 13, etc.). The further you go into the Fibonacci Series and divide a number in the series by the next number in the series, the closer you get to the ultimate value of phi. We learn that the square any given Fibonacci number is equal to one less than the product of adjacent numbers in the series. An example is the number 5 which is squared to become 25, and whose adjacent numbers of 3 and 8 multiply out to 24, which is, indeed, one less that the square. The Fibonacci Series can be expressed as a `continued fraction' of all ones, which is fascinating to look at, but is also found to be the most `irrational' of all numbers. (An irrational number is one that cannot be expressed as a fraction.) That really blew me away - this is the most irrational of all possible numbers! There are many other unique and surprising relationships that come out of the Fibonacci Series that were a treat for me, and that I am going to revisit in the days to come.

I was less enchanted with the philosophy at the end of the book that discussed whether order is something in our minds or something that really exists. How is it that we have matches between physics and math? I believe the coincidences should stir us to think about ultimate things. The book was on firmer ground when it stuck with the math.

You've probably gathered that you have to be a math geek to some degree to enjoy this book. I'm sure that is true. But if you are, read this book; it is the `rational' thing to do!

A lot of the early section about the history of mathematics, natural occurences, and the appearance of the phi is amazingly interesting.

Unfortunately, Livio then proceeds on a wild goose chase to mention many of the instances in which people have presumed (generally incorrectly) that various artists, writers, and other such public figures used phi in their work.

This book is certainly worth reading for the explanation of the origin of phi but I would recommend skimming the sections about artists, poetry, and musicians.

Well titled, this book is an adventure. It is clear, concise, interesting and thought provoking.
I used it as inspiration for a course on the philosophy of architecture.

There are many books explaining the places where the value PHI and the Golden Ratio show up. One discovers that maybe the ancient Greeks based their architecture upon it, and the value was one of the mysteries of the Pythagoreans. Some authors explain this golden ratio to the dimensions of the pyramids. Some books contain the view that the architecture of all life contains an undercurrent of the value of PHI. I have read many books about this subject and this is the book to read. It runs the whole gamut, history, architecture, art-history, and mathematics. The author, Mario Livio, cites many others books and studies, often critically showing that not all artists that knew of the golden ratio used it in their art and that any monument's dimensions when forced through enough mathematical gyrations will portray whatever is desired. The illustrations within the book are superb, right to the point and easy to understand. The author explains a difficult concept in an easy to read manner, even though it involves Fibonacci sequences and fractals. The author is Head of the Space Telescope Science Institute , he knows the subject of which he writes and more importantly can relate this knowledge to the reader. Kudos to the author, well done.

Even for non-mathematicians, this book is fun to read.
Numbers don't have to be static and boring.

Buy this book if you enjoy dry, boring, off-topic, meandering and convoluted writing. Seriously, don't waste your time. INSTEAD, go out and buy A Beginners Guide to Constructing the Universe. Do a search for it right now.

The Golden Ratio is supposed to be a book intended for casual readers to learn more about the almost magical properties of a number called Phi (1.613...). According to the author, this "most astonishing" number has been credited with all kinds of mystical and even divine properties and 250 pages are spent in trying to evaluate its properties.

However, after finishing it, I am left with the distinct impression that the point of writing this book had nothing to do with Phi. The book spend almost half of its pages on various topics related to Phi but most of them are debunking theories that tied Phi to various artifacts like the Pyramids, the Parthenon, various paintings, and the arrangement of the cosmos itself. The other half of the book is a meandering tale of the development of mathematics and particularly how geometry and astronomy continue to intersect. Since the author is an astronomist, I suppose that is not particularly surprising.

Phi is called the Golden Ratio, the Divine Ratio and various other names in this book. No matter what the name, it is simply one minor mathematical ratio that crops up occasionally and mostly accidentaly. If there is a magical component to it, or any reason why anyone would consider it to be "astonishing" remains to be proven by others as this volume certainly does not do so. As an example, the final chapter of the book - where one would expect the point to be made or at least recounted - is spent completely on a philosophical discussion on the importance of mathematics and why mathematics works in so many different fields.

Huh?

This leaves me with no idea why the book was written. Certainly its sub-title was not met and much of it has nothing to do with the subject matter. So, why should you spend your money on it?

Not recommended.

The most irrational of irrational numbers (phi) can be found throughout the universe. Everything from the spirals of galaxies to the proportions of the human body and even the coils of DNA is proportioned on the Golden Ratio. Surely the ubiquitous nature of such a seemingly odd number is a sign of one of the preferences of the creative unfolding of the universe; none other than the signature of God, here for our amusement and assurance.

I am not a mathematician, aspiring mathematician, scientist or otherwise left-brain person. I came across the concept of Phi from The DaVinci Code (what a tiresome read that was...) and simply wanted to know more about it. This book was perfect! I too have purchased many books on mathematics in my quest to learn more and the more I read the other books, the more I liked this one.

Mario Livio's approach is humble, his style direct, his concepts thorough and his examples concise. The book would be endless if he delved deeply into any one of the numerous references he provides. His quick references to the who, when and what of phi-related events were blessedly short and sweet. What references there are to god(s) are not in the least imposing or suggestive; they are historically relevant and important to note when contemplating the how's and why's of periodic developments.

I highly recommend this book to anyone wanting to know more about phi. If nothing else, the author provides many references to other books that are focused on specific areas of mathematics and physics.

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 an aspiring mathematician, I've been reading pretty much any book I can grab at the book store that has to do with mathematics. Phi is a number that has caught my attention before, and I was looking for a thorough look at different places where the number crops up. I was not, however, looking for any sort of religious discussion.

I wish I had known before I bought the book that the word "God" frequently cropped up in the book. If I had known, I probably would not have bought it. I was looking for a book of mathematics, not speculation into whether god is a mathematician. If that sort of discussion interests you, perhaps you'd like the book. However, if you are like me, hopefully you will not waste your money on it; get it from the library or find another book about phi.

I had high hopes going into this book. I had learned of the "divine proportion" in a linear algebra class I'd taken and it inspired me to pick this book up.

In the beginning I wasn't surprised, Livio outlined so many amazing applications of phi that I couldn't wait to get into the more in depth explanations. However, once I had, it seemed to me that Livio was convincing me that It wasn't so "divine." In any other applications other than simple geometry phi fails at being anything miraculous. For instance, when attempted to apply it into aesthetics it's useless.

The only reason I give this book 2 stars instead of 1 is that it gives a great overview of the history of Phi, which I appreciated. Other than that, it at least convinced me that Phi is just another way of looking for a set universal design. However, after further thought you should realize that if it weren't there, another mathematical structure would be. Because if we're working with simple Euclidean geometry, we're bound to come up with the same answers when looking for proportions in geometrical shapes.

Phi was an interesting read; Livio did a good job of explaining the history of the concept, and many applications in both science and nature.
He also spent a great deal of time dismantling the notion that artists throughout history used the golden ratio/golden rectangle in their work. I found this part of the book intriguing, although I did read it with a little bit of disappointment, being a lifelong fan of "Donald in Mathmagic Land".
Overall, I felt that the last sections of the book tapered off, and failed to continue to provide the same high quality of analysis that the author had offered in the first parts of the book. This was a pleasing read, although it grew a little dry and less creative towards the end.

Interesting subject, but written in such a boring fashion, I had to FORCE myself to finish it.

Chapter 9 of the "Golden Ratio" is labeled `Is God a Mathematician'; after studying the Fibonacci Sequence and its relation to Nature; one cannot but admit the statement to be an absolute truth. On the surface the universe may appear to be in a state of utter chaos; however on closer observation one may find the `golden ratio' to be the point of symmetry in this chaos. The value phi which is the undivided constant repeatedly appearing in nature. If one is to ask the relation between growth rates of rabbits, cows and bees; the nautilus shell; petals on the flowers; pine cone; leaf arrangement of plants; the structure of snow flakes; the spiral attack of a falcon on its prey. The list can be endless.

The "Golden Ratio" attempts to provide an introduction to the phi; most of the information listed is available for review through a quick search on the internet. The book does not add any supplemental information that raises eyebrows.

I was afraid when I got this book as a present that it would be some kind of new-age-hippy we're all ruled by PHI, "It's in the trees man!" kind of book.

In fact, it was quite the opposite. It was a light hearted little adventure through math history and it was fascinating to read. Try it.

It presents an interesting topic on the 'golden number' the 'golden rectangle' and phi in general.

It dispells many of the myths that ancient architecture and the use of phi before it was fully understood.

However, it tends to be repetative and it does have some gramatical problems throughout.

Interesting enough to keep you reading, but not the best book by any means.

This book provides an objective look at the number Phi and its presence in nature and human history. Very entertaining and educational for mathematics enthusiasts of all skill levels.

I feel that my life has mostly slipped by without knowing more of the Golden Ratio. I knew that windows in Oude Delft in Holland are Golden Rectangles, and I had some inkling of the shape of proportion in Hogarth, but Livio has sent me scurrying to get the marvelous new edition of Euclid (Amazon) and to find my compass and protractor. But learning on my own shows the shallowness of my understanding and the brilliance of this ultimate exposition of the liberal arts.

Phi is a number that is popping up more and more. There are 2 choices: Either it is an interesting coincidence OR it is somehow inextricably linked with the ongoing creation of the universe and a number whose importance is only going to emerge in times to come. You will need to read the book to decide.

Livio is undoubtedly the world's top authority on the number. Obviously intrigued by the enigma, he is simultaneously solidly grounded and debunks the leagues of over zealous golden ratio conspiracy theorists. All this in a very readable tome.

The fantastical theories are exciting though. How about this one (not in the book): "In the beginning was the Word". "Word" translated from the Greek "logos". Another meaning of Logos: "proportion/ratio". (Think "logarithm") We now have "In the beginning was the Ratio..." Thought provoking? Read the book!

The Golden Ratio is very well written, and anyone with a basic grasp of geometry and number theory can easily follow along.

(Note: Basic understanding of geometry goes beyond identifying the difference of a triangle and a circle, and number theory goes beyond counting)

I was a little disappointed that he spent so much time demystifying Phi and the Golden Ratio's use in art and music. I had always found that fascinating, and was a belief I held firmly...I guess I'm a Pythagorean at heart.

I own several books on dynamic symmetry, which includes the Golden Ratio. Unfortunately all of the other books (3 of them, some rare and out of print) are heavily mathematic in their texts. I would say at least 90% math and about 10% English. So I was stuck trying to decipher the text via the sparce English.
This book is not only fun to read (for a non-mathmetician type), it is fairly easy to understand (not a quick read, and at some points it takes some effort), and I would guesstimate about 80% English, 10% essential illustrations, and 10% math.
And, since I find all this dynamic symmetry and Phi stuff fascinating--and do not want to go back to school to learn advanced algebra and trig--this book is a godsend!
One more thing. This is written by a scientist that can write for non-scientists. He has done his research very thoroughly and covers the subject comprehensively.

"The Golden Ratio:...", by Mario Livio, NY: Broadway Books, 2002 - ISBN 0-7679-0816-3 - PB, 294 Pg. (8" x 5.125") includes 14 Pg. Append., 10 Pg. Further Ref., 12 Pg. Index, 4 Pg. Credits.

We are given 9 Chapters of mathematical pagentry, variably banal to sacerdotal & accompanied by figures, formula & Fibonacci sequences. Overall, an interesting semi-chronological history of numbers, mathematics & of those academicians (science, literature & arts) who invented, discovered, exploited, or expounded them. Special attention is given to Fibonacci Sequences & Golden Ratio (Section, Number, phi, etc.) as the author shuffles, leaps or waltzes from tantalizing tidbits of information we've long ago forgotten such as "Surely...breasts helped in the development of the abstract understanding of the number 2", or we "never (say) 'a yoke of dogs'."

Overall, we are told, despite rumors to the contrary, the Great Pyramid of Egypt & many other great artworks as those by Leonardo da Vinci, etc., or composer's as Mozart, Bartok, etc., etc., etc., probably did not incorporate Golden Ratio specifically but had semblance ratios c. 1.6 deemed pleasing. "Benford's Law was appropriately discussed but the half page quotation from Samuel Beckett's "Molloy" is incredulous (but funny).

For those having interests in mathematics & history of numbers, etc., this book will serve their purpose nicely. Livio knows & loves his numbers. The index helpful.

While I certainly didn't think this was going to be an easy read, I wasn't prepared for the tedious content I'd have to work through. I'm not a math enthusiast, so that probably explains why I didn't enjoy this book as much as the other reviewers. But coming from someone who considers himself a layman in terms of math, I expected more handholding from a text that presents itself as a "mainstream" book. That, of course, is based upon my expectations and should not count against the book as it stands by itself. I'm sure mathematicians will enjoy this work.

It is definitely thorough, but I didn't expect to wade through several detailed formulaic explanations to see the marvels of phi. In fact, the author is so thorough that it seems like he is just padding the book. Why waste a page listing phi to the 2,000th decimal place?

I ended up skimming much of the book in the end just to finish it.

This book is for mathematicians, not artists. Coming from both the science and art world myself I was disapointed a bit. Too much time is spent trying to debunk the use of the golden ratio in art stricly based on an analysis done in a manner suitable only if you are planning to launch a rocket but totally beside the point in the world of visual arts. In that field, the author misses the point entirely. Works of art are perceived with the eyes, often at a glance for architecture, and only in the context of the building on the field, not with an inch garduated ruler on a piece of paper. The section on the Parthenon is particularily wrong in that aspect and not even accurate in the materials used (the text and the pictures do not correlate). When treating art and architecture in this book, the author spends too much time analyzing if the artist/architect was consciously using the golden ratio on paper and doing a poor job at that. I don't care if he was. What I care about is the end result. Many artists probably do not think about mathematics when creating but the fact remains that many buildings and works of art follow the proportions of the golden ratio. And here the mistake the author makes is analyzing building in the eyes of a scientist, not in the eyes of an architect/artist. One example is when he says the Parthenon does not really follow the golden ratio proportions because if you include the lower stairs and edges of the roof in the rectangle, it doesn't fit anymore. Well, I have news for you, architecture is all about the overall appearance of a building, the main lines, what the human standing up at a distance from the building will see. The best test is, blurr your eyes at a frontal picture of the Parthenon taken on the field, from a distance and you will see the golden rectangle fits like a glove. If you want to get an idea of the artistic flair of this author, read the section on Le Corbusier where he compares the Modulor man to the Michelin man ;)

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.

This is a great book on phi. I like how Livio stayed objective and consistant throughout, disproving most of the New Age-y fluff sometimes associated with this number.

An easy to read, (as a math book can be), on this number as it came through antiquity and its significance to mankind and nature. Excellent!

I can't say I was overly impressed with this book, although it would be a reasonable introduction to the number phi and all the ways it shows up (and may not show up.) But I was looking for something more substantive. Still, it had enough bits and pieces that it was interesting, and it was short enough that it didn't take up too much time to read.

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.

Livio's book is really an interesting look at a number similar to pi in that's an irrational number which displays itself in various places in nature, from the arrangement of petals on a flower to the logarithmic spirals of galaxies.

Livio explains the original formulation of this number by Euclid and proceeds to address the various times in history in which it may have been employed by architects, artists and musicians.

I think this is a really good book if you're interested in reading about the most "irrational of all irrational numbers".

The book seems to have two purposes. First, it seeks to debunk the notion that the 'golden mean' is intuitively pleasing to all humans. Secondly, the book argues that we can best understand the cosmic meaning of life via numbers. From these two theses, the author establishes the 'genius' mathematician as mediator and priest between mankind and the cosmos.

Ok, enough mumbo-jumbo. If the above interests you, read the last chapter first. It should have been used as the introduction and will clarify the purpose of arguments presented throughout the book.

Getting back to the book's narrative, the early sections seek to debunk various claims that Egyptian and Mesopotamian civilization use the Fibonacci ratio. Reports of its use in China are ignored. Later, there is a section debunking the notion that great art 'uses' the Golden Mean. Scattered throughout are developments in number theory, starting with Pythagoras and continuing with the standard European mathematical genius roll call. The last chapters reveal the relationship between the Golden Mean and complexity/chaos theory/fuzzy logic/quantum theory/etc.

Livio's Fibonacci sequence, Penrose tilings, and quasi-crystals stories will probably engage the recreational mathematicians among us, and provide a handy 'all in one place' summary of such matters. Others will find the philosophic overtones tangential and/or distracting. Any successful philosophy of math needs to address the issues of cardinality and ordinality at the level of intuition. The two topics are dismissed by page 15, but they underlie the whole issue of 'intuitive acceptance of number theory'. This topic is discussed throughout the book as Livio seeks to explain why so many people see the Golden Mean in ancient works of architecture, modern art and stock market charts. Curiously, Livio argues they all got it wrong and only genius mathematicians have the gift of 'intiutive acceptance of number theory'. Not an argument which many will find satisfying, but for the passionately recreational mathematician, it may be seductive.

What a depressing book this turned out to be. I thought a book about the "golden section" would have been interesting but in the hands of Mario Livio it is pure pain. To give a few examples... The author discusses the theory that the golden ratio was used by the builders of the pyramids and refutes it easily. And then continues to refute it for page after page. Then he does the same thing with the Parthenon, destroying the theory using the exact same reasons he used for the pyramid, explaining them in the same level of detail. But he isn't done yet. We get to have the same discussion again when we look at Renaissance paintings. I didn't really care about the discussion when discussing the pyramids but by the time I heard the argument for the third time I was ready to find something else to read. As he discusses the history of the golden section he goes into side trips to discuss anyone who had even the slightest relationship with phi. Anyone who has never heard of Kepler may find this interesting even if it is irrelevant to phi but I just started skimming pages hoping for something a little meatier. There is a little spark here and there that kept me reading hoping for more but more never arrived. A writer with a greater interest in the mathematics of phi could have made this a fun and interesting book. Livio seems to think the math is boring so he avoids it like the plague and creates a book that completely misses the point and ends up being a total bore.

Bought this after reading Da Vinci Code, and I was looking for more of the same. it is a little dry. It is a quick read though, and how much do you want to know about 1.618?

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).

I picked this book up off the shelf in City Lights Bookstore in San Francisco thinking it looked interesting. As a physics/mathematics student, I knew phi existed, but didn't know too much about it, and thought this would be a good, entertaining, and still intellectual book. Now, after trudging through this slowly for a whole semester, I know more than I care to about phi, and how it may or may not have been used.

I feel that this book tries to do too much - entertain both the curious science-minded people, as well as the casual reader. This causes some conflict in some of his sections where he tries to explain mathematical or scientific concepts without getting too technical. In one section for instance, I felt a bit patronized and bored when he explained the wave-particle duality of light, and in other sections I wished for more equations and less hand-wavy math. This is the hardest thing to do when writing mathematical books for the not-necessarily math oriented person. He tried, and had some mathematical appendices when he didn't want to give equations and full proofs in the text. I appreciated these, and would have felt a bit cheated if he didn't give the proofs and equations there.

I also feel like he repeated the same idea a little too often. He gives multiple examples of writers, musicians, and painters, all who have been attributed with using the golden ratio in this work. Far too many times has he come to the conclusion that the golden ration may have been (but probably wasn't) used in this painting, or those pyramids, or that whatever. However, due to the arbitrary nature of where to draw the golden rectangle in the picture, or some other uncertain boundary, it's extremely hard to tell, and we hardly ever know what the creator was thinking when he or she made the object. Therefore, it's almost worthless to try and determine whether a certain artist did or didn't use the golden ratio, which makes about half of his book seem a little pointless.

On the other hand, I did enjoy reading about all the applications of the golden ratio in nature and in science, especially the section on Penrose tilings and their application to "quasi-crystals." That sort of stuff really interests me, and he did a relatively good job of explaining it simply. While sometimes an extra diagram or two might have made his point more clearly, it was easy enough to follow without much prior knowledge of the subject.

This book will also have plenty of good information for the person interested in the historical discovery of phi and, to some extent, of mathematics itself. It will take you to the ancient Greeks, to scientists of the Middle Ages, all the way to modern physicists. It really does cover a lot of historical material, though I certianly won't remember much of it. There are a lot of names that he covers, but with the index he gives, it should be easy to find out who did what regarding the golden ratio if I should ever need to in the future.

In short, it's an alright book; I probably wouldn't recommend it to you unless you really care about phi though.

This latest in what seems a deluge of science-history books, we have a good book bringing together the history of phi. At times there is an info overload, especially if you are familiar with some of the material, but informative if you're new to it.

In the last chapter, the author's philosophical discussion seems to conclude this, "I don't want to pick either option" even though he tries to come up with one from a merging of these two: Phi & math are inherent in the universe and are the same everywhere or math is simply a human construct describing what we see. Well, it is inherent in the universe (such as phi everywhere). We put words and figures to describe it, however other beings would use different figures to write 1+1=2, but the meaning would be the same. The author seems to want to accept this fact - and not accept it - at the same time. The last chapter, especially the last two pages, would have been more intelligent if he wasn't trying to make everyone happy.

See also Miranda Lundy's "Sacred Geometry," David Slatner's "The Joy of Pi," Robin Heath's "Sun, Moon & Earth," Gary Meisner's extensive website, "Phi: The Golden Number," and Dean's website "Mathematical Signatures in Nature: A Sign of Design."

Given how exciting the discovery of phi in everyday life can be, this book's approach is rather clinical. Spends the first few chapters debunking the finding of phi in historical monuments BEFORE explaining the actual places that it does crop up. Kind of boring. A shame, since the accounts I have read in articles by Isaac Asimov and Martin Gardner are so interesting - I would have hoped a book-length account would expand on these.

Let's us assume, that you have bought two newborn rabbits. I hope you have taken male and female. While all is correct. You will must feed them; do hardly look after them, and so on.

In third month you will have two pair rabbits, for the fourth month you will have 3 pairs, for the fifth month - 5 pair, on sixth - 8 pairs, and so on. See thevortextheory.com.
It is possible to speak about mysticism of a sequence Fhi very much and long.

I am sure, that the economists will see in it a rise in prices on the goods.

Physics will notice laws, which are shown in power transitions arising at transformations of elementary particles.

The chemists will find adequate parities in a structure of chemical connections.

I think that each of us will recollect a line of numbers from the life will find the numbers available in a sequence Phi. It is necessary only attentively to read this useful book.

I do not speak already that you can tell then to the relatives, friends and colleagues on work.

I am sure, that you will make on them indelible impression of educated and competently of man.

vavivlad-rvc@mtu-net.ru

This book is well written and does a good job of discussing the number phi. The author presents a lot of mathematical history and shows that he has done some in depth research concerning the use of this number throughout history. The book addresses many myths about where the number phi appears. One example is that some historians believe that phi was used in the construction of the pyramids, the author addresses these theories with a thorough examination of why he believes that this isn't so. Along the way he provides may citations and mathematical examples to support his ideas. This book covers the history of the golden ratio in a very thorough manner and is written so that people with both a mathematical background or a non-mathematical background can understand the concepts presented. I recommend this book to anyone interested in mathematics or mathematical history.