If you want to read a bit about the golden number, the golden ratio, the divine ratio, phi, or simply half of the sum of 1 and the square root of 5, then this might be the book for you.

Phi has been found all over the place in nature, architecture, art and science and Livio analyzes all those discoveries one by one, and refutes a number of historical claims.

What remains is that the number turns up in mathematics and geometry an astonishing number of times, but I am not convinced it makes it more magical than the square root of 2.

Mathematically phi is the positive solution to x^2 - x - 1 = 0, and that quadratic equation does turn up every now and then, maybe because of its simplicity.

The book got a bit tedious, maybe because I was more interested in the mathematics than the art. I would have wanted more like the the appendices with the proofs. Still, I can now be a bore at parties and claim that phi is not at all found in the pyramids. Unless you want very hard to find it.

( )

Well, I was expecting something a bit more exciting because of my natural love for Phi, simply because, you know... SPIRALS are EVERYWHERE, Dude.

Still, the author does a palatable job of giving me a fairly decent history of mathematics from the focus of the Golden Ratio, the Golden Triangle, the logarithmic spiral, the Fibonacci sequence... all of which is, of course, the same thing, expressed slightly different with a ton of additional cultural significances.

No surprise here. This is Phi.

However, I did take umbrage against some of the side explanations early on for why ancient or apparently unsophisticated tribes didn't have numbers that counted past four. I mean, sheesh, if we went purely by the mystical importance that the Pythagoreans placed upon the first couple of numbers, we might also believe they couldn't count past five. It's a mistake of the first order, taking a little bit of data and coming to enormous conclusions based on our own prejudices.

That's my problem, I suppose, and he does at least bring up the option that the ancient peoples might have been working on a base four mathematical system, but for me, it was too little, too late. I cultivated a little patience, waiting until we get further along the mathematical histories past the Greeks and into the Hindus and the Arabics where it got a lot more interesting, and then firmly into known territory with the Rennaisance.

Most interesting, but also rather sparse, was the Elliot wave and the modern applications of Phi. I wish we had spent a lot more time on that, honestly.

But as for the rest, giving us a piecemeal exploration of Phi in history, art, and math, this does its job rather well. ( )

The Golden Ratio dwells lovingly on each pearl in the necklace that is Phi. ( )

The universe is a strange place, with apparently built-in irrational numbers appearing in the weirdest of places. In addition to pi and e, there is phi (pronounced “fee”), also known as the “Golden Ratio,” with a precise value of 1.6180339887….

The Golden Ratio is probably best explained in a diagram like this one:



The Golden Ratio can be used to construct a Golden Rectangle:



The pink rectangle that results from taking away the blue square is also a Golden Rectangle. If a square is subtracted from that rectangle, the remaining rectangle will also be a Golden Rectangle, and so on, ad infinitum:



Each daughter Golden Rectangle will be smaller than the parent Golden Rectangle by a factor of phi.

The Golden Spiral is a special kind of logarithmic spiral with a growth factor of phi. It can be approximated using a Fibonacci spiral:



The numbers in the Fibonacci sequence (1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987…) have their own weird relationship with phi as well: as the Fibonacci numbers approach infinity, the ratio of two successive Fibonacci numbers will approach phi.

However, not all logarithmic spirals are Golden Spirals, something I looked up independently, which this book seemed to gloss over. So, while it’s true that “nature loves logarithmic spirals” and there are plenty of logarithmic spirals ranging in size from mollusk shells to spiral galaxies, not all of them are related to either phi or the Fibonacci sequence. However, many of the logarithmic spirals in plants are related to phi and the Fibonacci sequence. For example, pineapples usually have 5, 8, 13, or 21 spirals of increasing steepness on their surfaces, and all of these numbers are part of the Fibonacci sequence. Sunflower heads also display this pattern:

“Count the clockwise and counterclockwise spirals that reach the outer edge, and you'll usually find a pair of numbers from the sequence: 34 and 55, or 55 and 89, or—with very large sunflowers—89 and 144.” (From Science website).

Although this was not discussed in the book, the number of spirals on the head of Romanesco broccoli is also a Fibonacci number. Romanesco broccoli is also called “fractal broccoli” although it is only an approximate fractal.



(And no, this is not a messed-up product of genetic engineering; it’s been cultivated in Italy since the 1500’s.)

Speaking of fractals, phi shows up there as well. This isn’t surprising, since a fractal needs to be self-similar on different scales, and both the Golden Rectangle series of subdivisions and the Golden Spiral qualify. I had trouble with the Golden Sequence part but had a better time understanding the Golden Tree when it came to this chapter.

When I came to the last chapter, I learned that for some people, performing arithmetical calculations can trigger seizures. The condition is called epilepsia arithmetices, and fortunately it is very rare. For people with this condition, abnormal electrical activity is concentrated in the inferior parietal cortex, and damage to the same area also affects mathematical ability, writing, and spatial coordination.

And now I wish he would do a similar book for pi.
( )

The universe is a strange place, with apparently built-in irrational numbers appearing in the weirdest of places. In addition to pi and e, there is phi (pronounced “fee”), also known as the “Golden Ratio,” with a precise value of 1.6180339887….

The Golden Ratio is probably best explained in a diagram like this one:



The Golden Ratio can be used to construct a Golden Rectangle:



The pink rectangle that results from taking away the blue square is also a Golden Rectangle. If a square is subtracted from that rectangle, the remaining rectangle will also be a Golden Rectangle, and so on, ad infinitum:



Each daughter Golden Rectangle will be smaller than the parent Golden Rectangle by a factor of phi.

The Golden Spiral is a special kind of logarithmic spiral with a growth factor of phi. It can be approximated using a Fibonacci spiral:



The numbers in the Fibonacci sequence (1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987…) have their own weird relationship with phi as well: as the Fibonacci numbers approach infinity, the ratio of two successive Fibonacci numbers will approach phi.

However, not all logarithmic spirals are Golden Spirals, something I looked up independently, which this book seemed to gloss over. So, while it’s true that “nature loves logarithmic spirals” and there are plenty of logarithmic spirals ranging in size from mollusk shells to spiral galaxies, not all of them are related to either phi or the Fibonacci sequence. However, many of the logarithmic spirals in plants are related to phi and the Fibonacci sequence. For example, pineapples usually have 5, 8, 13, or 21 spirals of increasing steepness on their surfaces, and all of these numbers are part of the Fibonacci sequence. Sunflower heads also display this pattern:

“Count the clockwise and counterclockwise spirals that reach the outer edge, and you'll usually find a pair of numbers from the sequence: 34 and 55, or 55 and 89, or—with very large sunflowers—89 and 144.” (From Science website).

Although this was not discussed in the book, the number of spirals on the head of Romanesco broccoli is also a Fibonacci number. Romanesco broccoli is also called “fractal broccoli” although it is only an approximate fractal.



(And no, this is not a messed-up product of genetic engineering; it’s been cultivated in Italy since the 1500’s.)

Speaking of fractals, phi shows up there as well. This isn’t surprising, since a fractal needs to be self-similar on different scales, and both the Golden Rectangle series of subdivisions and the Golden Spiral qualify. I had trouble with the Golden Sequence part but had a better time understanding the Golden Tree when it came to this chapter.

When I came to the last chapter, I learned that for some people, performing arithmetical calculations can trigger seizures. The condition is called epilepsia arithmetices, and fortunately it is very rare. For people with this condition, abnormal electrical activity is concentrated in the inferior parietal cortex, and damage to the same area also affects mathematical ability, writing, and spatial coordination.

And now I wish he would do a similar book for pi.
( )

Un bel libro, che si legge quasi come un romanzo. Ricco di aneddoti, spazia dalla matematica alla fisica alla storia dell'arte (come richiesto dall'argomento) con competenza e senza perdere il filo del discorso. Assolutamente consigliato. ( )

The book explores the history of discovery of the Golden ratio, also known as phi, which is an irrational number that begins 1.6180. The author debunks that the Golden ratio was used to construct the pyramids and the Parthenon. he does cite some actual examples of the golden ration being used in art, music and architecture. the author also points out how the Golden ratio relates to the Fibonacci sequence and covers multiple areas where this sequence is found in nature. There is a lot of history of mathematicians which I didn't find very interesting and which to me took away from the point of the book. There is also a fair amount of math in the book that I found hard to follow. ( )

To write a whole book about the number zero requires a good deal of imagination and random association. That does not seem to be the case for the golden ratio. I've only gotten as far as Chapter 5, but there has been a good deal of interesting material presented. Chapter 2 introduces the Pythagoreans and many of their number theoretic discoveries. In this chapter the author gives a Babylonian formula for generating Pythagorean triples and argues that the existence of this formula and the fact that it generates a Pythagorean triple for every pair p, q where p > q proves that there are an infinite number of Pythagorean triples. I think that there are an infinite number, but the existence of the formula does not prove it. After all, the formula f(n) = (3 * n, 4 * n, 5 * n) yields an infinite number of Pythagorean triples, but they happen to be all in the same ratio, and therefore, essentially the same. Chapter 3 discusses assertions about the presence of the golden ratio in various ancient artifacts and buildings with scepticism. I liked the debunking, those pictures with a golden rectangle superimposed on the Parthenon always seemed questionable to me. Chapter 4 introduces lots of interesting properties of the ratio that were investigated before the dark ages.

For the interested reader there are a bunch of proofs of various theorems in the back, which is a nice touch. ( )

This was a great read, even for a math idiot like myself. ( )

For the love of Fibonacci numbers, roses, geometry and fractals. What do pineapples and spiral galaxies have in common? PHI! This is nerd candy. ( )

1.618033989... is a magic number. Its magic may not be as obvious as the most famous irrational, pi, nor as familiar as e (both of which are also transcendental), but its connection to the Fibonacci series (1, 1, 2, 3, 5, 8, ..., in which each element is the sum of the two previous) is a both intimate and surprising, and its role in the spiral of mollusc shells, inscribed pentagons, pineapple segments, fir cones, and the arrangement of seeds in a sunflower provides remarkable evidence that when nature speaks, she does so in the language of mathematics. Astrophysicist Mario Livio, who also wrote "The Equation That Couldn't Be Solved" (about group theory), takes a tempered approach to his subject. Claims have frequently been made that phi was the design principle of the Egyptian pyramids, the Parthenon, the works of Leonardo da Vinci, and in many other artistic creations. The evidence for most of them is weak - often based on a proportion that, when measured in a certain way, comes close to phi or its reciprocal. Livio is rightfully skeptical of most of these claims, but he also gives the Golden Ratio credit for works in which it is clearly implicated. This book provides a wonderful connection from science, art, music, and architecture to geometry and mathematics. ( )

The Golden Ratio, or phi, or even better, Φ, is an irrational number (equal to 1.618...) that pops up in many strange instances and has some odd properties. In its basic form, it is a simple ratio of a line segment divided in such a way that the proportion of the whole to the longer length is equal to the longer length to the shorter.

It pops up in the construction of a pentagon and the pentagram (the sides of the 'star' are divided in this ratio) and, supposedly, in the Parthenon of Athens. There is a tendency for leaves and twigs to branch off at angles determined by phi, the proportions of the human body as well as many animals are believed to be determined by Φ, and on and on.

The author goes through a long series of alleged uses of Φ, some of which are debunked (the Parthenon and the Pyramids) and some seem to be legitimate. As is to be expected, there is considerable information on the Fibonacci numbers, where the new term in the series is the sum of the two proceeding, this series converges to phi and has many appearances in nature, like the left and right hand spirals of a sunflower's seeds are always consecutive Fibonacci numbers. The math in the book is minimal, proofs are relegated to a series of appendices, and the bulk of things actually revolve around the concept of Φ being 'aesthetically pleasing', a long standing idea. Livio examines art that is claimed to use Φ in its dimensions somehow and tends to fail to find it.

The best way to sum up this book is that I drew out a logarithmic spiral for a friend and mentioned that any line drawn through the center of the spiral intersects the spiral over and over at the exact same angle. His response was, 'So what?', and yours probably is as well, but he was interested to know that a diving hawk follows this exact spiral so it can easily keep its eyes on its prey on the ground. I guess that's enough. ( )

If you divide a line so that the ratio of the smaller to the larger is equal to the ratio of the larger to the whole, you have the golden ratio, phi. There has been an abundance of literature on the presence of phi in a number of unexpected locations, and this book addresses many of these appearances intelligently. It is organized more or less historically, starting with the Pythagoreans' obsession with phi (due to its presence in the pentagon and other neat little number tricks) and continuing through the present. The author avoids doctoring numbers to fit phi into famous works of art and architecture, and indeed debunks several such cases. While some of the direct appearances of phi are pretty nifty (such as leaf growth patterns on plant stems), much of the book covers subjects that are only related to phi by a few generations, usually through the pentagon or the Fibonacci numbers. I do not fault the author for this; tangents are to be expected in books about such a narrow subject as a single number.

The final chapter, "Is God a Mathematician," includes leading theories in response to that question (yes, no, and sort of) and Livio's personal opinion. I understand the desire to address such a topic, since mathematics is pretty amazing and phi is no small example of this, but this chapter seemed sort of forced, like the author was at a loss on how to wrap up the book. The explanation of the dual nature of light was sort of random, and the rather unsubtle promotion of Stephen Wolfram's then-unpublished book (which was not well received by the math community) was sort of irritating. I imagine that Livio's desire was to instill a lingering thirst for knowledge in his reader, to encourage further study, but it felt more like an advertisement for a newfangled religion that will change the way you look at the world. Despite the final few pages, I found this book to be informative and quite readable, which is always high praise for a book about math. Perhaps if Livio had left out his personal opinion I would have finished it feeling more satisfied. ( )

This was a poor book. Aside from the many references to Livio's "friend"s, Livio accuses others of number-juggling while himself forgetting to explain how phi obtained the name "The Golden Ratio." He claims that there is no aesthetic appeal to the number, which may or may not be true; but in the process, explains only how it obtained the name "The Divine Proportion." Even so, he delves significantly off-topic, which was actually one of the better parts of the book. Which...says enough about the rest of it. ( )

Before I read this book, I'd heard about a lot of the astonishing mathematical properties of Φ, as well as the Golden Ratio's aesthetic appeal. What struck me reading Livio's book is not the math itself (as interesting as that was; I haven't studied math seriously in many years). No, what really caught my attention was the number of times that Φ has been cited as the basis for great works of art, that turned out to be pure B.S. Consider the following:

- Φ is not the ratio of the height of the Parthenon to its width.
- Φ has no role in the design of the Pyramids.
- While Da Vinci did illustrate a mathematical book on Φ (The Divine Proportion by Luca Pacioli), he did not use it as a guide to composing the Mona Lisa or anything else.
- Mozart and Mondrian didn't use it, either.

So, The Golden Ratio succeeds as a debunker's compendium. Livio makes the history of Golden Ratio fanaticism seem like so much Da Vinci Code-style overblown hokum. (All the more ironic that Dan Brown's praise for The Golden Ratio is given pride of place on the front cover.)

After that, the best part of the book for me was the end, where Livio digresses into fractal geometry and the enduring philosophical conundrum of why mathematics (a purely abstract human invention) mirrors the physical universe so precisely. These fundamental questions are more interesting to me than any laundry list of Φ trivia.

Original post on "All The Things I've Lost"

Most of what you ever wanted to know about the irrational number 1.61803 39887 49894 .... (The value through 2,000 decimal places is on p 81.) But glossed over is the fact that it, unlike e and pi and all but a countably infinite subset of the other real numbers, is merely algebraic and not transcendental. (It is one-half of 1+sqrt(5), a solution of x^2-x-1=0.)

An intersting look at a fascinating number. ( )

http://humdingerbooklist.blogspot.com/#7631591518690083194 ( )

ditto dusty's review. ( )