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.

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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.
( )