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explanation of the dual nature of light was sort of random, and the rather unsubtle promotion of Stephen Wolfram's then-unpublished book (which was not well received by the math community) was sort of irritating.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"

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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.)