Debunking Donald Duck--The Mysteries of Phi Φ

The Golden Ratio: The Story of Phi, the World's Most Astonishing Number by Mario Livio
My rating: 4 of 5 stars

The first time I heard of the "Golden Ratio," or the "Golden Rectangle," (the constant known as phi, Φ) it was in a Disney cartoon. Donald Duck in Mathmagic Land (https://youtu.be/U_ZHsk0-eF0) was actually a pretty fun video, with lots of cool math. Elementary school me didn't really understand what the ratio was, but it was amazing to see it in so many places, and many of the images stuck with me.

Mario Livio is here to destroy that memory. ;)

[Phi, if you don't know, is the relationship between two line segments, one longer and one shorter, where the ratio between the two (longer/shorter) equals the ratio you would get if you compared their sum to the longer line (longer + shorter/longer). The number is about 1.618.]

One of the things he does best here is debunk many of the supposed uses of the golden ratio in the ancient world up through modern times. Most of the oft-repeated examples of phi in architecture (like the pyramids or the Parthenon) or music or Renaissance art turns out to have very little evidence to support it, and sometimes is just plain wrong. Often, the ratios are merely close to phi, but not close enough to impress; other times, perfect golden rectangles are imposed on buildings or elements of paintings by arbitrarily deciding to include or exclude this edge or that top bit, jiggering it until it fits. Without evidence that the artist intended to use the ratio, such evidence is flimsy at best.

The author looks for evidence that the creators of art and architecture have intentionally used the golden ratio, and have said so in writing or to some other person, and does find some examples in the 19th and 20th Century, but little at any other time. He also debunks the notion of it creating the most beautiful rectangles, or I should say that he reports on researchers who have done so. When given an option, asked to pick the best rectangle, subjects don't choose this shape any more than any other.

I found this debunking (much more thorough than I've shown here) a useful contribution to knowledge but the least interesting part of the book. The careful evaluation of paintings and songs (counting measures or inspecting note sequences) that turn out NOT to be examples of anything is a bit of a dead end, dramatically speaking. Good scholarship sometimes leads to dull reading, alas. (3 nerdy stars for this impatient reader. My bad.)

The best part is showing the cool stuff about the number and related math: how it emerges in geometry as well as nature; how it can be calculated by a recursive square root or a recursive fraction; how it is useful in certain calculations; and so on. The most interesting thing for me, a non-math person IRL (though I was good at math things in school long long ago) is how it is related to the Fibonacci series (of "The Da Vinci Code" fame). Refresher:

1 1 2 3 5 8 13 21 34 55 etc.

You keep adding the last two numbers of the series to create the next number, and that is how the series continues infinitely. Whatever. (That's not the cool part. Hold on.) Now, if you take any two numbers and divide the larger by the smaller, you get an approximation of phi. The farther along in the series you go to pick your two numbers, the closer the approximation is. Why? Just cuz. And there are many more amazing connections between phi and Fibonacci, well worth checking out here. That's the part that gets a 5 star review. Good math fun.

Livio finishes with a sort of essay about what is truly beautiful in math, which he defines as whatever is surprising. I rather agree. I think this part would have made a better introduction, but I enjoyed the discussion. For example, the explanation, or more like the description, of Benford's Law (how have I never heard of this? Or did I only forget?) is amazing stuff. If you don't know what that is, you gotta google it. It's crazy. Anyway, yes, surprising stuff. And yes, beautiful. For reals. And I still don't get it.

Overall, an interesting read. And if you run on to it at Goodwill, like I did (don't judge me) it costs less than your morning coffee. The surprising beauty of the universe, yours for only $2.99......

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