Lies and Debunked Legends about the Golden Ratio

In my eyes, there’s a general pecking order for named mathematical constants. Pi is at the top, e gets a good amount of attention, and Tau, like a third-party candidate, sits by itself on the fringes while its supporters tell anyone who’ll listen that it’s a credible alternative to Pi. But somewhere in the middle is Phi, also known as the Golden Ratio. It’s no superstar, but it gets its fair share of credit in geometry and culture.

I was first introduced to Phi as a kid by watching the charming video Donald in Mathmagic Land. One of the things I remembered over the years is that the Greeks used the Golden Ratio in their paintings and architecture, particularly the Parthenon. Thanks to the power of the internet, I can share this piece of my childhood with you:

How brilliant and advanced of the Greeks, right? But there’s one problem…

It’s probably not true. My faith was first shaken reading Keith Devlin’s The Unfinished Game, where he entertained a quick digression:

Two other beliefs about this particular number [Phi] are often mentioned in magazines and books: that the ancient Greeks believed it was the proportion of the rectangle the eye finds most pleasing and that they accordingly incorporated the rectangle in many of their buildings, including the famous Parthenon. These two equally persistent beliefs are likewise assuredly false and, in any case, are completely without any evidence. For one thing, tests have shown that human beings who claim to have a preference at all vary in the rectangle they find most pleasing, both from person to person and often the same person in different circumstances. Also, since the golden ratio is actually not a ratio of two whole numbers, it is impossible to construct (by measurement) a rectangle having that proportion, even in theory.

What?! Donald, I trusted you! It was tempting to tell myself that the Greeks could have found ways to approximate the ratio, and that this is just one source, and I’ve heard it so many times it must be true, and la la la I don’t want Donald to have lied to me.

But I looked into it a bit more, checking out what Mario Livio had to say about it in his book The Golden Ratio. He acknowledges that it’s a very common belief, but ultimately backed Devlin up:

The appearance of the Golden Ratio in the Parthenon was seriously questioned by University of Maine mathematician George Markowsky in his 1992 College Mathematics Journal article “Misconceptions about the Golden Ratio.” Markowsky first points out that invariably, parts of the Parthenon (e.g. the edges of the pedestal [in a provided figure]) actually fall outside the sketched Golden Rectangle, a fact totally ignored by all the Golden Ratio enthusiasts. More important, the dimensions of the Parthenon vary from source to source, probably because different reference points are used in the measurements… I am not convinced that the Parthenon has anything to do with the Golden Ratio.
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So, was the Golden Ratio used in the Parthenon’s design? It is difficult to say for sure… However, this is far less certain than many books would like us to believe and is not particularly well supported by the actual dimensions of the Parthenon. [emphasis mine]

Alas, claims about the Greeks using Phi in their architecture seem overrated. Some sites bring you celebrity gossip, we bring gossip about celebrated mathematical constants. Welcome to Measure of Doubt!

Watching the video again, I can’t tell exactly how they decided where to overlay the Golden Rectangles. How much of the pedestal do we include in the rectangle? How much of the pillar? Does the waist start here, or there? It seems a bit arbitrary, as though we’re experiencing pareidolia and seeing the Golden Rectangle in everything.

Talk about disillusionment.