# Lies and Debunked Legends about the Golden Ratio

August 29, 2011 26 Comments

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

nota 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 Journalarticle “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.

…

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.

I don’t think you can consider if a lie if Donald had no intention to deceive. He might have believed the story himself, there having been no internet to reveal the truth to him. And furthermore, he might have been an imaginary character, with no ability to believe and no ability to intend. No way to know now, of course.

I also wonder what information was easily available to the writers back in 1959, when the film came out. The paper Mario Livio mentions came out in 1992, and the earliest “doubter” he references is

The Golden Number and the Scientific Aesthetics of Architecture– in 1958. The common opinion of the time was probably that the Golden Ratio was connected with the Parthenon, so even a fact-checker would have approved it.But a lot of this is guessing on my part, and does little to stem the hurt, betrayed feelings, heh.

NOOOOOOOO!

My entire world view has been crushed! How will I find the strength to carry on?

What Great Christina said. If you can’t trust Donald Duck, whom can you trust?

I agree, but I’m gobsmacked by the assertion of Devlin’s you quote:

“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.”

This asserts that the constructible numbers are a subset of the rational numbers, which is patently false. I have no idea what he means by “by measurement”, which I assume is his escape hatch for this outrageous claim.

John – that gave me pause too, glad I’m not the only one!

I assume he meant that for a span 1 arbitrary unit wide, they couldn’t measure the exact height that would create a Golden Rectangle. But it seems like there would be geometric ways to get damn good versions, like using the pentagram trick the shown in Donald in Mathmagic Land.

As it was, I decided to look for another authority and found Mario Livio – that went further to convince me of the disconnect between Phi and the Parthenon.

yes, that seemed rather mealy-mouthed to me as well. I mean, he is factually correct in saying that you cant geometrically construct it. However a good surveyor, architect, and builder–even in those days–could build something that your eye could not tell apart from a perfect phi.

Of course you can construct it.

Phi = (1+sqrt(5))/2

sqrt(5) is the hypotenuse of a right triangle whose legs have lengths 1 and 2.

Sorry, allow me to clarify. …cant build, according to geometric rules, something in meatspace that is exactly phi… Shouldnt have used the word “construct” in its building things out or wood, metal, stone, etc. meaning. Far too close to its mathematical definition.

I think the issue here is what is meant by “exactly”. Phi is the MOST irrational number, so when one constructs, one can get very “rigorous” as to whether something matches exactly. So, any critic can say, with enough resolution, that the ratio doesn’t match. For instance, if you construct a spiral based on the phi ratio and scale out to 10X original size and compare to an “ideal” spiral, you’ll notice that they won’t overlap. If they do, then you scale out 10X more and you’ll find a mismatch. Since the log spiral doesn’t change shape, this is possible.

Rick – perfect! I was going to end the post with something like “Next they’ll tell me Santa Claus doesn’t exist!” but thought it could come off as sarcastic. I have

noidea why anyone would consider me sarcastic, however.In 2008, the most popular aspect ratio for LCD monitors was 16:10, which is almost the Golden Ratio, but it’s been superseded by the 16:9 HD aspect ratio.

So then is it yet another urban legend that φ pops up in nature all over there place?

psh, next thing you know, the’ll say Pluto isnt a planet!

Naw, that would be silly. Everyone knows Pluto is a planet!

What’s that? Oh. I’m being told that this has already happened. *Cries theatrically in the corner*

I think the most important thing here is to recognize the golden ratio had its golden era in the Renaissance. As the name implies and its artists largely claimed, this movement was inspired by the art of the classical period, namely and foremost, that of ancient Greece. It is therefore not strange that today some misconceptions remain as to what was the original intention of the greek artists. The information we have today was filtered by the Renaissance conception and revival of ancient art. Furthermore today we are clearly almost as distant from the Renaissance worldview as the Renaissance artists were themselves from the Classical. The impact of “special” numbers and “golden proportions” on Renaissance architecture and a little beyond, is undeniable, though it will be most aparent for someone visiting Southern Europe than any other place. Remember these were the days of unveiling the unknown… Today we take reality for granted.

Cheers

The Fibonacci Series of whole numbers rapidly approaches Phi and is used to calculate(approximate) it. So the series goes 1,1,2,3,5,8,13,21,34,55… 13/8=1.625 21/13=1.615, 34/21=1.619, 55/34=1.618 (rounding up). Also, the ratio is easily constructed using compass/straightedge drawing, with no measurement whatsoever. Its similarity to the Fibonacci Series is what accounts for it showing up in nature frequently, as nature uses the FS in almost all growth patterns.

Let’s keep it straight – the FS describes natural growth patterns – nature doesn’t “use the FS” ;)

Hah! Yes, that’s a great way to phrase that.

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I am sorry for Mr.Devlin, but he is wrong when he says

“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.”

There exists a simple construction with rule and compass which gives the rigorous golden section for any segment

http://en.wikipedia.org/wiki/Golden_ratio#Geometry

Furthermore number Phi is irrational, but can be easily obtained from a rectangle having base that is twice its height. Diagonal of such a rectangle has a ratio (1/2)sqrt(5) with the base

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