Tuesday, January 15, 2013

Mythbusting the Golden Mean, Part 1

In architecture and design schools, it's common to hear the claim that "golden mean" geometry was used in the design of ancient buildings, especially the Parthenon. 


According to mathematician Keith Devlin of the Mathematical Association of America, this is a groundless myth, with no basis in fact whatsoever.



The golden mean (also known as the golden ratio or the divine proportion) refers to the relationship of 1:1.618..., an irrational number also known as "Phi." The ratio is found in nature, and has been championed in the last two centuries, but many other claims are unfounded.

Although Greek mathematicians knew about Phi, there is not a shred of evidence that any Greek architect used such a system as a design principle. Euclid's study of Phi occurred long after the Parthenon was already finished.

Devlin says:
"The oft repeated assertion that the Parthenon in Athens is based on the golden ratio is not supported by actual measurements. In fact, the entire story about the Greeks and golden ratio seems to be without foundation. Numerous tests have failed to show up any one rectangle that most observers prefer, and preferences are easily influenced by other factors. As to the Parthenon, all it takes is more than a cursory glance at all the photos on the Web that purport to show the golden ratio in the structure, to see that they do nothing of the kind. (Look carefully at where and how the superimposed rectangle - usually red or yellow - is drawn and ask yourself: why put it exactly there and why make the lines so thick?)"
In the examples above, the placement of the golden rectangle doesn't agree from one diagram to the next. In the top example, the sides of the rectangle hug the columns, and in the next, they're touching the edges of the pediment. In some, the bottom of some rectangles correspond to the bottom of the columns, while in others, they're several steps down the base. In the middle example above with the white lines, the source photo itself seems to be stretched vertically by about 15%.

According to University of Chicago math professor Phil Keenan, it doesn't matter how you arrange the diagram, because the lines in the Parthenon aren't straight or parallel anyway due to entasis and other factors. He says:
"One cannot define an exact rectangle on the front or back faces of the Parthenon. Even though the Parthenon is built to extremely accurate specifications, its curvature precludes rectangular measurements of any greater precision than about 1%. This built-in error precludes finding any Golden Mean rectangles, since the required accuracy is simply not attainable."

George Markowky elaborates:
"The dimensions of the Parthenon vary from source to source probably because different authors are measuring between different points. With so many numbers available a golden ratio enthusiast could choose whatever numbers gave the best result."
Keenan points out that, "the presence of the Golden Mean in the Parthenon was postulated by Adolf Zeising in the 1850s, and appears nowhere in ancient Greek architectural treatises."
Devlin concludes: "I am not convinced that the Parthenon has anything to do with the Golden Ratio."

Anticipating some questions and comments:
1. Does the golden mean appear in nature? Yes, and I'll get to that later in the series.
2. Is it a useful tool for composition or analysis? Sure, if it works for you. Busting this myth doesn't take away anyone's candy.
3. Do contemporary architects use it? Bauhaus training has reinforced both the myth and the practice.

22 comments:

Brett W. McCoy said...

You mean 'Donald Duck in Mathmagic Land' lied? Now my childhood is ruined ;-P

Jordan Beeston said...

I was actually amazed on how useful learning to draw a dynamic rectangle was, thanks to Myron Barnstone. It's a great way to deconstruct composition and it's even better for measuring proportion in observation.

Matthew Meyer said...

Thanks for busting this myth! I have to say I roll my eyes every time I hear it.

Craig Banholzer said...

Wow! Thanks, James, this looks like the start of a great series. I've long suspected that the Golden Ratio was overrated. I've done lots of tracing paper overlays on photos of old paintings trying to find it, and I've usually failed to do so. What I have found a great deal of are simple divisions of the picture plane based on the "armature of the rectangle."

Laura G. Young said...

Thanks for this; it makes one take a good look at other unquestioned geometric concepts retained since the 19th century Academy (which was of course still in the shadow of the Classicism before it.)

That being said, the golden mean remains a lovely concept. Order out of chaos and all that...

Michael Syrigos said...

Looking at the Parthenon almost every day, I never had even a fleeting thought as to the Golden Ratio being used in it's construction. The design has other foundations to it, and plenty of awesome craftsmanship, but no hocus pocus.

Plus, if you are actively looking for something (golden ratio, repetition of a number etc) you're bound to find ways of reproducing it.

Tom Hart said...

Like some others have said here, I never quite bought into the theory of the Golden Mean in art and architecture, but to be honest, I'm not sure that (in my case) that was at least partly justification for my glazed-over eyes.

That Devlin article is very intersting and fun to read! I never thought I'd find a scholarly mathematical article so entertaining.

Keith Parker said...

Very interesting read. People do seem to get attached to numbers, and assign strange significance to them...

I watched an episode of Nova a few weeks ago about the construction of medieval cathedrals and they worked all kinds of biblical numbers into those buildings on the show...but I was kind of skeptical, because there are so many numbers in the bible...and because they were having to switch between different units of measurement to make it work. Maybe they are right, or maybe they were forcing it because the wanted to find something of interest to report. Isn't that why people read, and watch fiction about explorers and adventurers? Because we want to discover something nobody knows about?

Keith Parker said...

I also find it funny that in every picture of the Parthenon above, the perspective is wrong. To actually test the GR against it wouldn't you need to take the picture from a slightly higher view point; straight on, instead of from underneath?

Terry said...

Ha! I knew it! I always thought people just saw what they wanted to see, and it never made sense to me. Now I feel better about "not getting it".

Scorchfield said...

Later, the scale of golden ratio was rediscover by Fibonacci.

Susan Sorger said...

The one that always stumped me was the image of the Nautilus. I am so glad that one is debunked here as well.
I never really saw a "Fit" but I kept saying to myself that "they" must mean "close enough". But really it isn't. I found an example on the web of a nautilus shell with a golden section spiral depicted above it. Interestingly, they did not superimpose the images, which I did in Photoshop. If the width of the 2 images match, then the height does not. If the size of the golden spiral is increased to match the height of the shell, then nothing else matches up. This isn't even "close enough". Why hadn't I actually followed up on my uneasiness in the past and just done that comparison. Too willing to "suspend disbelief", I guess.

JonInFrance said...

Fascinating - probaly true for most "rules" when it comes to art - but the thought that comes to me is that, all the same, the so-called rules help to shake one's mind up - look at things in a certain way. OK, it's not ultimately "true" - but if it helps one consider proportion and placement... ? Could consider the rules to be "crutches" informing (finally, onece discarded) the instinct... ??

etc, etc said...

Oh yeah? Donald Duck has never lied to me before, and he has those adorable little nephews Huey, Dewey, and Louie. Keith Devlin can't even spell "Botticelli" correctly.

Seriously, in the name of aesthetics, I'm willing to go down any rabbit hole one should point out. I do feel, however, that any valid theory of the aesthetic perception of beauty should have some phenomenological justification; in other words, it must be directly and readily visually comprehensible, and that would tend to disqualify complex mathematical relationships.

Rich said...

Good old Golden Mean. An attempt to schematise Nature & Universal Harmony into some mental concept, IMO.

Perhaps too schematic to be universally true, but probably not devoid of golden meanings.

António Araújo said...

As far as I can tell, the golden ratio mania is right up there ("down" there?) with numerology and astrology. There's a lot of ways that maths can enlighten you with relation to art (perspective, measurement, and a a rational theory of color are a few) but golden ratio mania doesn't seem to be one of them.

It really grates me when people measure in such a shabby (or even dishonest) way that they can "prove" anything they like. Another example is those people who divide actual human faces into thirds or into classical proportions or whatever (or, yes, into golden ratios). Such divisions are useful as *reference points*, but are not to be confused with actual realities in the individual sitter or (another fallacy debunked by actual studies) population averages.

I used to have a teacher who went on and on on how wonderful nature is, that it breaks down the face into simple ratios. Wonderful my butt - if you measure in such a sloppy way then you can get simple ratios just because your precision is so low. Yet, curiously, the very same type of person will find golden ratios everywhere, which isn't a "simple ratio" at all, but rather an irrational number, that requires infinite precision to be measured!

I guess the golden ratio is an irrational number that tends to be found by irrational people. :p

All of this is not to take away from the interesting geometric relations that actually involve golden ratios. Or from the possibility that indeed those relations may figure in Nature in some ways. But wherever they really figure they must be shown to do so with proper arguments, case by case.

David Simcox said...

While the idea of the mean has been skewed over the years and relegated to the laziness of dogma. I think the relevance comes into play during the time of idealization whereas artists were looking for the perfect form. The golden mean repeats, loosely, in nature and art throughout the world. It is not a rule, or a formula to create greatness...it is an observation of an idea...an approximation that I still think holds some merit.

Sit in a class of young grade-school students and ask them each to draw a perfect rectangle...you will see a decided preference toward a 1:1.6. This is not precise...but interesting.

António Araújo said...

By the way, one must be decent towards the other side also and admit that in the eventual cases where golden ratios or fibonacci numbers do figure in Nature, you cannot argue against them just because the actual measurements don't fit exactly. Why? Because whatever process is there will be perturbed by many other processes (say, in morphogenesis or whatever). So, paradoxically, a lack of a proper fit is also no argument. The true argument is that if you want to claim a relationship is there, then the burden of proof is upon you. Sure, vague measurements can inspire you, but they are not proof of anything. To "prove" it you have to do harder stuff than just staring at clouds and imagining unicorns - you have to make a strong case in probabilities or preferably demonstrate an actual mechanism for the ratio to be there (or in the case of the Parthenon, for instance, you might simply come up with historical documents that proved the intention was there).

Russell Dickerson said...

Purely out of curiosity, last year I tried to overlay the rectangle over some of my previous works. I just eyeball all of my compositions anyway, but I thought it would be a fun experiment. I found out that it does match in certain configurations, which I found fascinating. It doesn't change how I create my art or designs, but I did find it very interesting.

António Araújo said...

(for instance, how do we know that the ISO standard paper sizes (A4, A3, etc) follow the Lichtenberg ratio (often mistaken for the golden ratio)? We know it because we have documentary evidence. If you actually measure the paper sizes you can never prove it beyond a doubt because of course there will be a rational approximation to the irrational sizes, both in your measurements and in the manufacturing process, and so the intended "ratio" won't *ever* be actually there)

Andrea said...

Speaking as one who did her graduate work in neuroscience, I think that whether the Parthenon or other major works or art and architecture perfectly correspond to the Golden Mean is not quite the issue. It might be more useful to say that our visual system has evolved to extract information from a scene by moving our gaze in particular ways. There are certain proportions, certain combinations of lines, that conduct, stop, and start the gaze in patterns that evoke pleasure. (The reasons we perceive these patterns as pleasurable are a discussion for another occasion.) This pleasure is what we call "beauty." The Golden Mean simply moves our eyes and stimulates our perceptual system in pleasure-evoking ways. The proportions don't have to be precise and the correspondence with the Golden Mean doesn't have to be exact, because eye movements aren't precise. Nor does the architect/artist have to consciously understand the mathematics or the biopsychology underlying this principle; he just knows that he is making something beautiful, and the reason it is beautiful is unimportant. But as the human perceptual system has probably not evolved significantly in the past 2500 years, the beauty created in the Parthenon is valid today as it was when it was built.

MAQ911 said...

I'm so glad you're tackling this. When I was in art school this was a topic of fascination for me. Often professors would discuss phi, the fibonacci sequence, etc. But NEVER ONCE did they explain how it was used in art, just that the masters knew the secrets of this geometry. Let's crack this case open I say! Kudos to you!

I even created a blog for a school project discussing it. http://www.greatmastersmaterialsandtechniques.blogspot.com