Saturday, January 19, 2013

Part 5: Last question about the golden rectangle

I arrived at the end of my search for the truth about the golden mean. But I had one last question. Is the "golden rectangle" really more attractive than other rectangles? 

The answer was not what I expected, and not what I wanted to hear.

I had always accepted as an aesthetic axiom that the golden section rectangle (1.618 long by 1 wide) represented the ideal, even "divine" proportion. Was there any way to prove it?

In the 1860s, a psychologist named Gustav Fechner conducted experiments to explore this question. He presented subjects with an array of varying rectangles, and asked them which was their favorite.

The results showed that 76% of all choices focused on the three rectangles with ratios of 1.75:1, 1.62:1, and 1.50:1. The winner was the "Golden Rectangle" (D, above, with ratio of 1.618:1).

That seemed to settle the question for decades. Beauty, it appeared, could be defined in terms of a specific mathematical harmony of proportions.

Unfortunately, Flechner's conclusion unraveled as later scientists tested the hypothesis more rigorously. According to math expert Mario Livio,
"[University of Toronto professor Michael] Godkewitsch concluded from a study conducted in 1974 that the preference for the Golden Rectangle reported in the earlier experiments was an artifact of the rectangle's position in the range of rectangles presented to the subjects. He noted: 'The basic question whether there is or is not, in the Western world, a reliable verbally expressed aesthetic preference for a particular ratio between length and width of rectangular shapes can probably be answered negatively.'"
So, it seems, no rectangle stands out from the others as "golden" or uniquely beautiful. If a certain one gives us a warm feeling, maybe it's because we've trained ourselves to appreciate it.

The more I thought about it, the more it made sense. If the golden rectangle (1.618:1) really was the ideal shape, why didn't it appear everywhere in our carefully designed environment? Why don't we find it in the proportions of movie screens (1.37:1, 1.85:1, 2.35:1), photographs (1.50:1) television monitors, (1.33:1, 1.78:1) computer screens (1.33:1, 1.60:1, 1.78:1), credit cards (1.5858:1), not to mention iPhones, tablets, and office paper? Those rectangles, each so commonplace in our daily lives, vary greatly, and none of them quite matches the supposed ideal.

Perhaps there's a deeper aesthetic truth to be gleaned from all of this. A masterpiece, it turns out, does not issue from fixed mathematical rules. It comes from a happy mixture of all the elements of composition cohering with messy particularity. For one painting, a 3x4 rectangle might be the ideal choice; for another, a square might yield divine results. The picture's central idea must drive the decision. Just as there is no optimum running length for a film, no optimum key for a symphony, and no optimum structure for a poem, there's no optimum shape for a painting.

I welcomed these revelations as inspiring rather than disillusioning. Art cannot be reduced to any absolute formula. The golden conditions are situational, not preordained. A great creation pierces our hearts through an unexpected combination of factors. Beauty arrives in the night and hovers just outside our window, shifting and shimmering, floating just beyond the reach of our strings and calipers, unwilling to fit into any box we build for her.


Allen Garns said...

Fascinating post Mr. Gurney. Your thoughts have relieved me of 30 years of guilt!

Mario said...

While the idea of the golden ratio may have been inspiring for some artists (but other systems have been inspiring for many others), the claim that the look of the golden ratio has something special seems baseless to me. The golden ratio is an irrational number, which means that its decimal part is infinite and non-repeating. So, it can be described by a property or a formula, but it cannot be known with arbitrary precision. There is no way to check if a "real thing" is "as large as the golden ratio" (and, down to atomic level, it would be meaningless). A real thing can be "more or less" large as the golden ratio. But let's take the number 1.6: it's about 1% different from 1.618... and can be written as 8/5. Do you think anyone could spot a 1% difference in a painting or a temple? So, why not speculating about the "divine 8/5 ratio"? The beauty of the golden ratio lies in its definition: it's conceptual, not visual.

By the way: in nature, one can find a lot of "numbers", both rational and irrational. In a simple cubic lattice, the diagonal of each face is the square root of 2, and the diagonal of each cube is the square root of 3 (both irrational). Measuring the distance between different points, you can find many "magical numbers".

Mario said...

And, of course, I agree that every piece of art has its "right" proportions and measures, depending on its content.

James Gurney said...

Mario, thanks for those important points. Even the most fine-tuned artist's eye probably can't judge ratios of height to width closer than 1 or 2 percent.

And for those who aren't familiar with math terminology, when a number is "irrational" it doesn't mean crazy, it just describes a certain kind of number that can't be written as a simple fraction.

Another one is "pi," the ratio of the circumference to the diameter of a circle. That one has always fascinated me more than "phi" because I've always played around with wheels, anemometers, and other circular things.

Keith Parker said...

I was thinking that if the GR appears in nature so much it is likely more related to structural integrity than aesthetic appeal. Perhaps engineers, and physicists have already explored this thought. There are often very practical reasons for things to exist in the natural world.

Rich said...

There is hardly a given method, process or rule by which beauty can or ought to be created.

The more we develop vision and insight, the better; that's what I gather from these posts.

Gina Florio Sous said...

A great series of posts that I will share on my own blog. And I love the conclusion you come to in this last paragraph. Beautifully written!

mp said...

Way to wrap it up in grand style!

Diana Moses Botkin said...

Well, what a relief!

I read your posts on this subject the past few days with much interest. I'm very glad to be able to choose my compositions and painting dimensions according to what pleases me rather than a mathematical formula.

Sometimes painting can be a tedious enough. I think I would lose interest altogether if I had to first go through the mental gymnastics required for the golden mean in the planning stages. It wears me out just thinking about it.

Not to say that planning shouldn't be done, of course.

MLM247 said...

I have really appreciated your posts on this topic. My art education is quite limited and I find your writing easy to read. I am learning lots. I was wondering though if the concepts explored in the Golden Mean are present in visual art and architecture from other cultures. Has it been found in Japanese work, for instance? Or Aztec? Or Sudanese?

Scorchfield said...

It's wave, waterfall, tsunami, crescendo...

JC said...

I occasionally write for a photo blog, and some time back, did an essay on the "proper" size for photographs. Of course, there really is no such thing, but the question I was exploring involved the fact that photographic print sizes had always been constrained by mechanical aspects of the craft, such as grain size (which affects practical enlargement size) and the size of practical enlargers and so on. This changed with the rise of digital photography and digital printers. My question was, if you take an art that is viewed in very much the same way as wall-hung photographs, that do not have these mechanical constraints -- I was thinking of painting -- what size is most often chosen by artists? One immediate finding was that painters almost never painted as small as photographers printed -- that photographers might profitably consider this fact, and print larger. But I also took up the question of the Golden Mean. In looking at sizes and aspect ratio, I examined wall-hung paintings from the late Renaissance onward, as presented in a standard art history book. Of the dozens of paintings I examined, I found one (1) that was close to the Golden Mean in terms of aspect ratio. Almost all of them were much more square, whether they were displayed in portrait on landscape orientation. In fact, most paintings are much more square than the standard 35mm aspect ratio (1:1.5) with is itself more square than the GM.

mdmattin said...

James, thank you for this wonderful series of articles. I heartily concur with your conclusion that art cannot be reduced to a formula! The linked articles are fascinating as well, especially the Salingaros piece featured on Meandering Through Mathematics. He makes the point that should have been obvious - just because phi manifests itself in natural hierarchies of scale, why on earth should we assume that people will perceive and respond to that relationship in a rectangle with that side to end proportion?
That said, such rectangles do have interesting properties which are worth knowing about as artists. The rectangle is the format in which most of us do most of our work, so it makes sense to understand it, just as we work to understand human anatomy and the properties of pigments.
I did a bit of research over the last year into the more general questions of rectangular proportions, both as a historical phenomenon and a practical tool for artistic composition. Hambidge's Elements of Dynamic Symmetry is well worth reading for its geometric analysis, although his historical theories have been rather convincingly debunked. Back in 1921, Rhys Carpenter's article Dynamic Symmetry: A Criticism pointed out the arbitrariness of Hambidge's overlays on images of Greek vases purporting to show precise mathematical proportions, and the unlikelihood of humble potters knowing elaborate geometry, much less applying it to an imaginary 2D bounding box.
The historical role of both phi and the whole number divisions obtainable using the armature of the rectangle are more convincingly made in Charles Bouleau's beautiful book The Painter's Secret Geometry : a study of composition in art. Bouleau's thesis is that the medieval European artists were indeed fascinated by phi, attributing its irrational properties to a divine mystery, while the humanistic and rational impulses of the Renaissance led to a preference for simple whole number proportions.
The best account of this era may be in Rudolf Wittkower's Architectural Principles in the Age of Humanism. At once skeptical of overblown claims and cognizant of the documented role of numerical proportion in the art and architecture of the early Renaissance, he makes the case for a close relationship between the assumed divine virtues of various proportions and the actual practice of design.
My motive was reading all of this was not just historical curiosity; I really wanted find ways to strengthen my compositions. As a side project I created a tool for generating and measuring rectangular aspect ratios, which I have since found to be quite useful. I have posted it for free download here, and invite anyone who is interested to try it out. The accompanying instructions contain more background and ideas for how to integrate proportional concepts into making pictures.

J. R. Stremikis said...

great series here. thanks.

one of your points (above) puzzled me --

" If the golden rectangle (1.618:1) really was the ideal shape, why didn't it appear everywhere in our carefully designed environment? Why don't we find it in the proportions of movie screens (1.37:1, 1.85:1, 2.35:1), photographs (1.50:1) television monitors, (1.33:1, 1.78:1) computer screens (1.33:1, 1.60:1, 1.78:1), credit cards (1.5858:1), not to mention iPhones...?

so, I wonder why you chose from the "Overthinker"'s example for iPhone 4, compared to iPhone 4 Golden Ratio analyses - those that do support the GR in Apple's industrial designs and its logo: e.g.,
iPhone 4:
iPhone 4 camera:
the Apple logo:
a page from Apple's website:
anyway, doing a Google search for "the golden mean and corporate logos" - and specifying "Images" gives in fact many many pages of familiar logos.
as does "the golden rectangle and corporate logos" -- Apple's of course, Atari, Pepsi, Nissan, Toyota, BP's green and yellow concentric circles, KitKat candy bars, the Twitter logo.
there are many basic web design online guides (as well as print guides) that advocate use of Golden Ratio, Fibonacci numbers and Phi -- and it is employed everywhere - e.g., the Apple web page example above, The Fast Company,
granted, it's not the hard screens, it's what's on those screens.

Unknown said...

What if we look at this issue from another perspective?

Linear perspective is a form of geometry that can be made to be mathematically precise. We all know that the person off in the distance is not a Lilliputian but most of us have had to be taught a system for translating 3D information onto a 2D surface that reconciles with our shared experience. Yet, outside of academic exercises, how many of us still construct mathematically precise perspectives? Most of us use our experience with this tool to fudge things so that they look right. Sometimes we even distort the perspective to expand, contract or warp the space based on our intent.

All rectangles have a subset of topology that subdivide the whole. There are 8 (commonly used) root rectangles whose subdivisions relate proportionally back to the whole, the Phi Rectangle is one of them. The potential of these rectangles has less to do with their aspect ratio being some divine truth or aesthetic short cut and more to do with the mathematical truth that their proportional decomposition permits one to construct rhythm, hierarchy and balance within the rectangle that relates back to the whole. It may help to think of them like fractals.

Root rectangles are a geometric tool for organizing surface relationships (rhythm, movement and hierarchy) within a static image just as linear perspective is a geometric tool for organizing spatial relationships (depth) in a flat image. As with any tool its effective use or abuse is in the hand of the user. Using a tool because you believe in it’s magical power is not that different from dismissing a tool because you don’t believe in it’s magical power.

Cultivating the skill to use several tools permits greater flexibility in realizing your design. It also helps you recognize how other artists have constructed their design. Humans are very good at pattern recognition (objects, faces, rhythm, music, language and movement) and it may not be a coincidence that they are all processed in the temporal lobe.

When the image moves within the rectangle, as with the various screens available, the internal organizational geometry of a specific aspect ratio is of little use. That said, it’s worth noting that many of the aspect ratios cited as not Phi Rectangles are only a couple of percentage points off other root rectangles. For instance: 4:3 = 1.25:1 ≈ 1.27:1 = √Phi and 16:9 = 1.77:1 ≈ 1.73:1 = √3

After all who wants to buy a 1.73:1 TV when you can get the beefier 16:9 for the same price.

MCGuilmet said...

Well done, James.

While you have shined sunlight into some dark shadows surrounding phi, I do think this last commentator is touching upon the way in which phi, among other grid systems, may have been more accurately employed.

"Unknown", please send me a note at I want to show you something. Cheers.

Unknown said...

The Golden Rectangle (or any other compositional grid) is much misunderstood. All designers (painters, architects, graphic designers, et al.) work with a grid — the human eye/mind prefers things that line up. Pictorial composition is an abstract art that deals with unity, anomaly, counterpoint, fugue, etc, and the grid does not compose this for you: it mearly fine-tunes the placements within a chosen interval.

There are many grids beside phi and its derivatives: the most common in painting is the basic armature, but there are also the rebattement and the root series.

Unknown said...

In my opinion, "Unknown" is correct.

The golden ratio is much misunderstood: it is a geometrical grid that designers use to line-up elements in their compositions. All designers (painters, architects, graphic designers, et al.) use grids – the human eye/mind prefers things that line up. Pictorial composition, like music, is an abstract art that comprises unity, anomaly, counterpoint, fugue, value schemes, and hue choices, and a grid will not compose for you. What it does is fine-tune the placement of your elements within a chosen interval.

The golden-ratio grid (as well as its derivatives) is only one of several types. The most common in the history of representational painting is the basic armature, but there are also the rabatment and the root series.

When Maxfield Parrish was aked why he uses Dynamic Symmetry, he replied that his paintings look better when he uses it.

Underbitscot said...
This comment has been removed by the author.
Underbitscot said...

The Root Rectangles and the Rectangle of the Whirling Squares are just additional devices for the draftsman/designer to have in his or her tool bag. Obviously our compositions need some idea or structure in order to make them captivating to the viewer. Most laymen will not be able to tell you about a gamut or elements organized around a sphere conception, but they will feel the effects of such devices, and those effects will draw them in. Thus it is with the eyes one can use in these rectangles. I believe the masters used them as a natural response to their ideas not the other way around. Either way they are extremely helpful in placing coincidences in the most effective locations to create variety and balance at the 2D level. Like the book "Flatland" these rectangles help the artist design at the 2D level before moving into the third dimension.