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.

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Further reading:
Book: The Golden Ratio: The Story of PHI, the World's Most Astonishing Numberby Mario Livio
Article: "Golden Sales Pitch," (selling blue jeans with the golden mean) by Julie Rehmeyer
Mario Livio's website Math Plus, "The Golden Ratio and Aesthetics"
Meandering through Mathematics by Phil Keenan

More experimental results testing the golden rectangle

Iphone graphic from Overthinker

The complete golden mean series on GurneyJourney:

Part 1: Mythbusting the golden mean (the Parthenon)

Part 2: The golden mean and Leonardo
Part 3: How the golden mean caught on with artists
Part 4: The golden mean and the human body
Part 5: Last question about the golden rectangle