Ground truth · four catalogues, one measurement

Widen the Window Until It's True

Somebody will tell you that the great painters shaped their canvases to the golden ratio, 1.618. It is one of the few claims about art that can simply be settled, because museums write down what they measured. This page measures the proportions of 720,147 paintings from four catalogues that share no collection, and asks the question Markowsky said any such claim must answer: is 1.618 more crowded than the shapes right next to it. It is not. In all four it sits between 0.64 and 1.07 the median of its own neighbours, which is to say exactly ordinary, and it is not among the twenty-four most common proportions in any of them. What is there instead is a comb of whole numbers: 1:1, 6:5, 5:4, 4:3, 3:2, 2:1, standing two to six times above their surroundings everywhere. Then take the tolerance slider and make the claim come true yourself. At a wide enough window a tenth of all painting is golden, and so is a tenth of everything else. Gustav Fechner, who invented the golden-rectangle hypothesis, measured about 20,000 paintings in 1876 to test it and got 5:4 and 4:3. This is the same answer, from a different century and 720,000 pictures.

The golden ratio story arrives already believed. The great painters, it says, shaped their pictures to 1.6180, because that proportion is the one the eye is built for. Unlike most things said about art this one can simply be settled, because writing down how big a thing is has been part of a museum's job for a long time, and four large catalogues now publish that measurement openly.

It has also been settled before, twice, which is the part worth knowing first. In 1876 Gustav Fechner, the physicist who founded experimental aesthetics and who is the origin of the whole golden-rectangle idea in psychology, went and measured about twenty thousand paintings in twenty-two museums to see whether his hypothesis held in the world. It did not.

“Employing the method of use, Fechner collected data from some 20000 paintings in twenty-two museums and art galleries, in order to see if great works of art tended to be framed in golden proportions. Contrary to his prediction, however, the golden rectangle did not characterize the height-to-width ratio of the paintings. Vertical paintings, on average, displayed a 5:4 ratio, horizontal paintings a 3:4 ratio.” C D Green, "All that glitters: a review of psychological research on the aesthetics of the golden section", Perception 24 (1995), 943 to 944, summarising Fechner's Vorschule der Aesthetik (1876)

In 2015 Michael Trott ran it again with a computer, across more than a million paintings from a dozen collections, and reported the same thing: “The golden ratio is not an aspect ratio that occurs prominently in paintings”, while the maxima that are there sit at 6/5, 5/4, 9/7, 4/3 and 3/2. So the finding below is not new. It is a hundred and fifty years old, it has been confirmed at every scale anyone has tried, and the claim is in better health than ever, which is the actually interesting fact and the thing this page is built to show you.

The measurement

Here is every painting in four catalogues whose height and width can be read out of the record without guessing: 720,147 of them, reduced to one number each, the long side divided by the short side. A square is 1. A canvas twice as tall as wide is 2. The golden ratio is 1.6180. If the story is true, it should sit on a hill.

The test the claim has to pass

The eye can be argued with, so here is the arithmetic, and it is the arithmetic Markowsky asked for when he set out how a golden-ratio claim should be judged. He allowed the claim a generous band, 1.58 to 1.66, and then named the catch that band creates.

“Since the acceptance range includes infinitely many numbers near Φ it is necessary to justify the claim that Φ is the preferred number. Some other ratio coincidentally near Φ might be the important one.” George Markowsky, "Misconceptions about the Golden Ratio", The College Mathematics Journal 23:1 (1992), 2 to 19, at page 5

So: take a proportion, draw a window around it of plus or minus half a percent, count the paintings inside, and then do the same for every other proportion within twelve percent on either side. The question is not whether the golden ratio has paintings in it. Every proportion in that stretch of the axis has paintings in it. The question is whether it has more than its neighbours.

Widening to Markowsky's own generous band changes nothing that matters. In the largest catalogue, 3.51% of paintings fall between 1.58 and 1.66, and 34.1% of all equally wide bands elsewhere on the axis hold more paintings than that one does. The band is 5.06% wide. At that width you can put it almost anywhere and catch a comparable haul.

The comb is the finding the golden ratio was supposed to be. The square, 6:5, 5:4, 4:3, 3:2 and 2:1 stand two to six times above their surroundings, in every catalogue, in the same order. Those are the shapes you get when somebody works in whole units of something. Two irrational proportions with reputations of their own, the square root of 2 and the square root of 3, do no better than the golden ratio does. And the golden ratio's whole-number stand-in, 8:5, beats the golden ratio itself in all four, which is the tell: people make eight-by-five canvases. Nobody has ever cut a canvas to 1.618034.

Now make the claim true

Nothing above depended on the window being half a percent, and this is where the claim actually lives. Widen the window and the count at the golden ratio grows. Widen it far enough and a tenth of all painting is golden. The catch is that at that same width a tenth of all painting is also at 1.45, and at 1.72, and at 1.9. The window does not find the golden ratio. It manufactures it, at the same rate it manufactures everything else.

Two ways this could be an artefact, both tested

The rounding. A comb of whole-number ratios is what you would get for a dull reason. If a catalogue records a picture as 60 by 48 centimetres its ratio is 1.25 whatever the canvas really was, and a window half a percent wide is wider than that rounding. Trott flagged this himself: “the mostly centimeter-precise widths and heights do artificially amplify some simple fractions, such as 6/5, 5/4, and 3/2”.

Three of these four catalogues print every object twice, once in inches and once in centimetres, and one of those two is measured while the other is converted. A converted number is almost never round. So whichever unit comes out whole is the unit somebody actually held a rule in, and the records where neither unit is whole were made with sub-unit precision, where rounding cannot have moved a ratio onto a simple fraction.

The theory arriving late. Nobody could paint to the golden section before being told about it, and the telling is recent: the term itself appears in print in Ohm's textbook of 1835, Fechner's experiments are 1876, and the loud century is the twentieth. If the claim describes an influence rather than a law of the eye, the effect should be in the late work and absent from the early. It is in neither. The one cell that leans golden is the National Gallery before 1850, at ×1.60, which is the wrong side of the split for the story and one cell out of eight.

There is no rectangle the eye prefers

The strongest form of the story is not historical at all. It says the proportion is pleasing, that this is a fact about seeing. That version has a clean test, and one catalogue can run it, because the Metropolitan calls a Venetian altarpiece and a Chinese hanging scroll by the same word and measures both with the same hands.

The median painting in the Asian Art department is 1.93 times as long as it is wide. Only 4.8% of European paintings are that elongated. These are not two dialects of one preference. They are two settled and different answers to a question that a fact about the eye would have had to settle once.

Where the shape does come from

The proportions that are over-represented get more over-represented with time, which is Trott's other finding, and it survives here. Counting the works whose ratio falls within half a percent of a fraction with a denominator of five or less: the share climbs steadily and then sharply through the twentieth century.

The two-unit trick names the mechanism directly. A painting whose height and width are both a whole number of inches came off a rule marked in inches, which in practice means a stretcher somebody sold in a standard size. That share roughly triples through the twentieth century in all three museums, and it is the ready-made canvas arriving and taking over.

What this shows and what it does not

It shows that the outer proportions of paintings, in four large and independent catalogues, carry no trace of the golden ratio, and carry a strong trace of simple whole numbers. It shows that the trace does not appear after the theory did. It shows that two traditions measured by one institution disagree completely about what shape a painting is. None of that is new. Fechner had the first two in 1876 and Trott had all of it in 2015 at a larger scale. What is added here is the neighbour test, run on named proportions so the golden ratio is judged by the same yardstick as 4:3, the rounding control built from the catalogues' own double units, the split at 1850, and the whole thing in your hands rather than on a page.

It does not show that no painter ever used the golden ratio. Individual artists did, on purpose and by name, and a few dozen deliberate cases would be invisible in a distribution this size. Dali's The Sacrament of the Last Supper is the usual example and it is worth seeing precisely: the National Gallery records the canvas as 166.7 by 267 centimetres, which is 1.6017, about one percent under the golden ratio and comfortably inside Markowsky's acceptance range without being the number itself.

It does not test the other and more common form of the claim, that elements within a picture fall on golden-section divisions. That needs the pictures, not the catalogue, and it is a different study. It does not settle preference, because a museum's holdings are what somebody collected, not what was made, and all four catalogues lean European and American. And it inherits every error in the sources: where a museum and Wikidata both hold the same painting they disagree about its proportions by more than two percent 1.3% of the time, which is a floor under how precise any of this can be.

Works cited

1. C D Green, "All that glitters: a review of psychological research on the aesthetics of the golden section", Perception 24 (1995), 937 to 968. https://doi.org/10.1068/p240937. The Fechner passage quoted above is at 943 to 944.
2. Gustav Theodor Fechner, Vorschule der Aesthetik (Leipzig, 1876). The source of the method of use and of the 20,000-painting survey, reported here through Green.
3. George Markowsky, "Misconceptions about the Golden Ratio", The College Mathematics Journal 23:1 (1992), 2 to 19. https://www.jstor.org/stable/2686193. The acceptance range and the objection quoted above are on page 5.
4. Michael Trott, "Aspect Ratios in Art: What Is Better Than Being Golden?", Wolfram Blog, 18 November 2015. https://blog.wolfram.com/2015/11/18/aspect-ratios-in-art-what-is-better-than-being-golden-being-plastic-rooted-or-just-rational-investigating-aspect-ratios-of-old-vs-modern-paintings/
5. Art Institute of Chicago public API, https://api.artic.edu/docs/; The Metropolitan Museum of Art Open Access, https://github.com/metmuseum/openaccess; National Gallery of Art Open Data Program, https://github.com/NationalGalleryOfArt/opendata; Wikidata Query Service, https://query.wikidata.org/.
6. Dali's canvas dimensions are from the National Gallery of Art open data record for object 46590, accession 1963.10.115.