Is the golden ratio in art and architecture?

History

Euclid, Pacioli and Leonardo

The ratio was first written down as geometry, not as a rule for beauty. Euclid’s Elements defines it as cutting a line “in extreme and mean ratio”:

“A straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the less.”Euclid, Elements, Book VI, Definition 3 · translated by T. L. Heath

Euclid used it to build the pentagon faces of the dodecahedron. He never called it golden or divine.

The art link starts with a friar and a painter. Around 1496 the mathematician Luca Pacioli went to Milan, where he worked on a book about the ratio. MacTutor: “Pacioli began work on ... Divina proportione and the figures for the text were drawn by Leonardo.” It was printed in Venice in 1509.

That meeting came late in Leonardo’s career. Markowsky points out that Leonardo’s “acquaintance with the divine proportion dates from his meeting with Luca Pacioli, which occurred 13 years after he painted St. Jerome.” So a golden rectangle drawn over the earlier St. Jerome cannot be something Leonardo learned from Pacioli.

“Golden” came last of all. The German name goldener Schnitt, golden section, first appears in print in 1835, in a textbook by Martin Ohm. The letter φ was suggested in the early 1900s by Mark Barr.

Measured · Egypt

The Great Pyramid: φ fits, and so does π

The pyramid passes the test, but it passes a second one too. Markowsky used a base of 755.79 feet and a height of 481.4 feet. Cut the pyramid down the middle and the sloping face, the slant height, is 612.0 feet, while half the base is 377.9 feet. In his words: “612.01/377.90 ≈ 1.62 which differs from Φ by only 0.1%.”

Now try π with the same numbers. Four times the base divided by the height is 4 × 755.79 ÷ 481.4 = 6.28, and 2π = 6.283. Markowsky: it is “well within ±2% of 2π.” Both fits come from one slope, so the measurements alone cannot tell you which, if either, the builders meant.

No record ties φ to Egypt. The popular story that Herodotus described the golden ratio in the pyramid traces back to John Taylor’s 1859 pyramid book, and Markowsky found that Herodotus’s text “does not support the story.” His conclusion: “It does not appear that the Egyptians even knew of the existence of Φ much less incorporated it in their buildings.” Press Great Pyramid above to see the slant swung down onto the base.

Measured · Athens

The Parthenon: 2.25, not 1.618

The Parthenon’s front is much wider than a golden rectangle. With a width of 101 feet 3.75 inches and a height of 45 feet 1 inch, Markowsky found: “width/height ≈ 2.25 = 9/4 and length/width ≈ 2.25 which are well outside the acceptance range.”

Measuring to the roof peak does not rescue it. Up to the top of the triangular pediment, 59 feet: “101/59 ≈ 1.71 which also falls outside the acceptance range.” A golden rectangle as wide as the front would have to be 62.6 feet tall, higher than the peak.

Look closely at the famous diagrams. Most draw a golden rectangle over a photo of the front. Markowsky: “None of these authors is bothered by the fact that parts of the Parthenon are outside the golden rectangle.” Press Parthenon above: the dashed golden rectangle and the building do not line up.

No record · Florence

The Mona Lisa: no documentation

There is nothing to measure against, because no one knows where the rectangle is supposed to go. Mario Livio, writing for the University of Cambridge’s Plus Magazine, put it plainly: “No documentation exists to indicate that Leonardo consciously used the Golden Ratio in the Mona Lisa’s composition, nor to where precisely the rectangle should be drawn.”

That is the problem with most overlays on paintings. A face, a canvas edge or a hand gives dozens of possible starting points, and a golden rectangle or golden spiral can be slid around until it touches something. Leonardo drew the figures for Pacioli’s book, which shows he knew the ratio later in life. It does not show he painted with it.

On purpose · 1955

Dalí’s Last Supper: a golden canvas

Here the golden ratio is the size of the canvas itself. The Sacrament of the Last Supper, painted by Salvador Dalí in 1955, hangs in the National Gallery of Art in Washington. Its catalogue size is 166.7 × 267 cm, and 267 ÷ 166.7 = 1.6017, about 1% below φ and well inside the pass band.

He put a second golden object in it. Livio: “Dali also incorporated in the painting a huge dodecahedron (a twelve-faced Platonic solid in which each side is a pentagon) engulfing the supper table.” In a regular pentagon, each diagonal is φ times the side. Livio calls Dalí a painter “about whom there is very little doubt that he actually did deliberately include the Golden Ratio.”

On purpose · Architecture

Le Corbusier’s Modulor

The clearest case in architecture is a measuring system made for the purpose. The Fondation Le Corbusier says the architect “developed the Modulor between 1943 and 1950, with the help of his workshop (mainly Gerald Hanning).”

It is a scale of sizes built from the human body. “After being designed for a stature of 1.75 m (Le Corbusier’s own height), it was finally based on a man’s height of 1.83 m or 6 feet, inspired by the Fibonacci sequence and the golden section.” He published it in two books, Le Modulor (1950) and Modulor 2 (1955). Unlike the Parthenon, nobody has to guess here: the architect wrote it down.

Tested on people

Do people prefer golden rectangles?

Mostly not, when it is tested. Markowsky quotes a review by Schiffman and Bobko: “Research on the golden section proportion as an empirically demonstrable preference has most often been applied to the rectangle where the results, on the whole, are negative.”

In his own informal tests, the favourite was 1.83. When people picked the rectangle they liked best, “the most commonly selected rectangle is one with a ratio of 1.83”, outside the pass band. His verdict: “The various claims made about the esthetic importance of the golden ratio seem to be without foundation.” Press Favourite above to see how much longer 1.83 is than φ.

You can still use it as a guide. Nothing stops you using φ as a composition guide, the way photographers use the rule of thirds; it is a choice, not a law the eye obeys. To try it on your own pictures, load one into the Golden Ratio Lens; the photo stays on your device.