What Is the Golden Ratio? A Real Number With a Great Deal of Nonsense Attached

The ratio, usually written as the Greek letter phi, is the positive solution to a simple equation stating that a number equals one plus its own reciprocal, which gives the exact value of one plus the square root of five, all divided by two, and the decimal expansion 1.6180339887 and so on without repeating. Two properties follow immediately and are genuinely unusual. Squaring it gives the same number plus one, and taking its reciprocal gives the same number minus one, so the digits after the decimal point are identical in all three cases, which no other number does. It is also, in a precise sense, the number hardest to approximate by fractions, because its continued fraction expansion consists entirely of ones, and that fact is the reason it appears in nature rather than any aesthetic principle.

The sequence in which each number is the sum of the two before it, running 1, 1, 2, 3, 5, 8, 13, 21 and onward, has a direct relationship to the ratio: divide any term by the one before it and the answer oscillates around 1.618, converging on it ever more closely, so 13 divided by 8 gives 1.625 and 233 divided by 144 gives 1.6180555. This holds for any sequence built by the same rule regardless of the two starting numbers, which is a more surprising result than the Fibonacci case alone. The connection explains most appearances of both in nature at once, because the growth processes that produce Fibonacci counts are the same ones that approach the ratio.

The reliable case is phyllotaxis, the arrangement of leaves, seeds and florets around a stem or disc. A growing plant places each new element at a fixed angle from the previous one, and the angle that appears again and again, measured in real plants, is about 137.5 degrees, which is the circle divided in the golden proportion. The reason is optimisation rather than beauty:

• •Any angle that is a simple fraction of a turn causes new elements to line up in a small number of radial rows, overlapping and shading each other
• •The golden angle is the worst possible approximation by a fraction, so elements never line up and the packing is as even as it can be
• •The visible result is the interlocking spiral pattern in a sunflower head, a pine cone or a pineapple, and counting the spirals in each direction gives consecutive Fibonacci numbers
• •Computer simulations and physical experiments with repelling droplets reproduce the same angle from simple local rules, so no plant is calculating anything
• •Nautilus shells are frequently cited and are a poor example, since their spirals are logarithmic with a ratio that is usually not close to the golden one

The aesthetic tradition rests on much weaker ground. The Parthenon fits a golden rectangle only if you choose where to put the edges, and the choices made in the standard illustrations are not the building's actual structural boundaries; no Greek source mentions using the ratio in architecture. The great pyramid's proportions can be fitted to phi and equally well to pi or to simple whole-number slopes used by Egyptian builders, and no papyrus mentions it. Leonardo illustrated a book by Luca Pacioli about the ratio in 1509 and there is no evidence he composed paintings with it, despite a large industry of overlaid rectangles. The claim that the ratio appears in credit cards, book sizes or the human body relies on generous rounding, since a card is 1.586 and an A4 sheet is the square root of two. A 2015 review of the psychological literature on preference for golden rectangles, which has been tested since Fechner in 1876, found the effect weak, inconsistent and heavily dependent on how the question is asked.

The mathematics is real and attractive, the natural examples are genuine, and the combination gives the aesthetic claims an unearned credibility. The modern form of the myth largely dates to the nineteenth century, when the German writer Adolf Zeising asserted that the ratio was a universal law of proportion in nature and art, and to Mark Barr's assignment of the Greek letter in about 1909. Twentieth-century artists and architects then used it deliberately, which is a different matter from discovering it in older work: Le Corbusier built his Modulor system on it, Dali composed a painting around it explicitly, and Debussy has been argued to have used it structurally. Design practice retains it as a rule of thumb for pleasing proportion, which is defensible as a convention that produces consistent results and is not defensible as a discovered law. The honest summary is that a number with unusual mathematical properties and a clear role in plant growth has been retrofitted onto a great deal of art it had nothing to do with.

The ratio of about 1.618 is the solution to a number equalling one plus its reciprocal, and it is the hardest number to approximate by fractions, which is precisely why plants use the corresponding angle of 137.5 degrees to space leaves and seeds so that they never line up. Ratios of consecutive Fibonacci numbers converge on it. Claims about the Parthenon, the pyramids, Leonardo's compositions and human proportion rely on selective measurement, and psychological testing finds little consistent preference for golden rectangles.