The golden ratio: facts and myths

What the golden ratio is

Divide a line into a long part a and a short part b so that the whole line relates to a exactly as a relates to b. That ratio is φ:

The equation φ² = φ + 1 gives φ some curious properties. Its square is 2.618…, exactly one more than itself, and its reciprocal is 0.618…, exactly one less.

Where φ provably appears

• In the regular pentagon: the ratio of a diagonal to a side is exactly φ. The five-pointed star is full of golden ratios.
• In the Fibonacci sequence: the ratio of consecutive terms approaches φ.
• In continued fractions: φ = 1 + 1/(1 + 1/(1 + …)). Because every term is 1, φ is in a precise sense the number hardest to approximate by fractions.
• In the golden angle of about 137.5°, which divides a full turn in the golden ratio and appears in the arrangement of seeds in many plants.
• In geometry: the regular icosahedron and dodecahedron have coordinates built from φ.

Myth: the nautilus shell

The nautilus shell is a beautiful logarithmic spiral, and it is often described as a golden spiral. Measurements of real shells tell a different story: their growth rates vary between specimens and generally do not match the golden spiral. Many different logarithmic spirals exist, and the nautilus follows its own.

Myth: the Parthenon and the Mona Lisa

Golden rectangles drawn on photographs of the Parthenon or the Mona Lisa usually depend on where the lines are placed. With irregular shapes, thick outlines and a free choice of which points to measure, it is easy to find a ratio near 1.6 almost anywhere. There is no surviving evidence that the Parthenon’s builders designed with φ, and art historians have found no documentary support for Leonardo using it in the Mona Lisa.

Some later artists and architects, such as Le Corbusier, did use the golden ratio deliberately. Those are genuine examples, but they are modern choices rather than proof of an ancient secret.

Myth: the most beautiful proportion

Psychology experiments since the nineteenth century have asked people to choose their favourite rectangle. Results are mixed: preferences vary widely, depend on how the question is asked, and do not reliably peak at φ. Claims that faces or bodies are more attractive when they follow golden proportions are not supported by careful studies.

Computing φ yourself

Because φ = 1 + 1/φ, you can find it by repeating a single step: start with any positive number x and replace it with 1 + 1/x.

The values close in on 1.6180339…, alternating above and below it, exactly like the ratios of Fibonacci numbers. In the language of this site, φ is an attractor of the map x → 1 + 1/x, just as 6174 is an attractor of Kaprekar’s routine. The difference is that here the values only approach φ without ever reaching it, while Kaprekar’s routine lands on 6174 exactly.

Why the myths spread

Any ratio between about 1.5 and 1.7 looks “golden” if you want it to. Our brains are very good at spotting patterns, and a famous number makes a satisfying story. The same effect appears with 6174, which is sometimes described as a hidden constant of nature, as discussed in Is 6174 found in nature?.

The honest version is still remarkable: a single number links pentagons, Fibonacci ratios, continued fractions and plant growth. It does not need exaggeration.