Golden Ratio | Examples, Definition, & Facts - Encyclopedia Britannica

Mathematics

Golden Ratio in Nature and Design

The golden ratio shows up in some natural growth patterns. Spirals in sunflower heads and pinecones often count as consecutive Fibonacci numbers. These spirals reflect the golden angle—about 137.5°—obtained by dividing a circle according to the golden ratio, which helps leaves or seeds pack most efficiently around a stem.

The golden ratio appears in design and art. Spanish artist Salvador Dalí built the proportions and the background dodecahedron of his 1955 painting The Sacrament of the Last Supper on the golden ratio. Swiss architect Le Corbusier developed the Modulor, a scale of human-based measurements using ϕ, and applied it in projects such as the housing complex La Cité Radieuse in Marseille.

The golden ratio occurs in many mathematical contexts. It can be constructed using classical geometric methods—equivalent to straightedge and compass constructions—that go back to Euclid. For example, it can be shown in dividing a line or in forming a golden rectangle.

The golden ratio is closely connected with the Fibonacci sequence—1, 1, 2, 3, 5, 8, 13, …—where each number is the sum of the two before it. As the sequence grows, the ratio of one number to the previous one gets closer and closer to ϕ. Thus ϕ is the limit of these ratios. The golden ratio can also be expressed as an infinite continued fraction: ϕ = 1 + 1 1 + 1 1 + 1 1 + 1 1 + 1 1 + . . . . ​ .

Also known as the: golden section, golden mean, or divine proportion (Show more) Related Topics: number game pentagram number irrational number golden rectangle (Show more) On the Web: National Center for Biotechnology Information - PubMed Central - The golden ratio—dispelling the myth (Dec. 10, 2025) (Show more) See all related content

In modern mathematics, the golden ratio occurs in the description of fractals, figures that exhibit self-similarity and play an important role in the study of chaos and dynamical systems.