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

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• PBS - The Golden Ratio: Is It Myth or Math?
• Cornell University - Department of Mathematics - Golden Ratio
• CedarCommons - The Ubiquity of the Divine (Golden) Ratio and Fibonacci NumbersThe Ubiquity of the Divine (Golden) Ratio and Fibonacci Numbers Throughout the Heavens and EarthThroughout the Heavens and Earth (PDF)
• University of Montana Mansfield Library - ScholarWorks - An In-depth Investigation of the Divine Ratio (PDF)
• National Center for Biotechnology Information - PubMed Central - The golden ratio—dispelling the myth
• CORE - Golden ratio in science, as random sequence source,its computation and beyond
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• Story of mathematics - Golden Ratio � Explanation and Examples
• LiveScience - What is the Golden Ratio?

Ask the Chatbot a Question Also known as: 1.618, divine proportion, extreme and mean ratio, golden mean, golden section(Show More) Written by Stephan C. Carlson Professor of Mathematics, Rose-Hulman Institute of Technology, Terre Haute, Indiana. Author of Topology of Surfaces, Knots, and Manifolds: A First Undergraduate Course. Stephan C. CarlsonAll Fact-checked by Britannica Editors Encyclopaedia Britannica's editors oversee subject areas in which they have extensive knowledge, whether from years of experience gained by working on that content or via study for an advanced degree.... Britannica Editors Last updated Dec. 10, 2025 •History Table of Contents Table of Contents Ask the Chatbot Top Questions

What is the golden ratio?

The golden ratio is an irrational number, approximately 1.618, defined as the ratio of a line segment divided into two parts such that the ratio of the whole segment to the longer part is equal to the ratio of the longer part to the shorter part.

Who was the first to describe the golden ratio?

The ancient Greek mathematician Euclid, in his work Elements (c. 300 bce), described the division of a line “in extreme and mean ratio,” a proportion that corresponds to what is now called the golden ratio.

How is the golden ratio related to the Fibonacci sequence?

As the Fibonacci sequence grows, the ratio of one number to the previous one approaches the golden ratio, making the golden ratio the limit of these ratios.

How is the golden ratio used in modern mathematics?

In modern mathematics, the golden ratio appears in the description of fractals, which are self-similar figures important in the study of chaos and dynamical systems.

golden ratio, in mathematics, the irrational number 1 + √ 5 2 , often denoted by the Greek letters ϕ (phi) and sometimes τ (tau). Its value is approximately 1.618. The golden ratio is defined as the proportion in which a line segment is divided into two unequal parts such that the ratio of the whole segment to the longer part is equal to the ratio of the longer part to the shorter. In algebraic terms, if the shorter part of the segment is taken as one unit and the longer part as x units, this forms the equation ( x + 1 ) x = x 1 . This can be rearranged into the quadratic equation x 2 - x - 1 = 0 , for which the positive solution is x = 1 + √ 5 2 , the golden ratio.

History

The origins of the golden ratio can be traced to the ancient Greeks. Mathematician Euclid, in his work Elements (c. 300 bce), described the division of a line “in extreme and mean ratio,” a division in which the ratio of the whole length to the longer part equals the ratio of the longer part to the shorter—the proportion now known as the golden ratio. During the Renaissance, interest in the ratio was revived. Italian mathematician Luca Pacioli published De divina proportione (1509; “Divine Proportion”), a treatise that celebrated the ratio’s supposed harmony. The book was illustrated by polymath Leonardo da Vinci, whose works have often been analyzed for evidence of golden ratio proportions. German astronomer–mathematician Johannes Kepler, in Harmonices Mundi (1619; “Harmony of the Worlds”), referred to the ratio as proportio divina (“divine proportion”) and noted that the ratios of successive Fibonacci numbers—a sequence in which each number is the sum of the two before it—converge to the golden ratio.

In the 19th century German mathematician Martin Ohm introduced the term goldene Schnitt (“golden section”) in the second edition of his textbook Die Reine Elementar-Mathematik (1835; “The Pure Elementary Mathematics”). The first known English use of “golden section” was by English psychologist James Sully in his 1875 article on aesthetics for the ninth edition of the Encyclopaedia Britannica. At the beginning of the 20th century, American mathematician Mark Barr popularized the use of ϕ to represent the golden ratio, reportedly in reference to the Greek sculptor Phidias, who was thought by some art historians to have employed it in his works.

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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.

Stephan C. Carlson Anoushka Pant