golden ratio

golden ratio, in mathematics, the irrational number $\frac{1 + \surd 5}{2}$, 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 $\frac{\left(x + 1\right)}{x} = \frac{x}{1}$. This can be rearranged into the quadratic equation $x^2- x- 1 = \mathrm{<xref\; autoxref="y"\; topicid="656641"\; url="/science/zero-mathematics">0</xref>}$, for which the positive solution is $x = \frac{1 + \surd 5}{2}$, 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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Applications and interpretations

Mathematics

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: $\varphi =1 + \frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{1+....}}}}}$.

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.

Financial Markets

Some traders use ratios derived from the Fibonacci sequence—including 61.8 percent, the reciprocal of the golden ratio—as tools in financial technical analysis. Known as Fibonacci retracements and extensions, these methods mark percentage levels on price charts to suggest where prices might slow down, change direction, or continue moving.

Natural patterns and design

The golden ratio shows up in some natural growth patterns. Spirals in sunflower heads and pine cones 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.

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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.

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Fibonacci Trading Strategies

Fibonacci Sequence

Elements

Leonardo da Vinci