Golden ratio

In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities $$a$$ and $$b$$ with $$a > b > 0$$, $$a$$ is in a golden ratio to $$b$$ if $$ \frac{a+b}{a} = \frac{a}{b} = \varphi$$where the Greek letter phi ($\varphi$ or $$\phi$$) denotes the golden ratio. If the constraint on $$a$$ and $$b$$ each being greater than zero is lifted, then there are actually two solutions, one positive and one negative, to this equation. $$\varphi$$ is defined as the positive solution. The negative solution is $$-\varphi^{-1} = \tfrac12\bigl(1 - \sqrt5\bigr).$$ The sum of the two solutions is $$1$$, and the product of the two solutions is $$-1$$. The constant $$\varphi$$ satisfies the quadratic equation $$\varphi^2 = \varphi + 1$$ and is an irrational number with a value of $$\varphi = \frac{1+\sqrt5}{2} = $$1.618033988749....

The golden ratio was called the extreme and mean ratio by Euclid, and the divine proportion by Luca Pacioli, and also goes by several other names. Other names include the golden mean, golden section, golden proportion, golden number, medial section, and divine section.

Mathematicians have studied the golden ratio's properties since antiquity. It is the ratio of a regular pentagon's diagonal to its side and thus appears in the construction of the dodecahedron and icosahedron. A golden rectangle—that is, a rectangle with an aspect ratio of $$\varphi$$—may be cut into a square and a smaller rectangle with the same aspect ratio. The golden ratio has been used to analyze the proportions of natural objects and artificial systems such as financial markets, in some cases based on dubious fits to data. The golden ratio appears in some patterns in nature, including the spiral arrangement of leaves and other parts of vegetation.

Some 20th-century artists and architects, including Le Corbusier and Salvador Dalí, have proportioned their works to approximate the golden ratio, believing it to be aesthetically pleasing. These uses often appear in the form of a golden rectangle.

Quotes

 * In Euclid's Elements we meet the concept which later plays a significant role in the development of science. The concept is called the "division of a line in extreme and mean ratio" (DEMR). ...the concept occurs in two forms. The first is formulated in Proposition 11 of Book II. ...why did Euclid introduce different forms... which we can find in Books II, VI and XIII? ...Only three types of regular polygons can be faces of the s: the equilateral triangle... the square... and the regular pentagon. In order to construct the Platonic solids... we must build the two-dimensional faces... It is for this purpose that Euclid introduced the ... (Proposition II.11)... By using the "golden" isosceles triangle...we can  construct the regular pentagon... Then only one step remains to construct the ... which for Plato is one of the most important regular polyhedra symbolizing the universal harmony in his cosmology.
 * , Samuil Aranson, The “Golden” Non-Euclidean Geometry: Hilbert's Fourth Problem, “Golden” Dynamical Systems, and the Fine-Structure Constant (2016)


 * In Western Culture, starting from Phidias and the Parthenon, the Golden Section and the Golden Number are present, consciously or unconsciously, in very famous works. In the Renaissance, after the rediscovery of Fibonacci, it was a symbol of aesthetic perfection to be used in architecture and art with, among others, Leonardo da Vinci (1542-1519) and Albrecht Dürer (1471-1528). The Golden Number is in many geometric figures making them Golden. We have it among other things in the octagonal architecture of Castel del Monte. The Golden Ratio enters the pentagon which is Golden because the side of the star and the side of the pentagon are in the ratio of 38% and 62%, as required by the Golden Number.
 * As quoted in Antonino Zichichi, “Art and mathematics at the Bioparco, The Golden Number in animals”, edited by Antonino Zichichi and Victoria Noel-Johnson (April 22, 2024)