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. The figure on the right illustrates the geometric relationship. Expressed algebraically, for quantities a and b with a > b > 0,

a + b a = a b = def φ , {\displaystyle {\frac {a+b}{a}}={\frac {a}{b}}\ {\stackrel {\text{def}}{=}}\ \varphi ,}

where the Greek letter phi ( φ {\displaystyle \varphi } or ϕ {\displaystyle \phi } ) represents the golden ratio. Its value is:

φ = 1 + 5 2 = 1.6180339887 … . {\displaystyle \varphi ={\frac {1+{\sqrt {5}}}{2}}=1.6180339887\ldots .} A001622

The golden ratio is also called the golden mean or golden section (Latin: sectio aurea).^[1][2][3] Other names include extreme and mean ratio,^[4] medial section, divine proportion, divine section (Latin: sectio divina), golden proportion, golden cut,^[5] and golden number.^[6][7][8]

Some twentieth-century artists and architects, including Le Corbusier and Dalí, have proportioned their works to approximate the golden ratio—especially in the form of the golden rectangle, in which the ratio of the longer side to the shorter is the golden ratio—believing this proportion to be aesthetically pleasing. The golden ratio appears in some patterns in nature, including the spiral arrangement of leaves and other plant parts.

Mathematicians since Euclid have studied the properties of the golden ratio, including its appearance in the dimensions of a regular pentagon and in a golden rectangle, which may be cut into a square and a smaller rectangle with the same aspect ratio. The golden ratio has also been used to analyze the proportions of natural objects as well as man-made systems such as financial markets, in some cases based on dubious fits to data.^[9]