557: June 2010

Saturday, June 12, 2010

golden ratio

In mathematics and the arts, two quantities are in the golden ratio if the ratio of the sum of the quantities to the larger quantity is equal to (=) the ratio of the larger quantity to the smaller one. The golden ratio is an irrational mathematical constant, approximately 1.6180339887.^[1] Other names frequently used for the golden ratio are the golden section (Latin: sectio aurea) and golden mean.^[2]^[3]^[4] Other terms encountered include extreme and mean ratio,^[5] medial section, divine proportion, divine section (Latin: sectio divina), golden proportion, golden cut,^[6] golden number, and mean of Phidias.^[7]^[8]^[9] The golden ratio is often denoted by the Greek letter phi, usually lower case (φ).

The figure on the right illustrates the geometric relationship that defines this constant. Expressed algebraically:

$\frac{a+b}{a} = \frac{a}{b} = \varphi\,.$

This equation has as its unique positive solution the algebraic irrational number

$\varphi = \frac{1+\sqrt{5}}{2}\approx 1.61803\,39887\ldots\,$ ^[1]

At least since the Renaissance, many artists and architects 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. Mathematicians have studied the golden ratio because of its unique and interesting properties.