The Golden Rectangle

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A golden rectangle is a rectangle whose side lengths are in the golden ratio, 1: (one-to-phi), that is, approximately 1:1.618.

The large rectangle BA is a golden rectangle; that is, the proportion b:a is 1:. For any such rectangle, and only for rectangles of that specific proportion, if we remove square B, what is left, A, is another golden rectangle; that is, with the same proportions as the original rectangle.

A distinctive feature of this shape is that when a square section is removed, the remainder is another golden rectangle, that is, with the same proportions as the first. Square removal can be repeated infinitely, which leads to an approximation of the golden spiral.

According to astrophysicist and math popularizer Mario Livio, since the publication of Luca Pacioli's Divina Proportione in 1509, when "with Pacioli's book, the Golden Ratio started to become available to artists in theoretical treatises that were not overly mathematical, that they could actually use," many artists and architects have proportioned their works to approximate the form of the golden rectangle, which has been considered aesthetically pleasing. The proportions of the golden rectangle have been observed in works predating Pacioli's publication.

The Golden Mean

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In mathematics and the arts, two quantities are in the golden ratio if the ratio between the sum of those quantities and the larger one is the same as the ratio between the larger one and the smaller. The golden ratio is approximately 1.6180339887 (from the quadratic formula).

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.

The golden ratio can be expressed as a mathematical constant, usually denoted by the Greek letter (phi). The figure of a golden section illustrates the geometric relationship that defines this constant. Expressed algebraically:

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

Other names frequently used for or closely related to the golden ratio are golden section (Latin: sectio aurea), golden mean, golden number, and the Greek letter phi (). Other terms encountered include extreme and mean ratio, medial section, divine proportion (Italian: proporzione divina), divine section (Latin: sectio divina), golden proportion, golden cut, and mean of Phidias.

Construction of a golden rectangle:

1. Construct a unit square.
2. Draw a line from the midpoint of one side to an opposite corner.
3. Use that line as the radius to draw an arc that defines the long dimension of the rectangle.

• important part of any composition, in any medium