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Golden Ratio

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Golden ratio
The golden section is a line segment sectioned into two according to the golden ratio. The total length a+b is to the longer segment a as a is to the shorter segment b.
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.Not to be confused with golden mean (philosophy), the felicitous middle between two extremes, nor with golden numbers, an indicator of years in astronomy and calendar studies.
The golden ratio, usually denoted , expresses the relationship that the sum of two quantities is to the larger quantity as the larger is to the smaller. The golden ratio is the following algebraic irrational number with its numerical approximation:

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.

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 (φ). 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.

Aesthetics
Beginning in the Renaissance, a body of literature on the aesthetics of the golden ratio has developed. As a result, architects, artists, book designers, and others have been encouraged to use the golden ratio in the dimensional relationships of their works.

The first and most influential of these was De Divina Proportione by Luca Pacioli, a three-volume work published in 1509. Pacioli, a Franciscan friar, was known mostly as a mathematician, but he was also trained and keenly interested in art. De Divina Proportione explored the mathematics of the golden ratio and advocated its application to yield pleasing, harmonious proportions. Pacioli also saw Catholic religious significance in the ratio, which led to his work's title. Copiously illustrated by Leonardo Da Vinci, Pacioli's longtime friend and collaborator, De Divina Proportione was a major influence on generations of artists and architects. Da Vinci himself maintained that the human body has proportions that approximate the golden ratio.

The ratio is sometimes used in modern human-made constructions, such as stairs and buildings, and woodwork.


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