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A golden rectangle is a rectangle
Rectangle

In geometry, a rectangle is a Closed set planar quadrilateral with four right angles. A rectangle with vertices ABCD would be denoted as .A rectangle with adjacent sides of lengths a and b has area ab and diagonals of equal length ....
 whose side lengths are in the golden ratio
Golden ratio

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....
, 1: (one-to-phi
Phi (letter)

Phi , pronounced [] in Modern Greek language and as [] in English, is the 21st letter of the Greek alphabet. In modern Greek, it represents [], a voiceless labiodental fricative....
), that is, or approximately 1:1.618.

A distinctive feature of this shape is that when a square
Square (geometry)

In Euclidean geometry, a square is a regular polygon with four equal sides and four equal angles . A square with vertices ABCD would be denoted ....
 section is removed, the remainder is another golden rectangle; that is, with the same proportion
Proportionality (mathematics)

In mathematics, two quantity are called proportional if they vary in such a way that one of the quantities is a constant multiple of the other, or equivalently if they have a constant ratio....
s as the first. Square removal can be repeated infinitely, in which case corresponding corners of the squares form an infinite sequence of points on the golden spiral
Golden spiral

In geometry, a golden spiral is a logarithmic spiral whose growth factor b is related to φ, the golden ratio. Specifically, a golden spiral gets wider by a factor of φ for every quarter turn it makes....
, the unique logarithmic spiral
Logarithmic spiral

A logarithmic spiral, equiangular spiral or growth spiral is a special kind of spiral curve which often appears in nature. The logarithmic spiral was first described by Ren? Descartes and later extensively investigated by Jakob Bernoulli, who called it Spira mirabilis, "the marvelous spiral"....
 with this property.

According to astrophysicist and math popularizer Mario Livio
Mario Livio

Mario Livio is an astrophysics and an author of works that popularize science and mathematics. He is currently Senior Astrophysicist at the Space Telescope Science Institute....
, since the publication of Luca Pacioli
Luca Pacioli

Fra Luca Bartolomeo de Pacioli was an Italy mathematician and Franciscan friar, collaborator with Leonardo da Vinci, and seminal contributor to the field now known as accounting....
'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.






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Encyclopedia


A golden rectangle is a rectangle
Rectangle

In geometry, a rectangle is a Closed set planar quadrilateral with four right angles. A rectangle with vertices ABCD would be denoted as .A rectangle with adjacent sides of lengths a and b has area ab and diagonals of equal length ....
 whose side lengths are in the golden ratio
Golden ratio

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....
, 1: (one-to-phi
Phi (letter)

Phi , pronounced [] in Modern Greek language and as [] in English, is the 21st letter of the Greek alphabet. In modern Greek, it represents [], a voiceless labiodental fricative....
), that is, or approximately 1:1.618.

A distinctive feature of this shape is that when a square
Square (geometry)

In Euclidean geometry, a square is a regular polygon with four equal sides and four equal angles . A square with vertices ABCD would be denoted ....
 section is removed, the remainder is another golden rectangle; that is, with the same proportion
Proportionality (mathematics)

In mathematics, two quantity are called proportional if they vary in such a way that one of the quantities is a constant multiple of the other, or equivalently if they have a constant ratio....
s as the first. Square removal can be repeated infinitely, in which case corresponding corners of the squares form an infinite sequence of points on the golden spiral
Golden spiral

In geometry, a golden spiral is a logarithmic spiral whose growth factor b is related to φ, the golden ratio. Specifically, a golden spiral gets wider by a factor of φ for every quarter turn it makes....
, the unique logarithmic spiral
Logarithmic spiral

A logarithmic spiral, equiangular spiral or growth spiral is a special kind of spiral curve which often appears in nature. The logarithmic spiral was first described by Ren? Descartes and later extensively investigated by Jakob Bernoulli, who called it Spira mirabilis, "the marvelous spiral"....
 with this property.

According to astrophysicist and math popularizer Mario Livio
Mario Livio

Mario Livio is an astrophysics and an author of works that popularize science and mathematics. He is currently Senior Astrophysicist at the Space Telescope Science Institute....
, since the publication of Luca Pacioli
Luca Pacioli

Fra Luca Bartolomeo de Pacioli was an Italy mathematician and Franciscan friar, collaborator with Leonardo da Vinci, and seminal contributor to the field now known as accounting....
'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.

Construction

A golden rectangle can be constructed with only straightedge and compass by this technique:

  1. Construct a simple square
  2. Draw a line from the midpoint of one side of the square to an opposite corner
  3. Use that line as the radius to draw an arc that defines the height of the rectangle
  4. Complete the golden rectangle


Applications

  • Le Corbusier
    Le Corbusier

    Charles-?douard Jeanneret-Gris, who chose to be known as Le Corbusier , was a Swiss-French architect, designer, urbanist, writer and also Painting, who is famous for being one of the pioneers of what now is called Modern architecture or the International Style....
    's 1927 Villa Stein
    Villa Stein

    Villa Stein, designed by Le Corbusier, was built in 1927 at Garches, France. The building is also known as Villa Garches, Villa de Monzie, and Villa Stein-de Monzie....
     in Garches
    Garches

    Garches is a commune in France in the western suburbs of Paris, France. It is located . from the Kilometre Zero....
     features a rectangular ground plan, elevation, and inner structure that are closely approximate to golden rectangles.
  • Jan Tschichold
    Jan Tschichold

    Jan Tschichold was a typography, book designer, teacher and writer....
     describes the use of the golden rectangle in medieval book design
    Canons of page construction

    The canons of page construction are a set of principles in the field of book design used to describe the ways that page proportions, margins and type areas of books are constructed....
    s


See also

  • Fibonacci numbers
  • Kepler triangle
    Kepler triangle

    A Kepler triangle is a Special right triangle with edge lengths in geometric progression. The ratio of the edges of a Kepler triangle are linked to the golden ratio...
  • Leonardo of Pisa


External links


  • With interactive animation