The large rectangle BA is a golden rectangle; that is, the proportion b:a is 1:. If we remove square B, what is left, A, is another golden rectangle.

A golden rectangle is a rectangle whose side lengths are in the golden ratio, 1:, that is, approximately 1:1.618. Image File history File links This is a lossless scalable vector image. ... Image File history File links This is a lossless scalable vector image. ... In geometry, a rectangle is defined as a quadrilateral where all four of its angles are right angles. ... // Articles with similar titles include Golden mean (philosophy), the felicitous middle between two extremes, and Golden numbers, an indicator of years in astronomy and calendar studies. ...

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. For other uses, see Square. ... In mathematics, two quantities 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. ... A golden spiral is a logarithmic spiral whose growth factor b is related to phi, the golden ratio. ...

According to astrophysicist and math popularizer Mario Livio, since the publication of Luca Pacioli's Divina Proportione in 1509,^[1] 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,"^[2] 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.^[3] Mario Livio (born 1945) is an astrophysicist and an author of works that popularize science and mathematics. ... Painting of Luca Pacioli, attributed to Jacopo de Barbari, 1495 (attribution controversial[1]). Table is filled with geometrical tools: slate, chalk, compass, a dodecahedron model. ...

Constructing a golden rectangle

A method to construct a golden rectangle. The resulting dimensions are in the ratio 1:, the golden ratio.

A golden rectangle can be constructed with only straightedge and compass by this technique: Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ... Creating a regular hexagon with a ruler and compass Construction of a regular pentagon Compass and straightedge or ruler-and-compass construction is the construction of lengths or angles using only an idealized ruler and compass. ...

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 of the golden rectangle

Approximate golden spiral (green) made of quarter-circles in squares successively removed from a golden rectangle, compared to a true golden spiral (red)

• Some authors report that the dimensions of the façade of the Parthenon fits into a golden rectangle ^[4]
• Jan Tschichold describes the use of the golden rectangle in medieval book designs
• Le Corbusier's 1927 Villa Stein in Garches features a rectangular ground plan, elevation, and inner structure that are closely approximate to golden rectangles.^[5]
• The front view of the UN Headquarters building in New York City forms a golden rectangle.

UN Headquarters in New York City.

^{[citation needed]} Image File history File links This is a lossless scalable vector image. ... Image File history File links This is a lossless scalable vector image. ... The Parthenon seen from the hill of the Pnyx to the west. ... Titlepage for Typographische Gestaltung written and designed by Jan Tschichold using City Medium and Bodoni. ... Recto page from a rare Blackletter Bible (1497) In the field of book design, proportions of pages, type areas (print spaces), and margins of medieval books have been analyzed by scholars, and several canons of page construction have been described by them to represent the ways in which these books... Charles-Edouard Jeanneret, who chose to be known as Le Corbusier (October 6, 1887 – August 27, 1965), was a French, Swiss-born architect and writer, who is famous for his contributions to what now is called modernism, or the International Style. ... Year 1927 (MCMXXVII) was a common year starting on Saturday (link will display full calendar) of the Gregorian calendar. ... Villa Stein, designed by Le Corbusier, was built in 1927 at Garches, France. ... Garches is a city in suburban Paris in France Sites of interest The northern part of the suburban city wsa marked by the combat of January 19, 1871 when the Parisian besieged and tried to force the German blockade to join the French troops of Versailles. ... United Nations headquarters, view from East River United Nations headquarters in New York City The United Nations headquarters is a distinctive complex in New York City that has served as the United Nationss headquarters since its completion in 1952. ... Image File history File linksMetadata United_Nations_HQ_-_New_York_City. ... Image File history File linksMetadata United_Nations_HQ_-_New_York_City. ...

See also

• Golden ratio
• Leonardo of Pisa
• Fibonacci numbers

// Articles with similar titles include Golden mean (philosophy), the felicitous middle between two extremes, and Golden numbers, an indicator of years in astronomy and calendar studies. ... Drawing of Leonardo Pisano Leonardo of Pisa or Leonardo Pisano (Pisa, c. ... In mathematics, the Fibonacci numbers form a sequence defined recursively by: In words: you start with 0 and 1, and then produce the next Fibonacci number by adding the two previous Fibonacci numbers. ...

References

1. ^ Pacioli, Luca. De divina proportione, Luca Paganinem de Paganinus de Brescia (Antonio Capella) 1509, Venice.
2. ^ Livio, Mario (2002). The Golden Ratio: The Story of Phi, The World's Most Astonishing Number. New York: Broadway Books. ISBN 0-7679-0815-5. 
3. ^ Van Mersbergen, Audrey M., Rhetorical Prototypes in Architecture: Measuring the Acropolis with a Philosophical Polemic, Communication Quarterly, Vol. 46, 1998 ("a 'Golden Rectangle' has a ratio of the length of its sides equal to 1:1.61803+. The Parthenon is of these dimensions.")
4. ^ Hemenway, Priya – Divine Proportion pp101, Sterling Publishing, ISBN 1-4027-3522-7
5. ^ Le Corbusier, The Modulor, p. 35, as cited in Padovan, Richard, Proportion: Science, Philosophy, Architecture (1999), p. 320. Taylor & Francis. ISBN 0-419-22780-6: "Both the paintings and the architectural designs make use of the golden section".

External links

• Golden Ratio at MathWorld
• The Golden Mean and the Physics of Aesthetics
• Golden rectangle demonstration With interactive animation
• Golden Rectangle Construction Interactive website