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| {{about|the geometrical figure|the Indian highway project|Golden Quadrilateral}}
| | Jayson Berryhill is how I'm called and my wife doesn't like it at all. My day occupation is an information officer but I've already utilized for another 1. For many years he's been living in Mississippi and he doesn't strategy on altering it. What me and my family members love is bungee jumping but I've been taking on new things recently.<br><br>my web blog: [http://Wegmer.Co.kr/xe/rhz/361131 psychic phone readings] |
| [[Image:SimilarGoldenRectangles.svg|right|thumb|225px|A golden rectangle with longer side <span style="color:blue;">'''''a'''''</span> and shorter side <span style="color:red;">'''''b'''''</span>, when placed adjacent to a square with sides of length <span style="color:blue;">'''''a'''''</span>, will produce a [[Similarity (geometry)|similar]] golden rectangle with longer side <span style="color:green;">'''''a'' + ''b'''''</span> and shorter side <span style="color:blue;">'''''a'''''</span>. This illustrates the relationship <math> \frac{a+b}{a} = \frac{a}{b} \equiv \varphi\,.</math>]]
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| In [[geometry]], a '''golden rectangle''' is a [[rectangle]] whose side lengths are in the [[golden ratio]], <math>1 : \tfrac{1 + \sqrt{5}}{2}</math>, which is <math>1:\varphi</math> (the Greek letter [[Phi (letter)|phi]]), where <math>\varphi</math> is approximately 1.618.<!-- Note to editors: If you want to add more decimals, please consider that the longer side of a rectangle with a shorter side of 1 meter should be measured more accurately than to the nearest millimeter to make the next decimal meaningful! Note that 1.618... is an irrational number, and 1:φ is an exact number. Also I entered the number NOT inside math tabs so people can copy/paste it -->
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| ==Construction==
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| [[Image:Golden Rectangle Construction.svg|thumb|A method to construct a golden rectangle. The square is outlined in red. The resulting dimensions are in the golden ratio.]]
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| A golden rectangle can be [[Compass and straightedge constructions|constructed with only straightedge and compass]] by 4 simple steps:
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| * Construct a simple square
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| * Draw a line from the midpoint of one side of the square to an opposite corner
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| * Use that line as the radius to draw an arc that defines the height of the rectangle
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| * Complete the golden rectangle
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| ==Relation to regular polygons and polyhedra==
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| A distinctive feature of this shape is that when a [[square (geometry)|square]] section is removed, the remainder is another golden [[rectangle]]; that is, with the same [[aspect ratio]] 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]], the unique [[logarithmic spiral]] with this property.
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| [[File:Icosahedron-golden-rectangles.svg|thumb|Three golden rectangles in an icosahedron]]
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| An alternative construction of the golden rectangle uses three polygons [[Circumscribed circle|circumscribed]] by congruent circles: a regular [[decagon]], [[hexagon]], and [[pentagon]]. The respective lengths ''a'', ''b'', and ''c'' of the sides of these three polygons satisfy the equation ''a''<sup>2</sup> + ''b''<sup>2</sup> = ''c''<sup>2</sup>, so line segments with these lengths form a [[Special right triangles#Sides of regular polygons|right triangle]]. The ratio of the side length of the hexagon to the decagon is the golden ratio, so this triangle forms half of a golden rectangle.<ref>[http://aleph0.clarku.edu/~djoyce/java/elements/bookXIII/propXIII10.html Euclid, Book XIII, Proposition 10].</ref>
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| The [[convex hull]] of two opposite edges of a regular [[icosahedron]] forms a golden rectangle. The twelve vertices of the icosahedron can be decomposed in this way into three mutually-perpendicular golden rectangles, whose boundaries are linked in the pattern of the [[Borromean rings]].<ref>{{citation|title=The Heart of Mathematics: An Invitation to Effective Thinking|first1=Edward B.|last1=Burger|first2=Michael P.|last2=Starbird|publisher=Springer|year=2005|isbn=9781931914413|page=382|url=http://books.google.com/books?id=M-qK8anbZmwC&pg=PA382}}.</ref>
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| ==Applications==
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| According to astrophysicist and mathematics popularizer [[Mario Livio]], since the publication of [[Luca Pacioli]]'s ''Divina Proportione'' in 1509,<ref>Pacioli, Luca. ''De divina proportione'', Luca Paganinem de Paganinus de Brescia (Antonio Capella) 1509, Venice.</ref> 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,"<ref name="livio">{{cite book|last=Livio|first=Mario|year=2002|title=The Golden Ratio: The Story of Phi, The World's Most Astonishing Number|publisher=Broadway Books|location=New York|isbn=0-7679-0815-5}}</ref> many artists and architects have been fascinated by the presumption that the golden rectangle is considered aesthetically pleasing. The proportions of the golden rectangle have been observed in works predating Pacioli's publication.<ref>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.")</ref>
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| * [[Le Corbusier]]'s 1927 [[Villa Stein]] in [[Garches]] features a rectangular ground plan, elevation, and inner structure that closely approximate golden rectangles.<ref>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".</ref>
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| * The [[flag of Togo]] was designed to approximate a golden rectangle.<ref>{{cite web |url=http://www.fotw.us/flags/tg.html |title=Flag of Togo |accessdate=2007-06-09 |work=FOTW.us |publisher=Flags Of The World}}</ref>
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| ==See also==
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| * [[Fibonacci numbers]]
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| * [[Golden rhombus]]
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| * [[Kepler triangle]]
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| * [[Leonardo of Pisa]]
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| * [[Rabatment of the rectangle]]
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| * [[Silver ratio]]
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| ==References==
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| {{reflist|30em}}
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| ==External links==
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| {{Commons category|Golden rectangle}}
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| * [http://mathworld.wolfram.com/GoldenRatio.html Golden Ratio at MathWorld]
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| * [http://uk.arxiv.org/abs/physics/0411195 The Golden Mean and the Physics of Aesthetics]
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| * [http://www.mathopenref.com/rectanglegolden.html Golden rectangle demonstration] With interactive animation
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| [[Category:Elementary geometry]]
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| [[Category:Golden ratio]]
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| [[Category:Quadrilaterals]]
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| [[ja:黄金四角形]]
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| [[pt:Rectângulo de ouro]]
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| [[ru:Золотое сечение]]
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