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| A '''dynamic rectangle''' is a right-angled, four-sided figure (a [[rectangle]]) with [[dynamic symmetry]], which in this case, means that [[aspect ratio]] (height divided by width) is a distinguished value in '''dynamic symmetry''', a proportioning system and natural design methodology described in [[Jay Hambidge]]'s books. These dynamic rectangles begin with a [[Square (geometry)|square]], which is extended (using a series of arcs and cross points) to form the desired figure, which can be the [[golden rectangle]] (1 : 1.618...), the 2:3 rectangle, the double square (1:2), or a '''root rectangle''' (1:{{radic|φ}}, 1:{{radic|2}}, 1:{{radic|3}}, 1:{{radic|5}}, etc.).<ref>SKINNER, Stephen, ''Sacred Geometry Deciphering the Code'', New York City: Sterling Publishing Company, 2006, pp. 53</ref><ref name=hambidge>
| | If you present photography effectively, it helps you look much more properly at the globe around you. This means you can setup your mailing list and auto-responder on your wordpress site and then you can add your subscription form to any other blog, splash page, capture page or any other site you like. Should you go with simple HTML or use a platform like Wordpress. Keep reading for some great Word - Press ideas you can start using today. You can easily customize the titles of the posts in Word - Press blog in a way that only title comes in the new post link and not the date or category of posts. <br><br>Here is more information in regards to [http://bmehub.com.au/forum-topic/examining-effortless-worpress-methods wordpress dropbox backup] stop by our webpage. Any business enterprise that is certainly worth its name should really shell out a good deal in making sure that they have the most effective website that provides related info to its prospect. But as expected the level of support you get with them can be hit or miss based on the developer's free time and desire. With the free Word - Press blog, you have the liberty to come up with your own personalized domain name. By purchasing Word - Press weblogs you can acquire your very own domain title and have total command of your web site. Moreover, many Word - Press themes need to be purchased and designing your own WP site can be boring. <br><br>Minor and medium sized corporations also have a lot to obtain by shelling out in a very good website. Browse through the popular Wordpress theme clubs like the Elegant Themes, Studio Press, Woo - Themes, Rocket Theme, Simple Themes and many more. I've applied numerous Search engine optimization-ready Word - Press themes and I can say from knowledge that I consider the Genesis Search engine marketing panel one particular of the simplest to use. In crux the developer must have a detailed knowledge not only about the marketing tool but also about the ways in which it can be applied profitably. Premium vs Customised Word - Press Themes - Premium themes are a lot like customised themes but without the customised price and without the wait. <br><br>There has been a huge increase in the number of developers releasing free premium Word - Press themes over the years. I have compiled a few tips on how you can start a food blog and hopefully the following information and tips can help you to get started on your food blogging creative journey. Enterprise, when they plan to hire Word - Press developer resources still PHP, My - SQL and watch with great expertise in codebase. It's now become a great place to sell it thanks to Woo - Commerce. The Pakistani culture is in demand of a main surgical treatment. <br><br>Someone with a basic knowledge of setting up a website should be able to complete the process in a couple of minutes however even basic users should find they are able to complete the installation in around 20 minutes by following the step by step guide online. It can run as plugin and you can still get to that whole database just in circumstance your webhost does not have a c - Panel area. While deciding couple should consider the expertise of the doctor,clinics success rate,the costs of fertility treatment,including fertility tests and IVF costs and overall ones own financial budget. Word - Press is the most popular personal publishing platform which was launched in 2003. Verify whether your company has a team of developers or programmers having hands-on experience and knowledge about all Word - Press concepts. |
| {{cite book
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| |author=Jay Hambidge
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| |title=Dynamic Symmetry: The Greek Vase
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| |edition= Reprint of original Yale University Press edition
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| |publisher=Kessinger Publishing
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| |location=Whitefish, MT
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| |year=2003
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| |origyear=1920
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| |pages=19–29
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| |isbn=0-7661-7679-7
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| |url=http://books.google.com/books?id=Qq4gAAAAMAAJ&pg=PA19&dq=intitle:%22Dynamic+Symmetry%22+%22root+rectangles%22&ei=WTX2S5ffGYiIkgSNxoTJCQ&cd=1#v=onepage&q=intitle%3A%22Dynamic%20Symmetry%22%20%22root%20rectangles%22&f=false
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| }}</ref><ref name=ghyka>
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| {{cite book
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| |author = Matila Ghyka
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| |year = 1977
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| |title = The Geometry of Art and Life
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| |publisher=Courier Dover Publications
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| |pages =126–127
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| }}</ref>
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| ==Root rectangles==
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| [[File:Root rectangles Hambidge 1920.png|right|thumb|350px|Hambidge's 1920 illustration of the construction of root rectangles. The lengths of the horizontal sides of the original square and the four root rectangles derived from it, are respectively <math>\scriptstyle \sqrt{1}, \sqrt{2}, \sqrt{3}, \sqrt{4}, \sqrt{5} </math>.<ref name=hambidge/>]]
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| A root rectangle is a [[rectangle]] in which the ratio of the longer side to the shorter is the [[square root]] of an [[integer]], such as {{radic|2}}, {{radic|3}}, etc.<ref name=hambidge>
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| {{cite book
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| |author=Jay Hambidge
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| |authorlink=Jay Hambidge
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| |title=Dynamic Symmetry: The Greek Vase
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| |publisher=Yale University Press
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| |year=1920
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| |pages=19–29
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| |url=http://books.google.com/books?id=Qq4gAAAAMAAJ&pg=PA40&dq=dynamic-rectangles+inauthor:hambidge&lr=&as_brr=0&ei=3MFBSJeKLI7iiwHRtY2JBQ#PPA44,M1
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| }} (or 2003 reprint from Kessinger Publishing, Whitefish, MT, ISBN 0-7661-7679-7)</ref>
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| The root-2 rectangle (ACDK in Fig. 10) is constructed by extending two opposite sides of a [[Square (geometry)|square]] to the length of the square's diagonal. The root-3 rectangle is constructed by extending the two longer sides of a root-2 rectangle to the length of the root-2 rectangle's diagonal. Each successive root rectangle is produced by extending a root rectangle's longer sides to equal the length of that rectangle's diagonal.<ref name=jay2>Jay Hambidge. (1926, 1948, 1967)''[http://books.google.com/books?id=VYJK2F-dh2oC&pg=PA10&dq=intitle:dynamic+intitle:symmetry+inauthor:hambidge+root+dynamic-rectangles&lr=&as_brr=0&ei=b0BCSJHNBZW-jgG6gsGIBQ&sig=fE_8Kx_rO0GURo3g6JLizq1MUsI#PPA9,M1 The Elements of Dynamic Symmetry]''. Courier Dover Publications. pp. 9–10.</ref>
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| ===Properties===
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| *When a root-''N'' rectangle is divided into ''N'' congruent rectangles by dividing the longer edge into ''N'' segments, the resulting figures keep the root-''N'' proportion (as illustrated above).<ref>
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| {{cite book
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| | title = Book Design
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| | author = Andrew Haslam
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| | publisher = Laurence King Publishing
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| | year = 2006
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| | isbn = 1-85669-473-9
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| | pages = 48–49
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| | url = http://books.google.com/books?id=_Ri63jEKPfgC&pg=PT27&dq=root-rectangle&as_brr=3&ei=N2wySKOPL4uKtAOk2NnvAg&sig=98V_Ww1L3EgAKmPofmkscnvxMsk }}</ref>
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| *The root-3 rectangle is also called ''sixton'',<ref>Wim Muller (2001) ''Order and Meaning in Design''. Lemma Publishers, p. 49.</ref> and its short and longer sides are proportionally equivalent to the side and diameter of a [[hexagon]].<ref name=elam>
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| {{cite book
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| | author = Kimberly Elam
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| | year = 2001
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| | title = Geometry of Design: Studies in Proportion and Composition
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| | publisher = Princeton Architectural Press
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| | isbn = 1-56898-249-6
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| | pages = 34–41
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| | url = http://books.google.com/books?id=1KI0JVuWYGkC&pg=PA41&dq=root-5+intitle:geometry+inauthor:elam&lr=&as_brr=0&as_pt=ALLTYPES&ei=KC-iSe--J4HqkwSx6OCNAg#PPA33,M1
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| }}</ref>
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| *Since 2 is the square root of 4, the root-4 rectangle has a proportion 1:2, which means that it is equivalent to two squares side-by-side.<ref name=elam/>
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| *The root-5 rectangle is related to the [[golden ratio]] (φ). The longer side is equal to one plus two times 1/φ (0.618...).<ref name=elam/>
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| ==Root-φ rectangle==
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| [[File:Root phi rectangle.svg|right|thumb|A root-phi rectangle divides into a pair of [[Kepler triangle]]s (right triangles with edge lengths in geometric progression).]]
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| The root-φ rectangle is a dynamic rectangle but not a root rectangle. Its diagonal equals φ times the length of the shorter side. If a root-φ rectangle is divided by a diagonal, the result is two [[Congruence (geometry)|congruent]] [[Kepler triangle]]s.
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| ==Jay Hambidge==
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| [[Jay Hambidge]], as part of his theory of dynamic symmetry, includes the root rectangles in what he calls ''dynamic rectangles'', which have [[irrational number|irrational]] and geometric [[fraction (mathematics)|fractions]] as ratios, such as the [[golden ratio]] or square roots. Hambidge distinguishes these from rectangles with rational proportions, which he terms ''static rectangles''.<ref name=ghyka>
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| {{cite book
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| |author = Matila Ghyka
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| |year = 1977
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| |title = The Geometry of Art and Life
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| |publisher=Courier Dover Publications
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| |pages =126–127
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| |url=http://books.google.com/books?id=Qq4gAAAAMAAJ&pg=PA40&dq=dynamic-rectangles+inauthor:hambidge&lr=&as_brr=0&ei=3MFBSJeKLI7iiwHRtY2JBQ
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| }}</ref> According to him, root-2, 3, 4 and 5 rectangles are often found in Gothic and Classical Greek and Roman art, objects and architecture, while rectangles with aspect ratios greater than root-5 are seldom found in human designs.<ref name=jay2/>
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| According to [[Matila Ghyka]], Hambidge's dynamic rectangles
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| {{quote|can produce the most varied and satisfactory harmonic (consonant, related by symmetry) subdivisions and combinations, and this by the very simple process [...] of drawing inside the chosen rectangle a diagonal and the perpendicular to it from one of the two remaining vertices (thus dividing the surface into a reciprocal rectangle and its gnomon) and the drawing any network of parallels and perpendiculars to sides and diagonals. This produces automatically surfaces correlated by the characteristic proportion of the initial rectangle and also avoids (automatically again) the mixing of antagonistic themes like √2 and √3 or √5. √5 and Φ on the contrary are not antagonistic but consonant, also with √Φ, Φ<sup>2</sup>, et cetera.<ref name=ghyka/>}}
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| [[File:Root rectangles Caskey 1922.png|center|frame|Caskey's 1922 illustration of the property that a root-N rectangle divides into N reciprocal rectangles of the same proportions.<ref>{{cite book | title = Geometry of Greek Vases: Attic Vases in the Museum of Fine Arts Analysed According to the Principals of Proportion Discovered by Jay Hambidge | date=1922 | author=Lacey Davis Caskey | url = http://books.google.com/books?id=6qNWAAAAMAAJ&pg=PA4&dq=inauthor:Lacey+inauthor:Davis+inauthor:Caskey+reciprocal&lr=&as_brr=0&ei=JOdFSNTEBpK2sgPmxqGxDQ | publisher = Museum of Fine Arts, Boston}}</ref>]]
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| == The 12 orthogons of Wersin ==
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| According to [[Wolfgang von Wersin]]'s ''The Book of Rectangles, Spatial Law and Gestures of The Orthogons Described'' (1956), a set of 12 special ''orthogons'' (from the Gr. ''ορθος'', ''orthos'', "straight"<ref>''"Ortho-"'', ''Oxford dictionary of current English'', Oxford: Oxford University Press, 1998, pp. 627, 1071 p.</ref> and ''γονια'', ''gonia'', "angle"; "a right angled figure", which, as a consequence, is [[rectangle|rectangular]] and [[tetragon]]al<ref>CURTIS, Thomas, ''[http://books.google.com.mx/books?id=J3JMAAAAMAAJ&pg=PA356&dq=orthogon+geometry&as_brr=3&ei=fVVxSbfLKYbuMvTy1d8N&hl=en The London Encyclopaedia]'', 1829, pp. 356</ref>)
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| has been used historically by artists, architects and calligraphers to guide the placement and interaction of elements in a design.<ref name=ghyka/><ref>WERSIN, Wolfgang Von, ''Das Buch vom Rechteck Gesetz und Gestik des Raumlichen die Othogone-scheibe. Die Orthogone-scheibe'' (''The Book of Rectangles, Spatial Law and Gestures of The Orthogons Described. The Orthogons Described''), Ravensburg: Otto Maier Verlag Publishers, 1956</ref> These orthogons are:<ref>WERSIN, pp. 83</ref>
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| * Square (1:1 or 1:{{radic|1}})
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| * Diagon (1:{{radic|2}})
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| * Hecton or sixton (1:{{radic|3}})
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| * Doppelquadrat (1:2 or 1:{{radic|4}})
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| * Hemiolion (2:3)
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| * Auron (the [[golden rectangle]], 1:[[golden ratio|φ]])
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| * Hemidiagon (1:½{{radic|5}})
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| * Penton (1:{{radic|φ}})
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| * Trion (1:⅔{{radic|3}})
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| * Quadriagon
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| * Biauron (1:2φ)
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| * Bipenton
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| Wolfgang Von Wersin's book includes an extraordinary copy of text from the year 1558 ([[Renaissance]]), with diagrams of seven of the 12 orthogons and an invitation from the passage to pay careful attention as the "ancient" architects believed "nothing excels these proportions" as "a thing of the purest abstraction."<ref>WERSIN, op. cit., pp. 36</ref>
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| All 12 orthogons, when formed together, create an entire unit: a square that is developed into a double square.<ref>WERSIN, pp. 80</ref>
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| Perhaps the most popular among the ortogons is the ''auron'' or [[golden rectangle]], which is produced by projecting the diagonal that goes from the middle point of a side of a square to one of the opposite vertexes, until it is aligned with the middle point.
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| Four of these orthogons are harmonic rectangles: the ''diagon'' or ''[[square root of 2|root-2]] rectangle'' is produced by projecting the diagonal of a square; the ''sixton'', ''hecton'' or ''[[square root of 3|root-3]] rectangle'' is produced by projecting the diagonal of a diagon; the double square or ''[[square root of 4|root-4]] rectangle'' is produced by projecting the diagonal of an hecton; the ''[[square root of 5|root-5]] rectangle'' is produced by projecting the diagonal of a double square (or by projecting 180° both diagonals that go from the middle point of a side of a square to the opposite vertexes).
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| Two of the most complicated of these figures are; the ''penton'', with proportions 1:{{radic|φ}} is related to the section of the [[golden ratio#Mathematical pyramids and triangles|golden pyramid]], the ''bipenton'''s longer side is equal to the shorter multiplied by two thirds of the square root of three, longer side of the ''biauron'' is {{radic|5}} - 1 or 2τ times the shorter.
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| The ''quadriagon'' is related to the diagon in the sense that its longer side is produced by projecting the diagonal of a quarter of a square. The ''trion'' has the height of an equilateral triangle and the width of the side. The ''hemidiagon'' (1:½{{radic|5}}) longer side is half the one of the root-5 rectangle and is produced by projecting the diagonal of half a square until it is perpendicular with the origin.
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| Besides the square and the double square, the only other static rectangle included in the list is the ''hemiolion'', which is produced by projecting 90° or 180° half the side of a square.
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| === Constructing an orthogon ===
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| The dimensions of orthogons relate to each other and to the Orthogon as a whole. For this reason, use of Orthogons as a template or under-structure is of interest to artists, architects and designers.<ref>http://www.constructingtheuniverse.com/Volume4.html</ref>
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| Orthogons always begin with a square, any square. Once an individual Orthogon is constructed, additional related measurements are determined (small, medium, large). These measurements can then be used to guide the design (painting, architecture, pottery, furniture, calligraphy, auto, etc.).
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| Diagrams for all twelve orthogons are available.<ref>{{cite web | title = Constructie v/d harmonische Rechthoeken | url = http://alpha.uhasselt.be/~lucp1508/Harmonic/harmonic_rectangles.jpg}}</ref>
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| Wersin's book has very detailed explanations for creating individual Orthogons.<ref>WERSIN, pp. 82-85</ref> The measurements derived are then applied in a design. The artwork of [[Giorgio Morandi]] exemplifies how measurements of varying sizes (derived from an Orthogon) can create visual harmony.
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| === Orthogons and design ===
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| Use of dimensions related to an orthogon as an under-structure system (or template for a design) ensures that the various parts will relate to the design as a whole. Marcus [[Vitruvius]] Pollio in Book Three of "[[De Architectura]]" (known currently as "The Ten Books of Architecture") explains:
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| <blockquote>"Therefore, since nature has designed the human body so that its members are duly proportioned to the frame as a whole, it appears that the ancients had good reason for their rule, that in perfect buildings the different members must be in exact symmetrical relations to the whole general scheme. Hence, while transmitting to us the proper arrangements for buildings of all kinds, they were particularly careful to do so in the case of temples of the gods, buildings in which merits and faults usually last forever."
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| </blockquote>
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| Leonardo's drawing of the [[Vitruvian Man]] is an illustration of the concept of parts relating to the work as a whole.<ref>HEMENWAY, pp. 95</ref>
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| ==References==
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| {{reflist}}
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| * Hemenway, Priya; Divine Proportion, Phi in Art, Nature and Science; 2005, Sterling Publishing Co., Inc, NY, NY.
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| [[Category:Quadrilaterals]]
| |
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