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| {{Semireg dual polyhedra db|Semireg dual polyhedron stat table|dID}}
| | I'm Son and was born on 2 January 1978. My hobbies are Metal detecting and Kart racing.<br><br>Here is my webpage ... [http://blog.wetterstation-goettingen.de/2014/08/19/bookbyte-coupons-and-the-tips-that-will-help-you/ Bookbyte Promo Code] |
| In [[geometry]], the '''rhombic triacontahedron''' is a [[convex polyhedron]] with 30 [[rhombus|rhombic]] [[face (geometry)|faces]]. It has 60 [[edge (geometry)|edges]] and 32 [[vertex (geometry)|vertices]] of two types. It is a [[Catalan solid]], and the [[dual polyhedron]] of the [[icosidodecahedron]]. It is a [[zonohedron]].
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| {|style="float: right; border: 1px solid #BBB; margin: .5em 0 0 .2em;"
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| |- style="font-size: 88%;"
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| |valign="top" |[[File:GoldenRhombus.svg|240px]]<br/> A face of the rhombic triacontahedron. The lengths<br/> of the diagonals are in the [[golden ratio]].
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| |}
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| The ratio of the long diagonal to the short diagonal of each face is exactly equal to the [[golden ratio]], ''φ'', so that the [[Angle#Types of angles|acute angles]] on each face measure {{nowrap|2 tan<sup>−1</sup>(1/''φ'') {{=}} tan<sup>−1</sup>(2)}}, or approximately 63.43°. A rhombus so obtained is called a ''[[golden rhombus]]''.
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| Being the dual of an [[Archimedean solid]], the rhombic triacontahedron is ''[[face-transitive]]'', meaning the [[symmetry group]] of the solid acts [[transitive action|transitively]] on the set of faces. This means that for any two faces, A and B, there is a [[rotation]] or [[reflection (mathematics)|reflection]] of the solid that leaves it occupying the same region of space while moving face A to face B.
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| The rhombic triacontahedron is somewhat special in being one of the nine [[edge-transitive]] convex polyhedra, the others being the five [[Platonic solid]]s, the [[cuboctahedron]], the [[icosidodecahedron]], and the [[rhombic dodecahedron]].
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| The rhombic triacontahedron is also interesting in that it has all the vertices of eight [[tetrahedron]]s, four [[cube|hexahedrons]], or an [[icosahedron]] and a [[dodecahedron]].
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| ==Dimensions==
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| If the edge length of a rhombic triacontahedron is ''a'', surface area, volume, the [[radius]] of an [[inscribed sphere]] ([[tangent]] to each of the rhombic triacontahedron's faces) and midradius, which touches the middle of each edge are:<ref>[[Stephen Wolfram]], "[http://www.wolframalpha.com/input/?i=rhombic+triacontahedron]" from [[Wolfram Alpha]]. Retrieved January 7, 2013.</ref>
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| <math>S = a^2 \cdot 12\sqrt{5} \approx 26.8328 \cdot a^2</math>
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| <math>V = a^3 \cdot 4\sqrt{5+2\sqrt{5}} \approx 12.3107 \cdot a^3</math>
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| <math>r_i = a \cdot \frac{\varphi^2}{\sqrt{1 + \varphi^2}} = a \cdot \sqrt{1 + \frac{2}{\sqrt{5}}} \approx 1.37638 \cdot a</math>
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| <math>r_m = a \cdot \left(1+\frac{1}{\sqrt5{}}\right) \approx 1.44721 \cdot a</math>
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| where ''φ'' is the [[golden ratio]].
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| The plane of each face is perpendicular to the center of the rhombic triacontahedron, and is located at the same distance (short diagonals belong only to the edges of the inscribed regular dodecahedron, while long diagonals are included only in edges of the inscribed icosahedron).
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| Using one of the three orthogonal golden rectangles drawn into the inscribed [[icosahedron]] we can easily deduce the distance between the center of the solid and the center of its rhombic face.
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| ==Uses of rhombic triacontahedra==
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| Danish designer Holger Strøm used the rhombic triacontahedron as a basis for the design of his buildable lamp IQ-light. (IQ for "Interlocking Quadrilaterals")
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| [[File:IQ-light. Design by Holger Strøm.jpg|thumb|An example of the use of a rhombic triacontahedron in the design of a lamp. IQ stands for “Interlocking Quadrilaterals”.]]
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| Woodworker Jane Kostick builds boxes in the shape of a rhombic triacontahedron.<ref>[http://kosticks.com/triacontahedron-box.html triacontahedron box - KO Sticks LLC]</ref> The simple construction is based on the less than obvious relationship between the rhombic triacontahedron and the cube.
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| [[Roger von Oech]]'s "Ball of Whacks" comes in the shape of a rhombic triacontahedron.
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| In some [[roleplaying]] games, and for [[elementary school]] uses, the rhombic triacontahedron is used as the "[[Dice#Non-cubic|d30]]" thirty-sided die.
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| ==Related polyhedra==
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| {{Icosahedral truncations}}
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| This polyhedron is a part of a sequence of rhombic polyhedra and tilings with [''n'',3] [[Coxeter group]] symmetry. The cube can be seen as a rhombic hexahedron where the rhombi are also rectangles.
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| {{Quasiregular figure table}}
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| The rhombic triacontahedron forms the [[convex hull]] of one projection of a [[6-cube]] to 3 dimensions.
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| [[File:Rhombic tricontahedron cube tetrahedron.png|thumb|A rhombic triacontahedron with an inscribed tetrahedron (red) and cube (yellow).<br />[[:File:Rhombic tricontahedron cube tetrahedron.gif|(Click here for rotating model)]]]]
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| [[File:Rhombic tricontahedron icosahedron dodecahedron.png|thumb|A rhombic triacontahedron with an inscribed dodecahedron (blue) and icosahedron (purple).<br />[[:File:Rhombic tricontahedron icosahedron dodecahedron.gif|(Click here for rotating model)]]]]
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| {|class=wikitable width=480
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| |- valign=top
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| |[[File:6Cube-QuasiCrystal.jpg|240px]]<br>The 3D basis vectors [u,v,w] are:
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| :u = (1, ''φ'', 0, -1, ''φ'', 0)
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| :v = (''φ'', 0, 1, ''φ'', 0, -1)
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| :w = (0, 1, ''φ'', 0, -1, ''φ'')
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| |[[File:RhombicTricontrahedron.png|240px]]<br>Shown with inner edges hidden
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| |-
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| |colspan=2|There are 64 vertices and 192 unit length edges forming pentagonal symmetry along specific axis (as well as hexagonal symmetries on other axis).
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| |}
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| ===Stellations===
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| The rhombic triacontahedron has over 227 stellations.<ref>{{cite journal | last = Pawley | first = G. S. | title = The 227 triacontahedra | journal = Geometriae Dedicata | volume = 4 | issue = 2-4 | pages = 221-232 | publisher = Kluwer Academic Publishers | date = 1975 | issn = 1572-9168 | doi = 10.1007/BF00148756}}</ref><ref>{{cite journal | last = Messer | first = P. W. | title = Stellations of the Rhombic Triacontahedron and Beyond | journal = Structural Topology | volume = 21 | pages = 25-46 | date = 1995}}</ref>
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| ==See also==
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| *[[Truncated rhombic triacontahedron]]
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| *[[Rhombille tiling]]
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| *[[Golden rhombus]]
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| ==References==
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| <references />
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| {{More footnotes|date=December 2010}}
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| *{{The Geometrical Foundation of Natural Structure (book)}} (Section 3-9)
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| *{{Citation |last=Wenninger |first=Magnus |authorlink=Magnus Wenninger |title=Dual Models |publisher=[[Cambridge University Press]] |isbn=978-0-521-54325-5 |id={{MathSciNet|id=730208}} |year=1983}} (The thirteen semiregular convex polyhedra and their duals, Page 22, Rhombic triacontahedron)
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| *''The Symmetries of Things'' 2008, John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, ISBN 978-1-56881-220-5 [http://www.akpeters.com/product.asp?ProdCode=2205] (Chapter 21, Naming the Archimedean and Catalan polyhedra and tilings, page 285, Rhombic triacontahedron )
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| ==External links==
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| *{{Mathworld2 |urlname=RhombicTriacontahedron |title=Rhombic triacontahedron |urlname2=CatalanSolid |title2=Catalan solid}}
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| *[http://polyhedra.org/poly/show/39/rhombic_triacontahedron Rhombic Triacontrahedron] – Interactive Polyhedron Model
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| *[http://www.georgehart.com/virtual-polyhedra/vp.html Virtual Reality Polyhedra] – The Encyclopedia of Polyhedra
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| *[http://bulatov.org/polyhedra/rtc/ Stellations of Rhombic Triacontahedron]
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| *[http://www.vortexmaps.com/ EarthStar globe – Rhombic Triacontahedral map projection]
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| *[http://www.iqlight.com/ IQ-light]–Danish designer Holger Strøm's lamp
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| *[http://www.instructables.com/id/E7P30WYB6IEVYDYWFG/ Make your own]
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| *[http://www.kosticks.com/triacontahedron-box.html a wooden construction of a rhombic triacontahedron box] – by woodworker Jane Kostick
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| *''[http://demonstrations.wolfram.com/120RhombicTriacontahedra/ 120 Rhombic Triacontahedra]'', [http://demonstrations.wolfram.com/3012RhombicTriacontahedra/ 30+12 Rhombic Triacontahedra], and [http://demonstrations.wolfram.com/12RhombicTriacontahedra/ 12 Rhombic Triacontahedra] by Sándor Kabai, [[The Wolfram Demonstrations Project]]
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| *[http://www.pythagoras.nu/gallery/Escher98/selection/b0089.jpg A viper drawn on a rhombic triacontahedron].
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| {{Polyhedron navigator}}
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| {{Use dmy dates|date=December 2010}}
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| [[Category:Catalan solids]]
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| [[Category:Quasiregular polyhedra]]
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| [[Category:Zonohedra]]
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