View source for Rhombic triacontahedron

{{short description|Catalan solid with 30 faces}}
{{Semireg dual polyhedra db|Semireg dual polyhedron stat table|dID}}
[[File:Rhombic triacontahedron.stl|thumb|3D model of a rhombic triacontahedron]]

In [[Geometry|geometry]], the '''rhombic triacontahedron''', sometimes simply called the '''triacontahedron''' as it is the most common thirty-faced [[Polyhedron|polyhedron]], 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|dual polyhedron]] of the [[Icosidodecahedron|icosidodecahedron]]. It is a [[Zonohedron|zonohedron]].

{|style="float: right; border: 1px solid #BBB; margin: .5em 0 0 .2em;"
|- style="font-size: 88%;"
|valign="top" |[[File:GoldenRhombus.svg|240px]]<br/> A face of the rhombic triacontahedron. The lengths<br/> of the diagonals are in the [[Golden ratio|golden ratio]].
|}
[[File:Cube-R30.gif|thumb|This animation shows a transformation from a [[Cube|cube]] to a rhombic triacontahedron by dividing the square faces into 4 squares and splitting middle edges into new rhombic faces.]]
The ratio of the long diagonal to the short diagonal of each face is exactly equal to the [[Golden ratio|golden ratio]], {{mvar|φ}}, so that the [[Angle#Types of angles|acute angles]] on each face measure {{math|2 tan{{sup|−1}}({{sfrac|1|''φ''}}) {{=}} tan{{sup|−1}}(2),}} or approximately 63.43°. A rhombus so obtained is called a ''[[Golden rhombus|golden rhombus]]''.

Being the dual of an [[Archimedean solid]], the rhombic triacontahedron is ''face-transitive'', meaning the [[Symmetry group|symmetry group]] of the solid acts transitively on the set of faces. This means that for any two faces, {{mvar|A}} and {{mvar|B}}, there is a [[Physics:Rotation|rotation]] or [[Reflection (mathematics)|reflection]] of the solid that leaves it occupying the same region of space while moving face {{mvar|A}} to face {{mvar|B}}.

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|cuboctahedron]], the [[Icosidodecahedron|icosidodecahedron]], and the [[Rhombic dodecahedron|rhombic dodecahedron]].

The rhombic triacontahedron is also interesting in that its vertices include the arrangement of four Platonic solids. It contains ten [[Tetrahedron|tetrahedra]], five [[Cube|cube]]s, an [[Regular icosahedron|icosahedron]] and a [[Regular dodecahedron|dodecahedron]]. The centers of the faces contain five [[Octahedron|octahedra]].

It can be made from a [[Truncated octahedron|truncated octahedron]] by dividing the hexagonal faces into 3 rhombi:
[[File:Rhombic triacontahedron in truncated octahedron.png|thumb|A topological rhombic triacontahedron in [[Truncated octahedron|truncated octahedron]]]]

==Cartesian coordinates==

Let <math>\phi</math> be the [[Golden ratio|golden ratio]]. The 12 points given by <math>(0, \pm 1, \pm \phi)</math> and cyclic permutations of these coordinates are the vertices of a [[Regular icosahedron|regular icosahedron]]. Its dual [[Regular dodecahedron|regular dodecahedron]], whose edges intersect those of the icosahedron at right angles, has as vertices the 8 points <math>(\pm 1, \pm 1, \pm 1)</math> together with the 12 points <math>(0, \pm\phi, \pm 1/\phi)</math> and cyclic permutations of these coordinates. All 32 points together are the vertices of a rhombic triacontahedron centered at the origin. The length of its edges is <math>\sqrt{3-\phi}\approx 1.175\,570\,504\,58</math>. Its faces have diagonals with lengths <math>2</math> and <math>2/\phi</math>.

==Dimensions==
If the edge length of a rhombic triacontahedron is ''a'', surface area, volume, the [[Radius|radius]] of an [[Inscribed sphere|inscribed sphere]] ([[Tangent|tangent]] to each of the rhombic triacontahedron's faces) and midradius, which touches the middle of each edge are:<ref>[[Biography:Stephen Wolfram|Stephen Wolfram]], "[http://www.wolframalpha.com/input/?i=rhombic+triacontahedron]" from [[Physics:Wolfram Alpha|Wolfram Alpha]]. Retrieved 7 January 2013.</ref>

:<math>\begin{align}
S &= 12\sqrt{5}\,a^2 &&\approx 26.8328 a^2 \\
V &= 4\sqrt{5+2\sqrt{5}}\,a^3 &&\approx 12.3107 a^3 \\
r_\mathrm{i} &= \frac{\varphi^2}{\sqrt{1 + \varphi^2}}\,a = \sqrt{1 + \frac{2}{\sqrt{5}}}\,a &&\approx 1.37638 a \\
r_\mathrm{m} &= \left(1+\frac{1}{\sqrt{5}}\right)\,a &&\approx 1.44721 a
\end{align}</math>

where ''φ'' is the [[Golden ratio|golden ratio]].

The insphere is tangent to the faces at their face centroids. Short diagonals belong only to the edges of the inscribed regular dodecahedron, while long diagonals are included only in edges of the inscribed icosahedron.

== Dissection ==
The rhombic triacontahedron can be dissected into 20 golden rhombohedra: 10 acute ones and 10 obtuse ones.<ref>{{Cite web|url=http://www.cutoutfoldup.com/979-golden-rhombohedra.php|title = How to make golden rhombohedra out of paper}}</ref><ref>[http://www.georgehart.com/virtual-polyhedra/dissection-rt.html Dissection of the rhombic triacontahedron]</ref>
{| class=wikitable
!10 ||10
|- align=center
|[[File:Acute golden rhombohedron.png|105px]]<BR>Acute form
|[[File:Flat golden rhombohedron.png|120px]]<BR>Obtuse form
|}

==Orthogonal projections==
The rhombic triacontahedron has four symmetry positions, two centered on vertices, one mid-face, and one mid-edge. Embedded in projection "10" are the "fat" rhombus and "skinny" rhombus which tile together to produce the non-periodic tessellation often referred to as [[Penrose tiling]].
{|class=wikitable
|+ Orthogonal projections
|- align=center
!Projective<br>symmetry
|[2]
|[2]
|[6]
|[10]
|-
!Image
|[[File:Dual_dodecahedron_t1_v.png|120px]]
|[[File:Dual dodecahedron_t1_e.png|120px]]
|[[File:Dual dodecahedron_t1_A2.png|120px]]
|[[File:Dual dodecahedron_t1_H3.png|120px]]
|-
!Dual<BR>image
|[[File:Dodecahedron_t1_v.png|120px]]
|[[File:Dodecahedron_t1_e.png|120px]]
|[[File:Dodecahedron_t1_A2.png|120px]]
|[[File:Dodecahedron_t1_H3.png|120px]]
|}

==Stellations==
[[File:Rhombic_hexecontahedron.png|100px|thumb|Rhombic hexecontahedron]]
[[File:Construction of Rhombic hexecontahedron from Rhombic Triacontahedron.gif|thumb|An example of stellations of the rhombic triacontahedron.]]
{{See|Rhombic hexecontahedron}}
The rhombic triacontahedron has 227 fully supported 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| s2cid = 123506315 }}</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> 
Another stellation of the Rhombic triacontahedron is the [[Compound of five cubes|compound of five cubes]]. 
The total number of stellations of the rhombic triacontahedron is 358,833,097.

==Related polyhedra==

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.

<gallery widths="200px" heights="200px">
File:Spherical rhombic triacontahedron.png|Spherical rhombic triacontahedron
File:Rhombic tricontahedron cube tetrahedron.png|A rhombic triacontahedron with an inscribed tetrahedron (red) and cube (yellow).<br />[[:File:Rhombic tricontahedron cube tetrahedron.gif|(Click here for rotating model)]]
File:Rhombic tricontahedron icosahedron dodecahedron.png|A rhombic triacontahedron with an inscribed dodecahedron (blue) and icosahedron (purple).<br />[[:File:Rhombic tricontahedron icosahedron dodecahedron.gif|(Click here for rotating model)]]
File:StellaTruncRhombicTriaconta.png|Fully truncated rhombic triacontahedron
</gallery>

=== 6-cube ===
The rhombic triacontahedron forms a 32 vertex [[Convex hull|convex hull]] of one projection of a [[6-cube|6-cube]] to three dimensions.

{|class=wikitable width=480
<!--colspan=2|2D
!colspan=2|3D
|- 
!D<sub>6</sub> Coxeter plane
!H<sub>3</sub> Coxeter plane-->
|- valign=top
<!--|[[File:6-cube t0 B5.svg|240px]]<BR>[[6-cube|6-cube]]<BR>As 2 rings of 32 vertices
|[[File:Dual dodecahedron t1 H3.png|240px]]<BR>Rhombic triacontahedron<BR>32 vertices-->
|[[File:6Cube-QuasiCrystal.png|240px]]<br>The 3D basis vectors [u,v,w] are:
:u = (1, ''φ'', 0, -1, ''φ'', 0)
:v = (''φ'', 0, 1, ''φ'', 0, -1)
:w = (0, 1, ''φ'', 0, -1, ''φ'')
|[[File:RhombicTricontrahedron.png|240px]]<br>Shown with inner edges hidden<BR>20 of 32 interior vertices form a [[Regular dodecahedron|dodecahedron]], and the remaining 12 form an [[Regular icosahedron|icosahedron]].
|}

==Uses==
[[File:IQ-light. Design by Holger Strøm.jpg|thumb|An example of the use of a rhombic triacontahedron in the design of a lamp]]
Danish designer Holger Strøm used the rhombic triacontahedron as a basis for the design of his buildable lamp IQ-light (IQ for "Interlocking Quadrilaterals").

[[File:Rhombic_triacontahedron_box.stl|thumb|[[STL (file format)|STL model]] of a rhombic triacontahedral box made of six panels around a cubic hole &ndash; zoom into the model to see the hole from the inside]]
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.

Roger von Oech's "Ball of Whacks" comes in the shape of a rhombic triacontahedron.

The rhombic triacontahedron is used as the "d30" thirty-sided die, sometimes useful in some roleplaying games or other places.

==See also==
* [[Golden rhombus]]
* [[Rhombille tiling]]
* Truncated rhombic triacontahedron

==References==

<references />
* {{The Geometrical Foundation of Natural Structure (book)}} (Section 3-9)
* {{Citation |last=Wenninger |first=Magnus |title=Dual Models |publisher={{wipe|Cambridge University Press}} |isbn=978-0-521-54325-5 |mr=730208 |year=1983 |doi=10.1017/CBO9780511569371}} (The thirteen semiregular convex polyhedra and their duals, p.&nbsp;22, Rhombic triacontahedron)
* ''The Symmetries of Things'' 2008, John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, {{isbn|978-1-56881-220-5}} [https://web.archive.org/web/20100919143320/https://akpeters.com/product.asp?ProdCode=2205] (Chapter 21, Naming the Archimedean and Catalan polyhedra and tilings, p.&nbsp;285, Rhombic triacontahedron )

==External links==
* {{Mathworld2 |urlname=RhombicTriacontahedron |title=Rhombic triacontahedron |urlname2=CatalanSolid |title2=Catalan solid}}
* [https://web.archive.org/web/20070701185550/http://polyhedra.org/poly/show/39/rhombic_triacontahedron Rhombic Triacontrahedron] – Interactive Polyhedron Model
* [http://www.georgehart.com/virtual-polyhedra/vp.html Virtual Reality Polyhedra] – The Encyclopedia of Polyhedra
* [http://bulatov.org/polyhedra/rtc/ Stellations of Rhombic Triacontahedron]
* [http://www.vortexmaps.com/ EarthStar globe – Rhombic Triacontahedral map projection]
* [http://www.iqlight.com/ IQ-light]—Danish designer Holger Strøm's lamp
* [http://www.instructables.com/id/E7P30WYB6IEVYDYWFG/ Make your own] {{Webarchive|url=https://web.archive.org/web/20070717022841/http://www.instructables.com/id/E7P30WYB6IEVYDYWFG/ |date=17 July 2007 }}
* [http://www.kosticks.com/triacontahedron-box.html a wooden construction of a rhombic triacontahedron box] &ndash; by woodworker Jane Kostick
* ''[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
* [https://web.archive.org/web/20071219100912/http://www.pythagoras.nu/gallery/Escher98/selection/b0089.jpg A viper drawn on a rhombic triacontahedron].

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[[Category:Catalan solids]]
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{{Sourceattribution|Rhombic triacontahedron}}