- Rhombic triacontahedron
In

geometry , the**rhombic triacontahedron**is a convexpolyhedron with 30 rhombic faces. It is an Archimedean dual solid, or aCatalan solid . It is the polyhedral dual of theicosidodecahedron , and it is azonohedron .The ratio of the long diagonal to the short diagonal of each face is exactly equal to the

golden ratio , φ, so that the acute angles on each face measure 2 tan^{−1}(1/φ) = tan^{−1}(2), or approximately 63.43°.A rhombus so obtained is called a "golden rhombus ".Being the dual of an Archimedean polyhedron, the rhombic triacontahedron is "

face-transitive ", meaning thesymmetry group of the solid actstransitive ly on the set of faces. In elementary terms, this means that for any two faces A and B there is arotation or reflection of the solid that leaves it occupying the same region of space while moving face A to face B. The rhombic triacontahedron is also somewhat special in being one of the nineedge-transitive convex polyhedra, the others being the fivePlatonic solid s, thecuboctahedron , theicosidodecahedron , and therhombic dodecahedron .**Uses of rhombic triacontahedra**Danish designer Holger Strøm used the rhombic triacontahedron as a basis for the design of his buildable lamp IQ-light. (IQ for "Interlocking Quadrilaterals")

Woodworker Jane Kostick builds boxes in the shape of a rhombic triacontahedron. The simple construction is based on the less than obvious relationship between the rhombic triacontahedron and the cube.

In some

roleplaying games, and forelementary school uses, the rhombic triacontahedron is used as the "d30" thirty-sided die.**See also***

Truncated rhombic triacontahedron

*Quasiregular rhombic tiling **References*** (Section 3-9)

**External links***

* [*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.iqlight.com/ IQ-light*] –Danish designer Holger Strøm's lamp

* [*http://www.instructables.com/id/E7P30WYB6IEVYDYWFG/ Make your own*]

* [*http://web.mac.com/jjkostick/iWeb/Jane%20Kostick/triacontahedron%20box.html a wooden construction of a rhombic triacontahedron box*] – 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 .

*Wikimedia Foundation.
2010.*

### Look at other dictionaries:

**Medial rhombic triacontahedron**— Type Star polyhedron Elements F = 30, E = 60 V = 24 (χ = −6) Symmetry group Ih, [5,3], *532 … Wikipedia**Truncated rhombic triacontahedron**— The truncated rhombic triacontahedron is a convex polyhedron constructed from the rhombic triacontahedron by truncating the twelve vertices where five faces meet at their acute corners.The 12 vertices can be truncated such that all edges are… … Wikipedia**Rhombic dodecahedron**— The rhombic dodecahedron is a convex polyhedron with 12 rhombic faces. It is an Archimedean dual solid, or a Catalan solid. Its dual is the cuboctahedron. Properties It is the polyhedral dual of the cuboctahedron, and a zonohedron. The long… … Wikipedia**Rhombic enneacontahedron**— A rhombic enneacontahedron (plural: rhombic enneacontahedra) is a polyhedron composed of 90 rhombus shaped faces; with three, five, or six rhombi meeting at each vertex. It has 60 broad rhombi and 30 slim. The rhombic enneacontahedron is a… … Wikipedia**Disdyakis triacontahedron**— Click on picture for large version spinning version Type Catalan Face polygon scalene triangle Faces 120 Edges … Wikipedia**Truncated rhombic dodecahedron**— The truncated rhombic dodecahedron is a convex polyhedron constructed from the rhombic dodecahedron by truncating the 6 (order 4) vertices. The 6 vertices are truncated such that all edges are equal length. The original 12 rhombic faces become… … Wikipedia**Golden ratio**— For the Ace of Base album, see The Golden Ratio (album). Not to be confused with Golden number. The golden section is a line segment divided according to the golden ratio: The total length a + b is to the length of the longer segment a as the… … Wikipedia**Dodecadodecahedron**— Type Uniform star polyhedron Elements F = 24, E = 60 V = 30 (χ = −6) Faces by sides 12{5}+12{5/2} Wythoff symbol … Wikipedia**Compound of five cubes**— Type Regular compound Stellation core rhombic triacontahedron Convex hull Dodecahedron Index … Wikipedia**Cube**— This article is about the geometric shape. For other uses, see Cube (disambiguation). Regular Hexahedron (Click here for rotating model) Type Platonic solid Elements F = 6, E = 12 V = 8 (χ = 2) … Wikipedia