# Rhombic triacontahedron

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Rhombic triacontahedron

In geometry, the rhombic triacontahedron is a convex polyhedron with 30 rhombic faces. It is an Archimedean dual solid, or a Catalan solid. It is the polyhedral dual of the icosidodecahedron, and it is a zonohedron.

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 the symmetry group of the solid acts transitively on the set of faces. In elementary terms, this means that for any two faces A and B there is a rotation 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 nine edge-transitive convex polyhedra, the others being the five Platonic solids, the cuboctahedron, the icosidodecahedron, and the rhombic 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 for elementary school uses, the rhombic triacontahedron is used as the "d30" thirty-sided die.

*Truncated rhombic triacontahedron
*Quasiregular rhombic tiling

References

* (Section 3-9)

*
* [http://polyhedra.org/poly/show/39/rhombic_triacontahedron Rhombic Triacontrahedron] -- Interactive Polyhedron Model
* [http://www.georgehart.com/virtual-polyhedra/vp.html Virtual Reality Polyhedra] &ndash; The Encyclopedia of Polyhedra
* [http://bulatov.org/polyhedra/rtc/ Stellations of Rhombic Triacontahedron]
* [http://www.iqlight.com/ IQ-light] &ndash;Danish designer Holger Strøm's lamp
* [http://web.mac.com/jjkostick/iWeb/Jane%20Kostick/triacontahedron%20box.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.

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