- 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 .
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