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Rhombic triacontahedron
From Wikipedia, the free encyclopedia
| Rhombic triacontahedron | |
|---|---|
 (Click here for rotating model) |
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| Type | Catalan solid |
| Face type | rhombus |
| Faces | 30 |
| Edges | 60 |
| Vertices | 32 |
| Vertices by type | 20{3}+12{5} |
| Face configuration | V3.5.3.5 |
| Symmetry group | Ih |
| Dihedral angle | 144° |
| Dual | Icosidodecahedron |
| Properties | convex, face-transitive edge-transitive, zonohedron |
 Net |
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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.
The rhombic triacontahedron forms the (hull of) the projection of a 6-dimensional hypercube to 3 dimensions.[citation needed]
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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")
In some roleplaying games, and for elementary school uses, the rhombic triacontahedron is used as the "d30" thirty-sided die.
- Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. ISBN 0-486-23729-X. (Section 3-9)
- Eric W. Weisstein, Rhombic triacontahedron (Catalan solid) at MathWorld.
- Virtual Reality Polyhedra – The Encyclopedia of Polyhedra
- Stellations of Rhombic Triacontahedron