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| Rhombic triacontahedron | |
|---|---|
| Type | Catalan |
| Face polygon | rhombus |
| Faces | 30 |
| Edges | 60 |
| Vertices | 32 = 20 + 12 |
| Face configuration | V3.5.3.5 |
| Symmetry group | icosahedral (Ih) |
| Dual polyhedron | icosidodecahedron |
| Properties | convex, face-uniform, edge-uniform, zonohedron |
In geometry, the rhombic triacontahedron is a convex polyhedron with 30 rhombic faces. 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-uniform, 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-uniform 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.
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")
In some roleplaying games, and for elementary school uses, the rhombic triacontahedron is used as the "d30" thirty-sided die.
See also
External links
- Rhombic Triacontahedron – from MathWorld
- Virtual Reality Polyhedra – The Encyclopedia of Polyhedra
- IQ-light™–Danish designer Holger Strøm´s lamp
Catalan solids | Quasiregular polyhedra | Zonohedra
Rhombentriakontaeder | Triacontaedro rómbico | Romboëdrisch triacontaëder | 菱形三十面体 | Triacontaedro rômbico
"Rhombic triacontahedron"