Rhombic triacontahedron
| Rhombic triacontahedron | |
|---|---|
| Missing image Rhombictriacontahedron.jpg Rhombic triacontahedron Click on picture for large version. Click here for spinning version. | |
| Type | Catalan |
| Face polygon | rhombus |
| Faces | 30 |
| Edges | 60 |
| Vertices | 32 = 20 + 12 |
| Face configuration | 3,5,3,5 |
| Symmetry group | icosahedral (Ih) |
| Dual polyhedron | icosidodecahedron |
| Properties | convex, face/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), or approximately 63.43°.
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.
External links
- Rhombic Triacontahedron – from MathWorld
- Virtual Reality Polyhedra – The Encyclopedia of Polyhedra