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Rhombic Triacontahedron | bgcolor=#e7dcc3 colspan=2|Rhombic triacontahedron | lign=center colspan=2|
Click on picture for large version.
Click here for spinning version. | | gcolor=#e7dcc3|Type | Catalan | | gcolor=#e7dcc3|Face polygon | rhombus | | gcolor=#e7dcc3|Faces | 30 | | gcolor=#e7dcc3|Edges | 60 | | gcolor=#e7dcc3|Vertices | 32 = 20 + 12 | | gcolor=#e7dcc3|Face configuration | 3,5,3,5 | | gcolor=#e7dcc3|Symmetry group | icosahedral (Ih) | | gcolor=#e7dcc3|Dual polyhedron | icosidodecahedron | | gcolor=#e7dcc3|Properties | convex, face/edge-uniform, zonohedron | The Rhombic triacontahedron is a convex polyhedron with 30 rhombic faces. It is the polyhedral dual of the icosidodecahedron and a zonohedron. The ratio of 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. See also External links
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