Rhombic triacontahedron
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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

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

• Rhombic dodecahedron

External links

• Rhombic Triacontahedron – from MathWorld
• Virtual Reality Polyhedra – The Encyclopedia of Polyhedra