The Rhombic triacontahedron is a convexpolyhedron 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.
Being the dual of an Archimedean polyhedron, the rhombic dodecahedron is face-uniform, meaning the symmetry group of the solid acts transitively on the set of faces.
Honeybees use the geometry of rhombic dodecahedra to form honeycomb from a tessellation of cells each of which is a hexagonal prism capped with half a rhombic dodecahedron.
The rhombic dodecahedron forms the hull of the vertex-first projection of a tesseract to 3 dimensions.
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