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In geometry, the rhombic dodecahedron is a convex polyhedron with 12 congruent rhombic faces. It has 24 edges, and 14 vertices of 2 types. As a Catalan solid, it is the dual polyhedron of the cuboctahedron. As a parallelohedron, the rhombic dodecahedron can be used to tesselate its copies in space creating a rhombic dodecahedral honeycomb.
The vertices with the obtuse rhombic face angles have 4 cells. The vertices with the acute rhombic face angles have 6 cells. The rhombic dodecahedron can be twisted on one of its hexagonal cross-sections to form a trapezo-rhombic dodecahedron, which is the cell of a somewhat similar tessellation, the Voronoi diagram of hexagonal close-packing.
2.5 edges 1/2, vol. = 1/8 of 20 Duo-Tet Cube 3 24 MITEs Octahedron 4 dual of cube, spacefills w/ tet Rhombic Triacontahedron 5 radius = ~0.9994, vol. = 120 Ts Rhombic Triacontahedron 5+ radius = 1, vol. = 120 Es Rhombic Dodecahedron 6 space-filler, dual to cuboctahedron Rhombic Triacontahedron 7.5 radius = phi/sqrt(2) Icosahedron
The regular hyperbolic honeycombs thus include two with four or five dodecahedra meeting at each edge; their dihedral angles thus are π/2 and 2π/5, both of which are less than that of a Euclidean dodecahedron. Apart from this effect, the hyperbolic honeycombs obey the same topological constraints as Euclidean honeycombs and polychora.
The rhombic dodecahedron packs together to fill space. The rhombic dodecahedron can be seen as a degenerate pyritohedron where the 6 special edges have been reduced to zero length, reducing the pentagons into rhombic faces. The rhombic dodecahedron has several stellations, the first of which is also a parallelohedral spacefiller.
It shares its vertex arrangement with the small stellated truncated dodecahedron and the uniform compounds of 6 or 12 pentagrammic prisms.It additionally shares its edge arrangement with the rhombicosidodecahedron (having the square faces in common), and with the small dodecicosidodecahedron (having the decagonal faces in common).
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