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The rhombic dodecahedron can be seen as a degenerate limiting case of a pyritohedron, with permutation of coordinates (±1, ±1, ±1) and (0, 1 + h, 1 − h 2) with parameter h = 1. These coordinates illustrate that a rhombic dodecahedron can be seen as a cube with six square pyramids attached to each face, allowing them to fit together into a ...
Each vertex with the obtuse rhombic face angles is shared by 4 cells; each vertex with the acute rhombic face angles is shared by 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 ...
The rhombic dodecahedron, generated from four line segments, no two of which are parallel to a common plane. Its most symmetric form is generated by the four long diagonals of a cube. [2] It tiles space to form the rhombic dodecahedral honeycomb. The elongated dodecahedron, generated from five line segments, with two triples of coplanar segments.
The rhombicosidodecahedron shares its vertex arrangement with three nonconvex uniform polyhedra: the small stellated truncated dodecahedron, the small dodecicosidodecahedron (having the triangular and pentagonal faces in common), and the small rhombidodecahedron (having the square faces in common).
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).
Documented examples are rare. Two classes can be distinguished: Non-convex cells which pack without overlapping, analogous to tilings of concave polygons. These include a packing of the small stellated rhombic dodecahedron, as in the Yoshimoto Cube.
If a polygon can tile the plane, its prism is space-filling; examples include the cube, triangular prism, and the hexagonal prism. Any parallelepiped tessellates Euclidean 3-space, as do the five parallelohedra including the cube, hexagonal prism, truncated octahedron, and rhombic dodecahedron.
Additionally, two Catalan solids, the rhombic dodecahedron and rhombic triacontahedron, are edge-transitive, meaning their edges are symmetric to each other. [citation needed] Some Catalan solids were discovered by Johannes Kepler during his study of zonohedra, and Eugene Catalan completed the list of the thirteen solids in 1865. [4]