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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 ...
The illustration here shows the vertex figure (red) of the cuboctahedron being used to derive the corresponding face (blue) of the rhombic dodecahedron.. For a uniform polyhedron, each face of the dual polyhedron may be derived from the original polyhedron's corresponding vertex figure by using the Dorman Luke construction. [2]
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).
A rhombic dodecahedron has twelve faces, each of which is a rhombus. Paulingite is a perfect clear rhombic dodecahedron of 0.1 to 1.0 mm in diameter. Their attachment to vesicles causes a hemispherical shape exhibiting 5 to 6 planes of dodecahedral planes. In the vesicular walls, they appear to be dark brown to black.
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.
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]
Tetrahedron, Cube, Octahedron, Dodecahedron, Icosahedron; Regular spherical polyhedron. Dihedron, Hosohedron; Kepler–Poinsot polyhedron (Regular star polyhedra) Small stellated dodecahedron, Great stellated dodecahedron, Great icosahedron, Great dodecahedron; Abstract regular polyhedra (Projective polyhedron)
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 ...