Rhombic dodecahedron
The rhombic dodecahedron is a polyhedron with twelve rhombic faces, 14 vertices and 24 edges. On six of the corners borders four edges and at the other eight corners bordered by three edges.
It is a catalanischer body and dual to the cuboctahedron. The rhombic dodecahedron is also the enveloping body, which is described by the union of the penetration of a hexahedron (cube ) and an octahedron.
If a hexahedron " turned upside down ", results in a rhombic dodecahedron. Each side of the hexahedron describes a pyramid with the center of the hexahedron as top. These pyramids are the Hexaederseiten inside, composed (ie, on the set up Hexaederseiten ). The result is a rhombic dodecahedron with the hexahedron inscribed in hollow form. It follows that the volume of a rhombic dodecahedron is twice as large as that of a hexahedron with an edge length of the small diagonal of the side surfaces.
The rhombic dodecahedron is also formed by applying a similar process to the octahedron.
Several rhombododecahedral fill the space without gaps, when they - are joined together - as shown in the image opposite.
Related polyhedra
Deltoidalikositetraeder
Will the 12 boundary surfaces of the rhombic dodecahedron pyramids with edge lengths and placed, creating a general Hexakisoktaeder, if the following condition is satisfied:
- The special Hexakisoktaeder with the same dihedral angles at the edges and arises when is.
- If the maximum value previously mentioned, the Hexakisoktaeder degenerates into a Deltoidalikositetraeder with the edge lengths and.
Formulas
For the polyhedron
For the diamonds
Occurrence
- In nature, the rhombic dodecahedron is a typical crystal form when minerals of the garnet group, and is therefore also called Granatoeder. It can occur as a special form { 110} in all cubic crystal classes.
- The first Brillouin zone of the body-centered cubic lattice has the shape of a rhombic dodecahedron.
- The rhombic dodecahedron is the three-dimensional projection of a four-dimensional cube ( tesseract ).
Parallel projection of a tesseract