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

Comments

Pyramid (geometry) Deltoidal icositetrahedron Inscribed sphere Midsphere Inscribed circle Brillouin-Zone Andradite Single crystal Wolfram Research Triakis tetrahedron Triakis octahedron Pentagonal icositetrahedron Pentakis dodecahedron Triakis icosahedron Deltoidal hexecontahedron Pentagonal hexecontahedron
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