Rhombic dodecahedron

3D view of a rhombic dodecahedron ( animation )

Parquet of the room with rhombic dodecahedra

The rhombic dodecahedron is a polyhedron with twelve rhombic faces, 14 corners and 24 edges. Four edges adjoin six of the corners and three edges adjoin the other eight corners.

It is a Catalan solid and dual to the cuboctahedron . The rhombic dodecahedron is also the envelope , which is described by the union of the penetration of a hexahedron (cube) and an octahedron .

If a hexahedron is "turned inside out", a rhombic dodecahedron is created. Each side of the hexahedron describes a pyramid with the center of the hexahedron as the tip. These pyramids are put together with the hexahedron sides facing inwards (i.e. placed on the hexahedron sides). The result is a rhombic dodecahedron with the inscribed hexahedron as a hollow shape. It follows that the volume of a rhombic dodecahedron is twice as large as that of a hexahedron with the edge length of the small diagonals of the side surfaces.

The rhombic dodecahedron is also created by applying a similar process to the octahedron.

Several rhombic dodecahedra fill the space without any gaps when they are joined together - as shown in the adjacent graphic.

Related polyhedra

• 

  Hexakis octahedron

• 

  Deltoidal icositetrahedron

If pyramids with the flank lengths and are placed on the 12 boundary surfaces of the rhombic dodecahedron , a general hexakis octahedron is created , provided the following condition is met: ${\displaystyle b}$ ${\displaystyle c\,(<b)}$

${\displaystyle {\tfrac {a}{3}}{\sqrt {6}}<b<{\tfrac {2}{9}}a{\sqrt {15}}}$

• The special hexakis octahedron with equal surface angles at the edges and arises when is. ${\displaystyle a}$ ${\displaystyle b}$ ${\displaystyle b=2a\,({\sqrt {2}}-1)}$
• Assumes the aforementioned maximum value, the hexakis octahedron degenerates into a deltoidal icosity tetrahedron with the edge lengths and . ${\displaystyle b}$ ${\displaystyle a}$ ${\displaystyle b}$

Formulas

For the polyhedron Sizes of a rhombic dodecahedron volume ${\displaystyle V={\frac {16}{9}}\,a^{3}{\sqrt {3}}}$ Surface area ${\displaystyle A_{O}\,=8\,a^{2}{\sqrt {2}}}$ Inc sphere radius ${\displaystyle \rho ={\frac {a}{3}}{\sqrt {6}}}$ Edge ball radius ${\displaystyle r={\frac {2}{3}}\,a\,{\sqrt {2}}}$ Face angle  = 120 ° ${\displaystyle \cos \,\alpha =-{\frac {1}{2}}}$ Surface-edge angle  ≈ 125 ° 15 ′ 52 ″ ${\displaystyle \cos \,\beta =-{\frac {1}{3}}{\sqrt {3}}}$ 1. Corners solid angle  = π (3 surfaces) ${\displaystyle \cos \,\Omega _{1}=-1}$ 2. Corner solid angle  = 2/3 π (4 surfaces) ${\displaystyle \cos \,\Omega _{2}=-{\frac {1}{2}}}$

For the rhombuses Sizes of the diamonds Area ${\displaystyle A={\frac {2}{3}}a^{2}{\sqrt {2}}}$ Inscribed radius ${\displaystyle r={\frac {a}{3}}{\sqrt {2}}}$ Long diagonal ${\displaystyle e={\frac {2}{3}}a{\sqrt {6}}=f{\sqrt {2}}}$ Short diagonal ${\displaystyle f={\frac {2}{3}}a{\sqrt {3}}}$ Acute angle (2) ≈ 70 ° 31 ′ 44 ″ ${\displaystyle \cos \,\alpha ={\frac {1}{3}}}$ Obtuse angles (2) ≈ 109 ° 28 ′ 16 ″ ${\displaystyle \cos \,\beta =-{\frac {1}{3}}}$

Occurrence

• In nature, the rhombic dodecahedron occurs as a typical crystal form in minerals of the garnet group and is therefore also called garnet heather . 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 a three-dimensional projection of a four-dimensional cube ( tesseract ).

• 

  Andradite - single crystal as a representative of the garnet group

• 

  Parallel projection of a tesseract

Remarks

1. ↑ ^{a b c} edge length a

Web links

Commons : Rhombic dodecahedron  - collection of images, videos and audio files

Wiktionary: Rhombic dodecahedron  - explanations of meanings, word origins, synonyms, translations

• Eric W. Weisstein : Rhombic dodecahedron . In: MathWorld (English).
• Mineral Atlas: Rhombic dodecahedron Interactive display of the rhombic dodecahedron in the mineral atlas