The Rhombic dodecahedron reference article from the English Wikipedia on 24-Jul-2004 (provided by Fixed Reference: snapshots of Wikipedia from wikipedia.org)

Rhombic dodecahedron

Rhombic dodecahedron
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Type	Catalan
Face polygon	rhombus
Faces	12
Edges	24
Vertices	14 = 6 + 8
Face configuration	3,4,3,4
Symmetry group	octahedral (Oh)
Dual polyhedron	cuboctahedron
Properties	convex, face/edge-uniform, zonohedron

The Rhombic dodecahedron is a convex polyhedron with 12 rhombic faces. It is the polyhedral dual of the cuboctahedron and a zonohedron. The long diagonal of each face is exactly √2 times the length of the short diagonal, so that the acute angles on each face measure 2 tan⁻¹(1/√2), or approximately 70.53°.

Being the dual of an Archimedean polyhedron, the rhombic dodecahedron is face-uniform, meaning the symmetry group of the solid acts transitively on the set of faces. In elementary terms, this means that for any two faces A and B there is a rotation or reflection of the solid that leaves it occupying the same region of space while moving face A to face B. The rhombic dodecahedron is also somewhat special in being one of the nine edge-uniform convex polyhedra, the others being the five Platonic solids, the cuboctahedron, the icosidodecahedron and the rhombic triacontahedron.

The rhombic dodecahedron can be used to tessellate 3-dimensional space. This tessellation can be seen as the Voronoi tessellation of the face-centred cubic lattice. Honeybees use the geometry of rhombic dodecahedra to form honeycomb from a tessellation of cells each of which is a hexagonal prism capped with half a rhombic dodecahedron.

See also

• Rhombic triacontahedron

External links

• Rhombic Dodecahedron – from MathWorld
• Virtual Reality Polyhedra – The Encyclopedia of Polyhedra

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