Rhombic dodecahedron

Rhombic dodecahedron
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Rhombicdodecahedron.jpg
Rhombic dodecahedron

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TypeCatalan
Face polygonrhombus
Faces12
Edges24
Vertices14 = 6 + 8
Face configuration3,4,3,4
Symmetry groupoctahedral (Oh)
Dual polyhedroncuboctahedron
Propertiesconvex, 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.

The rhombic dodecahedron forms the (hull of the) projection of a tesseract to 3 dimensions.

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