# Rhombic dodecahedron

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. Multiples of it can be stacked to fill a space, luch like an hexagon fill a plane, honey combs have a shape similar to the Rhombic dodecahedron cut in half.

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.

File:Rhombic dodecahedra.jpg
Part of a tessellation of space using rhombic dodecahedra

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 vertex-first projection of a tesseract to 3 dimensions. There are exactly two ways of decomposing a rhombic dodecahedron into 4 congruent parallelopipeds, giving 8 possible parallelopipeds. The 8 cells of the tesseract under this projection map precisely to these 8 parallelopipeds.