Properties
The rhombic dodecahedron is a
zonohedron. Its polyhedral
dual is the
cuboctahedron. 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 arccos(1/3), or approximately 70.53°.
Being the dual of an
Archimedean polyhedron, the rhombic dodecahedron is
face-transitive, 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 one of the nine
edge-transitive 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 three-dimensional space. It can be stacked to fill a space much like
hexagons fill a plane.
This
polyhedron in a space-filling tessellation can be seen as the
Voronoi tessellation of the
face-centered cubic lattice. It is the Brillouin zone of body centered cubic (bcc) crystals. Some minerals such as
garnet form a rhombic dodecahedral
crystal habit.
Honey bees use the geometry of rhombic dodecahedra to form
honeycombs from a tessellation of cells each of which is a
hexagonal prism capped with half a rhombic dodecahedron. The rhombic dodecahedron also appears in the unit cells of
diamond and
diamondoids. In these cases, four vertices (alternate threefold ones) are absent, but the chemical bonds lie on the remaining edges.^{
[1]}
The graph of the rhombic dodecahedron is
nonhamiltonian.
A rhombic dodecahedron can be
dissected with its center into 4
trigonal trapezohedra. These rhombohedra are the cells of a
trigonal trapezohedral honeycomb. This is analogous to the dissection of a
regular hexagon dissected into
rhombi, and tiled in the plane as a
rhombille.
Rhombic dodecahedron |
Hexagon |