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The rhombic dodecahedron is a space-filling polyhedron, meaning it can be applied to tessellate three-dimensional space: it can be stacked to fill a space, much like hexagons fill a plane. It is a parallelohedron because it can be space-filling a honeycomb in which all of its copies meet face-to-face. [ 7 ]
The vertices with the obtuse rhombic face angles have 4 cells. The vertices with the acute rhombic face angles have 6 cells. The rhombic dodecahedron can be twisted on one of its hexagonal cross-sections to form a trapezo-rhombic dodecahedron, which is the cell of a somewhat similar tessellation, the Voronoi diagram of hexagonal close-packing.
The cube is the only Platonic solid that can fill space, although a tiling that combines tetrahedra and octahedra (the tetrahedral-octahedral honeycomb) is possible. Although the regular tetrahedron cannot fill space, other tetrahedra can, including the Goursat tetrahedra derived from the cube, and the Hill tetrahedra.
The illustration here shows the vertex figure (red) of the cuboctahedron being used to derive the corresponding face (blue) of the rhombic dodecahedron.. For a uniform polyhedron, each face of the dual polyhedron may be derived from the original polyhedron's corresponding vertex figure by using the Dorman Luke construction. [2]
Download QR code; Print/export Download as PDF; Printable version; In other projects ... rhombic dodecahedron: 2|3 4 3.4.3.4 O h: U07 K12 12 24 14
Non-convex cells which pack without overlapping, analogous to tilings of concave polygons. These include a packing of the small stellated rhombic dodecahedron, as in the Yoshimoto Cube. Overlapping of cells whose positive and negative densities 'cancel out' to form a uniformly dense continuum, analogous to overlapping tilings of the plane.
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