Rhombic dodecahedral honeycomb
The rhombic dodecahedral honeycomb is a space-filling tessellation in Euclidean 3-space. It is the Voronoi diagram of the face-centered cubic sphere-packing, which has the densest possible packing of equal spheres in ordinary space.
Geometry
It consists of copies of a single cell, the rhombic dodecahedron. All faces are rhombi, with diagonals in the ratio 1:. Three cells meet at each edge. The honeycomb is thus cell-transitive, face-transitive, and edge-transitive; but it is not vertex-transitive, as it has two kinds of vertex. 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.
Colorings
Cells can be given 4 colors in square layers of 2-colors where neighboring faces have different colors, and 6 colors in hexagonal layers of 3 colors where same-colored cells have no contact at all.| 4-colors | 6-colors |
| Alternate square layers yellow, blue with red and green | Alternate hexagonal layers of red,green,blue and magenta, yellow, cyan. |
Related honeycombs
The rhombic dodecahedral honeycomb can be dissected into a trigonal trapezohedral honeycomb with each rhombic dodecahedron dissected into 4 trigonal trapezohedrons. Each rhombic dodecahedra can also be dissected with a center point into 12 rhombic pyramids of the rhombic pyramidal honeycomb.Trapezo-rhombic dodecahedral honeycomb
The trapezo-rhombic dodecahedral honeycomb is a space-filling tessellation in Euclidean 3-space. It consists of copies of a single cell, the trapezo-rhombic dodecahedron. It is similar to the higher symmetric rhombic dodecahedral honeycomb which has all 12 faces as rhombi.Related honeycombs
It is a dual to the vertex-transitive gyrated tetrahedral-octahedral honeycomb.Rhombic pyramidal honeycomb
The rhombic pyramidal honeycomb or half oblate octahedrille is a uniform space-filling tessellation in Euclidean 3-space.This honeycomb can be seen as a rhombic dodecahedral honeycomb, with the rhombic dodecahedra dissected with its center into 12 rhombic pyramids.