| bgcolor=#e7dcc3 colspan=2 | Rhombic dodecahedral honeycomb | |
|---|---|---|
| bgcolor=#ffffff align=center colspan=2 | ||
| Type | convex uniform honeycomb dual | |
| Coxeter-Dynkin diagram | = | |
| Cell type | Rhombic dodecahedron V3.4.3.4 | |
| Face types | Rhombus | |
| Space group | Fmm (225) | |
| Coxeter notation | ½ {\tilde{C}}3 {\tilde{B}}3 {\tilde{A}}3 | |
| Dual | tetrahedral-octahedral honeycomb | |
| Properties | edge-transitive, face-transitive, cell-transitive |
The rhombic dodecahedral honeycomb (also dodecahedrille) is a space-filling tessellation (or honeycomb) 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 (see Kepler conjecture).
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.
The tiling's cells can be 4-colored in square layers of 2 colors each, such that two cells of the same color touch only at vertices; or they can be 6-colored in hexagonal layers of 3 colors each, such that same-colored cells have no contact at all.
| 4-coloring | 6-coloring | |
|---|---|---|
| Alternate square layers of yellow/blue and red/green | Alternate hexagonal layers of red/green/blue and magenta/yellow/cyan |
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.
| bgcolor=#e7dcc3 colspan=2 | Trapezo-rhombic dodecahedral honeycomb | |
|---|---|---|
| bgcolor=#ffffff align=center colspan=2 | ||
| Type | convex uniform honeycomb dual | |
| Cell type | trapezo-rhombic dodecahedron VG3.4.3.4 | |
| Face types | rhombus, trapezoid | |
| Symmetry group | P63/mmc | |
| Dual | gyrated tetrahedral-octahedral honeycomb | |
| Properties | edge-uniform, face-uniform, cell-uniform |
It is a dual to the vertex-transitive gyrated tetrahedral-octahedral honeycomb.
| bgcolor=#e7dcc3 colspan=2 | Rhombic pyramidal honeycomb | |
|---|---|---|
| bgcolor=#ffffff align=center colspan=2 | (No image) | |
| Type | Dual uniform honeycomb | |
| Coxeter-Dynkin diagrams | ||
| Cell | rhombic pyramid | |
| Faces | Rhombus Triangle | |
| Coxeter groups | [4,3<sup>1,1</sup>], {\tilde{B}}3 [3<sup>[4]], {\tilde{A}}3 | |
| Symmetry group | Fmm (225) | |
| vertex figures | ,, | |
| Dual | Cantic cubic honeycomb | |
| Properties | Cell-transitive |
This honeycomb can be seen as a rhombic dodecahedral honeycomb, with the rhombic dodecahedra dissected with its center into 12 rhombic pyramids.
It is dual to the cantic cubic honeycomb: