# Rhombic dodecahedral honeycomb

Rhombic dodecahedral honeycomb :
| Type | convex uniform honeycomb dual |
| Coxeter-Dynkin diagram | = |
| Cell type | Rhombic dodecahedron V3.4.3.4 |
| Face types | Rhombus |
| Space group | Fm3m (225) |
| Coxeter notation | ½${\tilde {C}}_{3}$, [1$^+$,4,3,4] ${\tilde {B}}_{3}$, [4,3$^{1,1}$] ${\tilde {A}}_{3}$×2, <[3$^{[4]}$]> |
| 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).

## Geometry

It consists of copies of a single cell, the rhombic dodecahedron. All faces are rhombi, with diagonals in the ratio 1:√2. 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. Each vertex with the obtuse rhombic face angles is shared by 4 cells; each vertex with the acute rhombic face angles is shared by 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 honeycomb can be derived from an alternate cube tessellation by augmenting each face of each cube with a pyramid. | The view from inside the rhombic dodecahedral honeycomb. |

### Colorings

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 |

## 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

Trapezo-rhombic dodecahedral honeycomb :
| Type | convex uniform honeycomb dual |
| Cell type | trapezo-rhombic dodecahedron VG3.4.3.4 |
| Face types | rhombus, trapezoid |
| Symmetry group | P6$_3$/mmc |
| Dual | gyrated tetrahedral-octahedral honeycomb |
| Properties | edge-uniform, face-uniform, cell-uniform |
The trapezo-rhombic dodecahedral honeycomb is a space-filling tessellation (or honeycomb) 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

Rhombic pyramidal honeycomb :
| (No image) ||
| Type | Dual uniform honeycomb |
| Coxeter-Dynkin diagrams | |
| Cell | rhombic pyramid |
| Faces | Rhombus Triangle |
| Coxeter groups | [4,3$^{1,1}$], ${\tilde {B}}_{3}$ [3$^{[4]}$], ${\tilde {A}}_{3}$ |
| Symmetry group | Fm3m (225) |
| vertex figures | , , |
| Dual | Cantic cubic honeycomb |
| Properties | Cell-transitive |
The rhombic pyramidal honeycomb or half oblate octahedrille is a uniform space-filling tessellation (or honeycomb) 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.
| rhombic dodecahedral honeycomb | Rhombohedral dissection | Within a cube |

#### Related honeycombs

It is dual to the cantic cubic honeycomb: