# Rhombic dodecahedral honeycomb

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).

Rhombic dodecahedral honeycomb | |
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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, [1^{+},4,3,4], [4,3 ^{1,1}]×2, <[3 ^{[4]}]> |

Dual | tetrahedral-octahedral honeycomb |

Properties | edge-transitive, face-transitive, cell-transitive |

## 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. 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 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

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 |
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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

Trapezo-rhombic dodecahedral honeycomb | |
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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.

### Rhombic pyramidal honeycomb

Rhombic pyramidal honeycomb | |
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(No image) | |

Type | Dual uniform honeycomb |

Coxeter-Dynkin diagrams | |

Cell | rhombic pyramid |

Faces | Rhombus Triangle |

Coxeter groups | [4,3^{1,1}], [3 ^{[4]}], |

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 |

## References

- Williams, Robert (1979).
*The Geometrical Foundation of Natural Structure: A Source Book of Design*. Dover Publications, Inc. p. 168. ISBN 0-486-23729-X.

## External links

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