# Rhombic dodecahedron

Rhombic dodecahedron | |
---|---|

Rank | 3 |

Type | Uniform dual |

Notation | |

Bowers style acronym | Rad |

Coxeter diagram | o4m3o () |

Conway notation | jC |

Elements | |

Faces | 12 rhombi |

Edges | 24 |

Vertices | 6+8 |

Vertex figure | 6 squares, 8 triangles |

Measures (edge length 1) | |

Inradius | |

Volume | |

Dihedral angle | 120° |

Central density | 1 |

Number of external pieces | 12 |

Level of complexity | 2 |

Related polytopes | |

Army | Rad |

Regiment | Rad |

Dual | Cuboctahedron |

Conjugate | None |

Abstract & topological properties | |

Flag count | 96 |

Euler characteristic | 2 |

Surface | Sphere |

Orientable | Yes |

Genus | 0 |

Properties | |

Symmetry | B_{3}, order 48 |

Flag orbits | 2 |

Convex | Yes |

Nature | Tame |

The **rhombic dodecahedron** is one of the 13 Catalan solids. It has 12 rhombi as faces, with 6 order-4 and 8 order-3 vertices. It is the dual of the uniform cuboctahedron.

It can also be obtained as the convex hull of a cube and an octahedron scaled so that their edges are orthogonal. For this to happen, the octahedron's edge length must be times that of the cube's edge length. Each edge of the cube or octahedron corresponds to one of the diagonals of the faces.

It can also be constructed by augmenting a cube with square pyramids of height 0.5.

Each face of this polyhedron is a rhombus with longer diagonal times the shorter diagonal, with acute angle and obtuse angle .

The rhombic dodecahedron is the only Catalan solid that can tile 3D space by itself, forming the rhombic dodecahedral honeycomb.

## Vertex coordinates[edit | edit source]

A rhombic dodecahedron of edge length 1 has vertex coordinates given by all permutations of

- ,
- .

Scaling these coordinates by a factor of gives integral coordinates for the rhombic dodecahedron.

## Related polytopes[edit | edit source]

The rhombic dodecahedron can tile 3-space as the rhombic dodecahedral honeycomb, which is isotopic.

### Variations[edit | edit source]

The rhombic dodecahedron can be varied to remain isohedral under tetrahedral symmetry. In the process the rhombic faces turn into kites, and the resulting polyhedron can be called a deltoidal dodecahedron.

The Bilinski dodecahedron is a variant of the rhombic dodecahedron with golden rhombi for faces. This variant can be dissected into 4 acute golden rhombohedra and 4 obtuse golden rhombohedra.^{[1]}

## External links[edit | edit source]

- Klitzing, Richard. "Rad".
- Wikipedia contributors. "Rhombic dodecahedron".
- McCooey, David. "Rhombic Dodecahedron"
- Quickfur. "The Rhombic Dodecahedron".

## References[edit | edit source]

- ↑ Bardos, Laszlo. "Golden Rhombohedra".