# Dodecahedron atop small rhombicosidodecahedron

Dodecahedron atop small rhombicosidodecahedron | |
---|---|

Rank | 4 |

Type | Segmentotope |

Notation | |

Bowers style acronym | Doasrid |

Coxeter diagram | xx5oo3ox&#x |

Elements | |

Cells | 20 tetrahedra, 30 triangular prisms, 12 pentagonal prisms, 1 dodecahedron, 1 small rhombicosidodecahedron |

Faces | 20+60 triangles, 30+60 squares, 12+12 pentagons |

Edges | 30+60+60+60 |

Vertices | 20+60 |

Vertex figures | 20 triangular antipodiums, edge lengths 1 (base 1), √2 (sides), and (1+√5)/2 (base 2) |

60 isosceles trapezoidal pyramids, base edge lengths 1, √2, (1+√5)/2, √2, side edge lengths 1, 1, √2. √2 | |

Measures (edge length 1) | |

Circumradius | |

Hypervolume | |

Dichoral angles | Tet–3–trip: |

Pip–4–trip: | |

Doe–5–pip: 162° | |

Srid–3–tet: | |

Srid–4–trip: | |

Srid–5–pip: 18° | |

Height | |

Central density | 1 |

Related polytopes | |

Army | Doasrid |

Regiment | Doasrid |

Dual | Icosahedral-deltoidal hexecontahedral tegmoid |

Conjugate | Great stellated dodecahedron atop quasirhombicosidodecahedron |

Abstract & topological properties | |

Euler characteristic | 0 |

Orientable | Yes |

Properties | |

Symmetry | H_{3}×I, order 120 |

Convex | Yes |

Nature | Tame |

**Dodecahedron atop small rhombicosidodecahedron**, or **doasrid**, is a CRF segmentochoron (designated K-4.152 on Richard Klitzing's list). As the name suggests, it consists of a dodecahedron and a small rhombicosidodecahedron as bases, connected by 20 tetrahedra, 30 triangular prisms, and 12 pentagonal prisms.

It is also sometimes referred to as a **dodecahedral cupola**, as one generalization of the definition of a cupola is to have a polytope atop an expanded version.

It can be obtained as a dodecahedron-first cap of the small disprismatohexacosihecatonicosachoron.

## Segmentochoron display[edit | edit source]

## Vertex coordinates[edit | edit source]

The vertices of a dodecahedron atop small rhombicosidodecahedron segmentochoron of edge length 1 are given by all permutations of the first three coordinates of:

Plus all even permutations of:

## External links[edit | edit source]

- Klitzing, Richard. "doasrid".
- Wikipedia contributors. "Dodecahedral cupola".