# Icosahedron atop dodecahedron

Icosahedron atop dodecahedron | |
---|---|

Rank | 4 |

Type | Segmentotope |

Notation | |

Bowers style acronym | Ikadoe |

Coxeter diagram | xo5oo3ox&#x |

Elements | |

Cells | 20+30 tetrahedra, 12 pentagonal pyramids, 1 icosahedron, 1 dodecahedron |

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

Edges | 30+30+60 |

Vertices | 12+20 |

Vertex figures | 12 pentagonal antiprisms, edge length 1 |

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

Measures (edge length 1) | |

Circumradius | |

Hypervolume | |

Dichoral angles | Tet–3–tet: |

Tet–3–ike: | |

Tet–3–peppy: | |

Doe–5–peppy: 72° | |

Height | |

Central density | 1 |

Related polytopes | |

Army | Ikadoe |

Regiment | Ikadoe |

Dual | Dodecahedral-icosahedral tegmoid |

Conjugate | Great icosahedron atop great stellated dodecahedron |

Abstract & topological properties | |

Euler characteristic | 0 |

Orientable | Yes |

Properties | |

Symmetry | H_{3}×I, order 120 |

Convex | Yes |

Nature | Tame |

The **icosahedron atop dodecahedron**, or **ikadoe**, is a CRF segmentochoron (designated K-4.78 on Richard Klitzing's list). As the name suggests, it consists of a dodecahedron and an icosahedron as bases, connected by 12 pentagonal pyramids and 20+30 tetrahedra.

It is also commonly referred to as a dodecahedral or icosahedral antiprism, as the two bases are a pair of dual polyhedra.

The icosahedron atop dodecahedron can also be obtained from the hexacosichoron as a monostratic stack. This is more readily seen from the hexacosichoron's vertex-first projection (where the two bases are concentric) or its edge-first projection (where the two bases are flattened).

## Segmentochoron display[edit | edit source]

## Vertex coordinates[edit | edit source]

Coordinates for the vertices of an icosahedron atop dodecahedron of edge length 1 are given by:

- and all permutations of the first three coordinates,
- and all permutations of the first three coordinates,

## External links[edit | edit source]

- Klitzing, Richard. "ikadoe".
- Hi.gher.Space Wiki Contributors. "Rhodohedral antiprism".