# Dodecahedral prism

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Dodecahedral prism | |
---|---|

Rank | 4 |

Type | Uniform |

Space | Spherical |

Notation | |

Bowers style acronym | Dope |

Coxeter diagram | x x5o3o () |

Elements | |

Cells | 12 pentagonal prisms, 2 dodecahedra |

Faces | 30 squares, 24 pentagons |

Edges | 20+60 |

Vertices | 40 |

Vertex figure | Triangular pyramid, edge lengths (1+√5)/2 (base), √2 (legs) |

Measures (edge length 1) | |

Circumradius | |

Hypervolume | |

Dichoral angles | Pip–4–pip: |

Doe–5–pip: 90° | |

Height | 1 |

Central density | 1 |

Number of external pieces | 14 |

Level of complexity | 4 |

Related polytopes | |

Army | Dope |

Regiment | Dope |

Dual | Icosahedral tegum |

Conjugate | Great stellated dodecahedral prism |

Abstract & topological properties | |

Flag count | 960 |

Euler characteristic | 0 |

Orientable | Yes |

Properties | |

Symmetry | H_{3}×A_{1}, order 240 |

Convex | Yes |

Nature | Tame |

The **dodecahedral prism** or **dope** is a prismatic uniform polychoron that consists of 2 dodecahedra and 12 pentagonal prisms. Each vertex joins 1 dodecahedron and 3 pentagonal prisms. It is a prism based on the dodecahedron. As such it is also a convex segmentochoron (designated K-4.74 on Richard Klitzing's list).

## Gallery[edit | edit source]

Card with cell counts, verf, and cross-sections

Segmentochoron display, doe atop doe

## Vertex coordinates[edit | edit source]

The vertices of a dodecahedral prism of edge length 1 are given by all permutations and changes of sign of the first three coordinates of:

along with all even permutations and all sign changes of:

## Representations[edit | edit source]

A dodecahedral prism has the following Coxeter diagrams:

- x x5o3o (full symmetry)
- xx5oo3oo&#x (bases considered separately)

## External links[edit | edit source]

- Bowers, Jonathan. "Category 19: Prisms" (#891).

- Klitzing, Richard. "Dope".

- Wikipedia Contributors. "Dodecahedral prism".