# Icosahedral prism

Icosahedral prism | |
---|---|

Rank | 4 |

Type | Uniform |

Space | Spherical |

Notation | |

Bowers style acronym | Ipe |

Coxeter diagram | x o5o3x () |

Elements | |

Cells | 20 triangular prisms, 2 icosahedra |

Faces | 40 triangles, 30 squares |

Edges | 12+60 |

Vertices | 24 |

Vertex figure | Pentagonal pyramid, edge lengths 1 (base), √2 (legs) |

Measures (edge length 1) | |

Circumradius | |

Hypervolume | |

Dichoral angles | Trip–4–trip: |

Ike–3–trip: 90° | |

Height | 1 |

Central density | 1 |

Number of external pieces | 22 |

Level of complexity | 4 |

Related polytopes | |

Army | Ipe |

Regiment | Ipe |

Dual | Dodecahedral tegum |

Conjugate | Great icosahedral 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 **icosahedral prism** or **ipe** is a prismatic uniform polychoron that consists of 2 icosahedra and 20 triangular prisms. Each vertex joins 1 icosahedron and 5 triangular prisms. It is a prism based on the icosahedron. As such it is also a convex segmentochoron (designated K-4.36 in Richard Klitzing's list).

## Gallery[edit | edit source]

Card with cell counts, verf, and cross-sections

Segmentochoron display, ike atop ike

## Vertex coordinates[edit | edit source]

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

## Representations[edit | edit source]

An icosahedral prism has the following Coxeter diagrams:

- x o5o3x (full symmetry)
- x2s3s4o () (bases as pyritohedral symmetry)
- x2s3s3s () (as snub tetrahedral prism)
- oo5oo3xx&#x (bases seen separately)
- xxxx oxoo5ooxo&#xt (H
_{2}×A_{1}axial, edge-first)

## Related polychora[edit | edit source]

An icosahedral prism can be cut into a central pentagonal antiprismatic prism augmented with 2 pentagonal pyramidal prisms.

The regiment of the icosahedral prism also contains the great dodecahedral prism.

## External links[edit | edit source]

- Bowers, Jonathan. "Category 19: Prisms" (#892).

- Klitzing, Richard. "Ipe".

- Wikipedia Contributors. "Icosahedral prism".