# Petrial icosahedron

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Petrial icosahedron | |
---|---|

Rank | 3 |

Type | Regular |

Space | Spherical |

Notation | |

Schläfli symbol | |

Elements | |

Faces | 6 skew decagons |

Edges | 30 |

Vertices | 12 |

Vertex figure | Pentagon, edge length 1 |

Related polytopes | |

Petrie dual | Icosahedron |

Conjugate | Petrial great icosahedron |

Convex hull | Icosahedron |

Abstract properties | |

Flag count | 120 |

Euler characteristic | -12 |

Schläfli type | {10,5} |

Topological properties | |

Orientable | No |

Genus | 14 |

Properties | |

Convex | No |

The **petrial icosahedron** is a regular skew polyhedron and is the Petrie dual of the icosahedron, and so it shares both of its vertices and edges with the icosahedron. It consists of 6 skew decagons and has an Euler characteristic of -12.

## Vertex coordinates[edit | edit source]

The vertices of a petrial icosahedron of edge length 1 centered at the origin are the same as the icosahedron, being all cyclic permutations of:

## Related polyhedra[edit | edit source]

The rectification of the petrial icosahedron is the small dodecahemidodecahedron, which is uniform.

## External links[edit | edit source]

- Wikipedia Contributors. "Petrie dual".
- Hartley, Michael. "{10,5}*120b".

Wikipedia Contributors. "Petrie dual".