# Petrial tetrahedron

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Petrial tetrahedron | |
---|---|

Rank | 3 |

Type | Regular |

Notation | |

Schläfli symbol | |

Elements | |

Faces | 3 skew squares |

Edges | 6 |

Vertices | 4 |

Vertex figure | Triangle, edge length 1 |

Petrie polygons | 4 triangles |

Measures (edge length 1) | |

Circumradius | |

Related polytopes | |

Army | Tet |

Regiment | Tet |

Dual | Hemioctahedron (abstract) |

Petrie dual | Tetrahedron |

Orientation double cover | Cube |

Abstract properties | |

Flag count | 24 |

Net count | 1 |

Euler characteristic | 1 |

Schläfli type | {4,3} |

Topological properties | |

Surface | Real projective plane |

Orientable | No |

Genus | 1 |

Properties | |

Symmetry | A_{3}, order 24 |

The **Petrial tetrahedron** is a regular skew polyhedron. It is composed of 3 skew squares It is the Petrie dual of the regular tetrahedron.

## Hemicube[edit | edit source]

The **hemicube** is a tiling of the real projective plane which is abstractly equivalent to the Petrial tetrahedron. It's double cover is a cube and it can be seen as a cube with antipodal points identified. In other words it a quotient of the cube.

The hemicube is also related to the triangular hosohedron, as if the opposite edges and vertices of each square are identified, the result is a triangular hosohedron.

## Related polyhedra[edit | edit source]

The rectified petrial tetrahedron is the tetrahemihexahedron, a uniform polyhedron.

## External links[edit | edit source]

- Hartley, Michael. "{4,3}*24".
- Wikipedia Contributors. "Hemicube (geometry)".