# Tetrahedral prism

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Tetrahedral prism | |
---|---|

Rank | 4 |

Type | Uniform |

Space | Spherical |

Notation | |

Bowers style acronym | Tepe |

Coxeter diagram | x x3o3o () |

Tapertopic notation | 1^{2}1 |

Elements | |

Cells | 2 tetrahedra, 4 triangular prisms |

Faces | 8 triangles, 6 squares |

Edges | 4+12 |

Vertices | 8 |

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

Measures (edge length 1) | |

Circumradius | |

Hypervolume | |

Dichoral angles | Tet–3-trip: 90° |

Trip–4–trip: | |

Heights | Tet atop tet: 1 |

Dyad atop trip: | |

Square atop ortho square: | |

Central density | 1 |

Number of external pieces | 6 |

Level of complexity | 4 |

Related polytopes | |

Army | Tepe |

Regiment | Tepe |

Dual | Tetrahedral tegum |

Conjugate | None |

Abstract & topological properties | |

Flag count | 192 |

Euler characteristic | 0 |

Orientable | Yes |

Properties | |

Symmetry | A_{3}×A_{1}, order 48 |

Convex | Yes |

Nature | Tame |

The **tetrahedral prism** or **tepe** is a prismatic uniform polychoron that consists of 2 tetrahedra and 4 triangular prisms. Each vertex joins 1 tetrahedron and 3 triangular prisms. As the name suggests, it is a prism based on a tetrahedron, and as such is also a segmentochoron (designated K-4.9 in Richard Klitzing's list).

## Gallery[edit | edit source]

Segmentochoron display, tet atop tet

Segmentochoron display, square atop orthogonal square

Card with cell counts, verf, and cross-sections

## Vertex coordinates[edit | edit source]

The vertices of a tetrahedral prism of edge length 1 are given by all even sign changes of the first three coordinates of:

## Representations[edit | edit source]

A tetrahedral prism has the following Coxeter diagrams:

- x x3o3o (full symmetry)
- x2s4o3o () (prism of alternated cube)
- x2s2s4o () (prism of alternated square prism)
- x2s2s2s () (prism of alternated cuboid)
- xx3oo3oo&#x (bases considered separate)
- xx ox3oo&#x (A
_{2}×A_{1}axial, dyad atop triangular prism) - xx xo ox&#x (A
_{1}×A_{1}×A_{1}axial, square atop orthogonal square) - oox xxx&#x (base has one symmetry axis only)
- xxxx&#x (irregular bases)
- xxoo ooxx&#xr (A
_{1}×A_{1}axial) - oxxo3oooo&#xr (A
_{2}axial)

## External links[edit | edit source]

- Bowers, Jonathan. "Category 19: Prisms" (#889).

- Klitzing, Richard. "Tepe".

- Wikipedia Contributors. "Tetrahedral prism".