# Triangular cupolic prism

Triangular cupolic prism | |
---|---|

Rank | 4 |

Type | Segmentotope |

Notation | |

Bowers style acronym | Tricupe |

Coxeter diagram | xx ox3xx&#x |

Elements | |

Cells | 1+3 triangular prisms, 3 cubes, 2 triangular cupolas, 1 hexagonal prism |

Faces | 2+6 triangles, 3+3+3+6+6 squares, 2 hexagons |

Edges | 3+6+6+6+6+12 |

Vertices | 6+12 |

Vertex figures | 6 rectangular pyramids, base edge lengths 1 and √2, side edge length √2 |

12 irregular tetrahedra, edge lengths 1 (1), √2 (4), and √3 (1) | |

Measures (edge length 1) | |

Circumradius | |

Hypervolume | |

Dichoral angles | Trip–4–cube: |

Tricu–3–trip: 90° | |

Tricu–4–cube: 90° | |

Tricu–6–hip: 90° | |

Trip–4–hip: | |

Cube–4–hip: | |

Heights | Tricu atop tricu: 1 |

Trip atop hip: | |

Central density | 1 |

Related polytopes | |

Army | Tricupe |

Regiment | Tricupe |

Dual | Semibisected hexagonal trapezohedral tegum |

Conjugate | None |

Abstract & topological properties | |

Euler characteristic | 0 |

Orientable | Yes |

Properties | |

Symmetry | A_{2}×A_{1}×I, order 12 |

Convex | Yes |

Nature | Tame |

The **triangular cupolic prism**, or **tricupe**, is a CRF segmentochoron (designated K-4.45 on Richard Klitzing's list). It consists of 2 triangular cupolas, 4 triangular prisms, 3 cubes, and 1 hexagonal prism.

As the name suggests, it is a prism based on the triangular cupola. As such, it is a segmentochoron between two triangular cupolas. It can also be viewed as a segmentochoron between a hexagonal prism and a triangular prism.

A variant, known as the **triangular cupolic wedge**, has the same symmetry as the triangular cupolic prism, which occurs if the lacing edge lengths of the two prisms are unequal.

Two triangular cupolic prisms can be joined at their hexagonal prismatic cells in opposite orientations to form a cuboctahedral prism. By rotating one of the triangular cupolic prisms before joining, one can instead form a triangular orthobicupolic prism.

## Vertex coordinates[edit | edit source]

Coordinates of the vertices of a triangular cupolic prism of edge length 1 centered at the origin are given by:

## External links[edit | edit source]

- Klitzing, Richard. "tricupe".