# Elongated triangular cupola

The **elongated triangular cupola** (OBSA: **etcu**) is one of the 92 Johnson solids (J_{18}). It consists of 1+3=4 triangles, 3+3+3=9 squares, and 1 hexagon. It can be constructed by attaching a hexagonal prism to the hexagonal base of the triangular cupola.

Elongated triangular cupola | |
---|---|

Rank | 3 |

Type | CRF |

Notation | |

Bowers style acronym | Etcu |

Coxeter diagram | oxx3xxx&#xt |

Elements | |

Faces | |

Edges | 3+3+3+3+3+6+6=27 |

Vertices | 3+6+6=15 |

Vertex figures | 3 rectangles, edge lengths 1 and √2 |

6 trapezoids, edge lengths 1, √2, √2, √2 | |

6 isosceles triangles, edge lengths √3, √2, √2 | |

Measures (edge length 1) | |

Volume | |

Dihedral angles | 3–4 join: |

4–4 join: | |

3–4 cupolaic: | |

4–4 prismatic: 120° | |

4–6: 90° | |

Central density | 1 |

Number of external pieces | 14 |

Level of complexity | 18 |

Related polytopes | |

Army | Etcu |

Regiment | Etcu |

Dual | Hexakis order-6 truncated semibisected trigonal trapezohedron |

Conjugate | None |

Abstract & topological properties | |

Flag count | 108 |

Euler characteristic | 2 |

Surface | Sphere |

Orientable | Yes |

Genus | 0 |

Properties | |

Symmetry | A_{2}×I, order 6 |

Flag orbits | 18 |

Convex | Yes |

Nature | Tame |

If a second cupola is attached to the other hexagonal base of the prism in the same orientation, the result is the elongated triangular orthobicupola. If the second cupola is rotated 60º instead, rhe result is the elongated triangular gyrobicupola.

## Vertex coordinates edit

An elongated triangular cupola of edge length 1 has the following vertices:

- ,
- ,
- ,
- .

## Related polytopes edit

The elongated triangular cupola can be tunnelled with a triangular cupola and a triangular prism to form a quasi-convex Stewart toroid, the tunnelled elongated triangular cupola. This is the only quasi-convex Stewart toroid with the elongated triangular cupola as its convex hull.

## External links edit

- Klitzing, Richard. "etcu".
- Quickfur. "The Elongated Triangular Cupola".

- Wikipedia contributors. "Elongated triangular cupola".
- McCooey, David. "Elongated Triangular Cupola"