# Square cupola

Square cupola | |
---|---|

Rank | 3 |

Type | CRF |

Notation | |

Bowers style acronym | Squacu |

Coxeter diagram | ox4xx&#x |

Stewart notation | Q_{4} |

Elements | |

Faces | |

Edges | 4+4+4+8 |

Vertices | 4+8 |

Vertex figures | 4 isosceles trapezoids, edge lengths 1, √2, √2, √2 |

8 scalene triangles, edge lengths 1, √2, √2+√2 | |

Measures (edge length 1) | |

Circumradius | |

Volume | |

Dihedral angles | 3–4: |

4–4: 135° | |

3–8: | |

4–8: 45° | |

Height | |

Central density | 1 |

Number of external pieces | 10 |

Level of complexity | 10 |

Related polytopes | |

Army | Squacu |

Regiment | Squacu |

Dual | Semibisected tetragonal trapezohedron |

Conjugate | Retrograde square cupola |

Abstract & topological properties | |

Flag count | 80 |

Euler characteristic | 2 |

Surface | Sphere |

Orientable | Yes |

Genus | 0 |

Properties | |

Symmetry | B_{2}×I, order 8 |

Flag orbits | 10 |

Convex | Yes |

Nature | Tame |

The **square cupola** is one of the 92 Johnson solids (J_{4}). It consists of 4 triangles, 1+4 squares, and 1 octagon. It is a cupola based on the square, and is one of three Johnson solid cupolas, the other two being the triangular cupola and the pentagonal cupola.

It can be obtained as a segment of the small rhombicuboctahedron, when considered as an elongated square orthobicupola.

## Vertex coordinates[edit | edit source]

A square cupola of edge length 1 has vertices given by the following coordinates:

- ,
- ,
- .

These can be obtained from placing a square and octagon in parallel planes.

## Representations[edit | edit source]

A square cupola has the following Coxeter diagrams:

- ox4xx&#x
- so8ox&#x
- oqxw qowx&#xr (bases in digonal symmetry)

## Related polyhedra[edit | edit source]

Two square cupolas can be attached at their octagonal bases in the same orientation to form a square orthobicupola. If the second cupola is rotated by 45º the result is the square gyrobicupola.

An octagonal prism can be attached to the square cupola's octagonal base to form the elongated square cupola. If an octagonal antiprism is attached instead, the result is the gyroelongated square cupola.

## External links[edit | edit source]

- Klitzing, Richard. "squacu".
- Quickfur. "The Square Cupola".

- Weisstein, Eric W. "Square Cupola" ("Johnson solid") at MathWorld.

- Wikipedia contributors. "Square cupola".
- McCooey, David. "Square Cupola"
- Hi.gher.Space Wiki Contributors. "Square cupola".