# Square antiprism

Square antiprism | |
---|---|

Rank | 3 |

Type | Uniform |

Notation | |

Bowers style acronym | Squap |

Coxeter diagram | s2s8o () |

Conway notation | A4 |

Elements | |

Faces | 8 triangles, 2 squares |

Edges | 8+8 |

Vertices | 8 |

Vertex figure | Isosceles trapezoid, edge lengths 1, 1, 1, √2 |

Measures (edge length 1) | |

Circumradius | |

Volume | |

Dihedral angles | 3–3: |

4–3: | |

Height | |

Central density | 1 |

Number of external pieces | 10 |

Level of complexity | 4 |

Related polytopes | |

Army | Squap |

Regiment | Squap |

Dual | Square antitegum |

Conjugate | Square retroprism |

Abstract & topological properties | |

Flag count | 64 |

Euler characteristic | 2 |

Surface | Sphere |

Orientable | Yes |

Genus | 0 |

Properties | |

Symmetry | (I_{2}(8)×A_{1})/2, order 16 |

Convex | Yes |

Nature | Tame |

The **square antiprism**, or **squap**, is a prismatic uniform polyhedron. It consists of 8 triangles and 2 squares. Each vertex joins one square and three triangles. As the name suggests, it is an antiprism based on a square.

## Vertex coordinates[edit | edit source]

A square antiprism of edge length 1 has vertex coordinates given by:

## Representations[edit | edit source]

A square antiprism has the following Coxeter diagrams:

- s2s8o (alternated octagonal prism)
- s2s4s (alternated ditetragonal prism)
- xo4ox&#x (bases considered separately)

## General variant[edit | edit source]

The square antiprism has a general isogonal variant of the form xo4ox&#y that maintains its full symmetry. This variant uses isosceles triangles as sides.

If the base edges are of length b and the lacing edges are of length l, its height is given by and its circumradius by .

The bases of the square antiprism are rotated from each other by an angle of 45°. If this angle is changed the result is more properly called a square gyroprism.

A notable case occurs as the alternation of the uniform octagonal prism. This specific case has base edges of length and side edges of length .

## In vertex figures[edit | edit source]

A square antiprism with base edges of length 1 and side edges of length occurs as the vertex figure of the small prismatotetracontoctachoron. One using side edges of length occurs as vertex figures of the great distetracontoctachoron.

## Related polyhedra[edit | edit source]

A square pyramid can be attached to a base of the square antiprism to form the gyroelongated square pyramid. If a second square pyramid is attached to the other base, the result is the gyroelongated square bipyramid.

Two non-prismatic uniform polyhedron compounds are composed of square antiprisms:

- Great snub cube (3)
- Great disnub cube (6)

There are also an infinite amount of prismatic uniform compounds that are the antiprisms of compounds of squares.

Thirteen square antiprisms can be joined in a ring at their triangular faces if the faces are very slightly distorted.^{[1]} This is because where is the dihedral angle between the square and the triangle for the uniform square antiprism.

## References[edit | edit source]

- ↑ McNeill, Jim. "A Ring of Anti-prisms."

## External links[edit | edit source]

- Klitzing, Richard. "squap".
- Quickfur. "The Square Antiprism".

- Wikipedia contributors. "Square antiprism".
- McCooey, David. "Square Antiprism"
- Hi.gher.Space Wiki Contributors. "Square antiprism".