# Digonal-square prismantiprismoid

Digonal-square prismantiprismoid | |
---|---|

Rank | 4 |

Type | Isogonal |

Space | Spherical |

Notation | |

Bowers style acronym | Dispap |

Coxeter diagram | x4s2s4o () |

Elements | |

Cells | 4 tetragonal disphenoids, 8 wedges, 4 rectangular trapezoprisms |

Faces | 16 isosceles triangles, 16 isosceles trapezoids, 4+4 rectangles |

Edges | 8+8+8+16 |

Vertices | 16 |

Vertex figure | Mirror-symmetric notch |

Measures (as derived from unit-edge square-octagonal duoprism) | |

Edge lengths | Short edges of rectangles (8): 1 |

Side edges (8+16): | |

Long edges of rectangles (8): | |

Circumradius | |

Central density | 1 |

Related polytopes | |

Army | Dispap |

Regiment | Dispap |

Dual | Digonal-square tegmantitegmoid |

Abstract & topological properties | |

Euler characteristic | 0 |

Orientable | Yes |

Properties | |

Symmetry | (B_{2}×B_{2})/2, order 32 |

Convex | Yes |

Nature | Tame |

The **digonal-square prismantiprismoid** or **dispap**, also known as the **edge-snub digonal-square duoprism**, **2-4 prismantiprismoid**, **digonal duoexpandoprism**, or **digonal duotruncatoalterprism**, is a convex isogonal polychoron and the first member of the duoexpandoprism, duotruncatoprism, and duotruncatoalterprism families. It consists of 4 rectangular trapezoprisms, 4 tetragonal disphenoids, and 8 wedges. 1 tetrgonal disphenoid, 2 rectangular trapezoprisms, and 3 wedges join at each vertex. It can be obtained through the process of alternating one class of edges of the square-octagonal duoprism so that the octagons become rectangles. However, it cannot be made uniform, as it generally has 4 edge lengths, which can be minimized to no fewer than 2 different sizes.

A variant with regular tetrahedra and squares can be vertex-inscribed into a rectified tesseract.

Using the ratio method, the lowest possible ratio between the longest and shortest edges is 1: ≈ 1:1.36603.

## Vertex coordinates[edit | edit source]

The vertices of a digonal-square prismantiprismoid, assuming that the tetragonal disphenoids are regular and are connected by squares of edge length 1, centered at the origin, are given by:

An alternate set of coordinates, assuming that the edge length differences are minimized, centered at the origin, are given by:

An additional variant based on a unit square-octagonal duoprism has vertices given by: