# Pyritohedral icosahedral antiprism

Pyritohedral icosahedral antiprism | |
---|---|

Rank | 4 |

Type | Isogonal |

Notation | |

Bowers style acronym | Pikap |

Coxeter diagram | |

Elements | |

Cells | 24 sphenoids, 6 tetragonal disphenoids, 8 triangular gyroprisms, 2 pyritohedral icosahedra |

Faces | 48 scalene triangles, 24+24 isosceles triangles, 16 triangles |

Edges | 12+12+24+48 |

Vertices | 24 |

Vertex figure | Triangular-pentagonal antiwedge |

Measures (as derived from unit-edge truncated octahedral prism) | |

Edge lengths | Diagonals of original squares (12+12+24): |

Edges of equilateral triangles (48): | |

Circumradius | |

Central density | 1 |

Related polytopes | |

Army | Pikap |

Regiment | Pikap |

Dual | Pyritohedral antitegum |

Abstract & topological properties | |

Euler characteristic | 0 |

Orientable | Yes |

Properties | |

Symmetry | (B_{3}/2×2×A_{1})/2, order 48 |

Convex | Yes |

Nature | Tame |

The **pyritohedral icosahedral antiprism**, **pyritohedral icosahedral alterprism** or **pikap**, also known as the **alternated truncated octahedral prism** or **omnisnub tetrahedral antiprism**, is a convex isogonal polychoron that consists of 2 pyritohedral icosahedra, 8 triangular gyroprisms, 6 tetragonal disphenoids, and 24 sphenoids. 1 pyritohedral icosahedron, 2 triangular antiprisms, 1 tetragonal disphenoid, and 4 sphenoids join at each vertex. It can be obtained through the process of alternating the truncated octahedral prism. However, it cannot be made uniform, as it generally has 4 edge lengths, which can be minimized to no fewer than 2 different sizes..

Using the ratio method, the lowest possible ratio between the longest and shortest edges is 1: ≈ 1:1.09325. In this variant, none of the cells are regular, though the sphenoids and tetragonal disphenoids become identical to each other.

This polychoron has a kind of pyritohedral antiprismatic symmetry. A variant with chiral tetrahedral antiprismatic symmetry, called the snub tetrahedral antiprism, also exists.

## Vertex coordinates[edit | edit source]

Vertex coordinates for a pyritohedral icosahedral antiprism, created from the vertices of a truncated octahedral prism of edge length 1, are given by all even permutations in the first 3 coordinates of:

A variant where the icosahedra and the tetragonal disphenoids are regular of edge length 1, centered at the origin, is given by all even permutations in the first 3 coordinates of:

A variant using regular octahedra of edge length 1, centered at the origin, is given by all even permutations in the first 3 coordinates of:

An alternate set of coordinates, assuming that the edge length differences are minimized, centered at the origin, are given by all even permutations in the first 3 coordinates of:

## Isogonal derivatives[edit | edit source]

Substitution by vertices of these following elements will produce these convex isogonal polychora:

- Sphenoid (24): Pyritohedral icosahedral antiprism
- Triangle (16): Tesseract
- Isosceles triangle (24): Pyritohedral icosahedral antiprism
- Scalene triangle (48): Pyritosnub alterprism
- Edge (12): Octahedral prism
- Edge (24): Pyritohedral icosahedral antiprism
- Edge (48): Pyritosnub alterprism

## External links[edit | edit source]

- Klitzing, Richard. "pikap".
- Wikipedia contributors. "Omnisnub tetrahedral antiprism".