# Snub cubic antiprism

Snub cubic antiprism | |
---|---|

File:Snub cubic antiprism.png | |

Rank | 4 |

Type | Isogonal |

Notation | |

Bowers style acronym | Sniccap |

Coxeter diagram | s2s4s3s () |

Elements | |

Cells | 48 irregular tetrahedra, 12 rhombic disphenoids, 8 triangular gyroprisms, 6 square gyroprisms, 2 snub cubes |

Faces | 48+48+48+48 scalene triangles, 16 triangles, 12 squares |

Edges | 24+24+24+24+48+48 |

Vertices | 48 |

Vertex figure | Triangular-pentagonal gyrowedge |

Measures (as derived from unit-edge great rhombicuboctahedral prism) | |

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

Edges of equilateral triangles (48): | |

Edges of squares (48): | |

Circumradius | |

Central density | 1 |

Related polytopes | |

Army | Sniccap |

Regiment | Sniccap |

Dual | Pentagonal icositetrahedral antitegum |

Abstract & topological properties | |

Euler characteristic | 0 |

Orientable | Yes |

Properties | |

Symmetry | (B_{3}×A_{1})+, order 48 |

Convex | Yes |

Nature | Tame |

The **snub cubic antiprism**, **omnisnub cubic antiprism**, or **sniccap**, also known as the **alternated great rhombicuboctahedral prism**, is a convex isogonal polychoron that consists of 2 snub cubes, 6 square gyroprisms, 8 triangular gyroprisms, 12 rhombic disphenoids, and 48 irregular tetrahedra. 4 tetrahedra and one each of the other 4 types of cells join at each vertex. It can be obtained through the process of alternating the great rhombicuboctahedral prism. However, it cannot be made uniform, as it generally has 6 edge lengths, which can be minimized to no fewer than 3 different sizes.

Using the ratio method, the lowest possible ratio between the longest and shortest edges is 1: ≈ 1:1.11803.

## Vertex coordinates[edit | edit source]

Vertex coordinates for a snub cubic antiprism, created from the vertices of a great rhombicuboctahedral prism of edge length 1, are given by all even permutations with an even number of sign changes, plus all odd permutations with an odd amount of sign changes in the first three coordinates, of:

A variant with uniform snub cubes as bases is given by all even permutations with an even number of sign changes, plus all odd permutations with an odd amount of sign changes excluding the last coordinate of:

where

An alternate set of coordinates, assuming that the edge length differences are minimized, centered at the origin, are given by all even permutations with an even number of sign changes, plus all odd permutations with an odd amount of sign changes excluding the last coordinate of:

where

which has rhombic disphenoids (via the absolute value method), or

where the ratio of the largest edge length to the smallest edge length is lowest (via the ratio method).

## External links[edit | edit source]

- Klitzing, Richard. "sniccap".
- Wikipedia contributors. "Omnisnub cubic antiprism".