# Snub square antiprism

Snub square antiprism | |
---|---|

Type |
Johnson J _{84} - J - J_{85}_{86} |

Faces |
8+16 triangles 2 squares |

Edges | 40 |

Vertices | 16 |

Vertex configuration |
8(3^{5}) 8(3 ^{4}.4) |

Symmetry group |
D_{4d} |

Dual polyhedron | - |

Properties | convex |

Net | |

In geometry, the **snub square antiprism** is one of the
Johnson solids (*J*_{85}).

A Johnson solid is one of 92 strictly convex polyhedra that have regular faces but are not uniform (that is, they are not Platonic solids, Archimedean solids, prisms or antiprisms). They were named by Norman Johnson, who first listed these polyhedra in 1966.^{[1]}

It is one of the elementary Johnson solids that do not arise from "cut and paste" manipulations of the Platonic and Archimedean solids, although it is a relative of the icosahedron that has fourfold symmetry instead of threefold.

## Snub antiprisms

The *snub square antiprism* is constructed as its name suggests, a square antiprism which is snubbed, and represented as ss{2,8}, with s{2,8} as a square antiprism.^{[2]} Similarly constructed the ss{2,6} is a *snub triangular antiprism* (a lower symmetry octahedron), and result as a regular icosahedron. A *snub pentagonal antiprism*, ss{2,10}, or higher n-antiprisms can be similar constructed, but not as a convex polyhedron with equilateral triangles. The preceding Johnson solid, the snub disphenoid also fits constructionally as ss{2,4}, but you have to retain two degenerate digonal faces (drawn in red) in the digonal antiprism.

Symmetry | D_{2d}, [2^{+},4], (2*2) |
D_{3d}, [2^{+},6], (2*3) |
D_{4d}, [2^{+},8], (2*4) |
D_{5d}, [2^{+},10], (2*5) |
---|---|---|---|---|

Antiprisms | s{2,4} (v:4; e:8; f:6) |
s{2,6} (v:6; e:12; f:8) |
s{2,8} (v:8; e:16; f:10) |
s{2,10} (v:10; e:20; f:12) |

Truncated antiprisms |
ts{2,4} (v:16;e:24;f:10) |
ts{2,6} (v:24; e:36; f:14) |
ts{2,8} (v:32; e:48; f:18) |
ts{2,10} (v:40; e:60; f:22) |

Symmetry | D_{2}, [2,2]^{+}, (222) |
D_{3}, [3,2]^{+}, (322) |
D_{4}, [4,2]^{+}, (422) |
D_{5}, [5,2]^{+}, (522) |

Snub antiprisms |
J84 | Icosahedron | J85 | Concave |

ss{2,4} (v:8; e:20; f:14) |
ss{2,6} (v:12; e:30; f:20) |
ss{2,8} (v:16; e:40; f:26) |
ss{2,10} (v:20; e:50; f:32) |

## References

- ↑ Johnson, Norman W. (1966), "Convex polyhedra with regular faces",
*Canadian Journal of Mathematics*,**18**: 169–200, doi:10.4153/cjm-1966-021-8, MR 0185507, Zbl 0132.14603. - ↑ Snub Anti-Prisms