Snub square antiprism

Snub square antiprism

Type	Johnson J₈₄ - J₈₅ - J₈₆
Faces	8+16 triangles 2 squares
Edges	40
Vertices	16
Vertex configuration	8(3⁵) 8(3⁴.4)
Symmetry group	D_4d
Dual polyhedron	-
Properties	convex
Net

In geometry, the snub square antiprism is one of the Johnson solids (J₈₅).

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 snub square antiprism, 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.

Snub antiprisms
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)
Snub antiprisms	J84	Icosahedron	J85	Concave
Snub antiprisms	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)