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 D4d
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.

It can also be constructed as a square gyrobianticupolae, connecting two anticupolae with gyrated orientations.

Construction

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]

It can also be constructed in Conway polyhedron notation as sY4 (snub square pyramid).[3]

Snub antiprisms

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 D2d, [2+,4], (2*2) D3d, [2+,6], (2*3) D4d, [2+,8], (2*4) D5d, [2+,10], (2*5)
Antiprisms
s{2,4}
A2

(v:4; e:8; f:6)
s{2,6}
A3

(v:6; e:12; f:8)
s{2,8}
A4

(v:8; e:16; f:10)
s{2,10}
A5

(v:10; e:20; f:12)
Truncated
antiprisms
ts{2,4}
tA2
(v:16;e:24;f:10)
ts{2,6}
tA3
(v:24; e:36; f:14)
ts{2,8}
tA4
(v:32; e:48; f:18)
ts{2,10}
tA5
(v:40; e:60; f:22)
Symmetry D2, [2,2]+, (222) D3, [3,2]+, (322) D4, [4,2]+, (422) D5, [5,2]+, (522)
Snub
antiprisms		 J84 Icosahedron J85 Concave
sY3 = HtA3 sY4 = HtA4 sY5 = HtA5

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

  1. ↑ 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 .
  2. ↑ Snub Anti-Prisms
  3. ↑ https://levskaya.github.io/polyhedronisme/?recipe=C100sY4
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