Snub square antiprism

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Snub square antiprism
Snub square antiprism.png
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
Johnson solid 85 net.png

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[edit]

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]

Snub antiprisms[edit]

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 Digonal antiprism.png
s{2,4}
CDel node h.pngCDel 2x.pngCDel node h.pngCDel 4.pngCDel node.png
(v:4; e:8; f:6)
Trigonal antiprism.png
s{2,6}
CDel node h.pngCDel 2x.pngCDel node h.pngCDel 6.pngCDel node.png
(v:6; e:12; f:8)
Square antiprism.png
s{2,8}
CDel node h.pngCDel 2x.pngCDel node h.pngCDel 8.pngCDel node.png
(v:8; e:16; f:10)
Pentagonal antiprism.png
s{2,10}
CDel node h.pngCDel 2x.pngCDel node h.pngCDel 10.pngCDel node.png
(v:10; e:20; f:12)
Truncated
antiprisms
Truncated digonal antiprism.png
ts{2,4}
(v:16;e:24;f:10)
Truncated octahedron prismatic symmetry.png
ts{2,6}
(v:24; e:36; f:14)
Truncated square antiprism.png
ts{2,8}
(v:32; e:48; f:18)
Truncated pentagonal antiprism.png
ts{2,10}
(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
Snub digonal antiprism.png
ss{2,4}
(v:8; e:20; f:14)
Snub triangular antiprism.png
ss{2,6}
(v:12; e:30; f:20)
Snub square antiprism colored.png
ss{2,8}
(v:16; e:40; f:26)
Snub pentagonal antiprism.png
ss{2,10}
(v:20; e:50; f:32)

References[edit]

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

External links[edit]