Snub square antiprism


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

Construction

The snub square antiprism is constructed as its name suggests, a square antiprism which is snubbed, and represented as ss, with s as a square antiprism. It can be constructed in Conway polyhedron notation as sY4.
It can also be constructed as a square gyrobianticupolae, connecting two anticupolae with gyrated orientations.

Cartesian coordinates

Let k ≈ 0.82354 be the positive root of the cubic polynomial
Furthermore, let h ≈ 1.35374 be defined by
Then, Cartesian coordinates of a snub square antiprism with edge length 2 are given by the union of the orbits of the points
under the action of the group generated by a rotation around the z-axis by 90° and by a rotation by 180° around a straight line perpendicular to the z-axis and making an angle of 22.5° with the x-axis.
We may then calculate the surface area of a snub square of edge length a as
and its volume as
where ξ ≈ 3.60122 is the greatest real root of the polynomial

Snub antiprisms

Similarly constructed, the ss is a snub triangular antiprism, and result as a regular icosahedron. A snub pentagonal antiprism, ss, 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, but one has to retain two degenerate digonal faces in the digonal antiprism.
SymmetryD2d, , D3d, , D4d, , D5d, ,
Antiprisms
s
A2

s

s

s
Truncated
antiprisms
ts
tA2

ts

ts

ts
SymmetryD2, +, D3, +, D4, +, D5, +,
Snub
antiprisms		J84IcosahedronJ85Concave
Snub
antiprisms		 = HtA3 = HtA4 = HtA5
Snub
antiprisms
ss

ss

ss

ss