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Concept
Snub square antiprism
Summary
In geometry, the snub square antiprism is one of the Johnson solids (J_85).
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.
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. It can be constructed in Conway polyhedron notation as sY4 (snub square pyramid).
It can also be constructed as a square gyrobianticupolae, connecting two anticupolae with gyrated orientations.
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 antiprism of edge length a as
and its volume as
where ξ ≈ 3.60122 is the greatest real root of the polynomial
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 one has to retain two degenerate digonal faces (drawn in red) in the digonal antiprism.
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