Snub square antiprism explained

Type:	Johnson
Faces:	24 triangles 2 squares
Edges:	40
Vertices:	16
Symmetry:	D_4d
Vertex Config:	8 x 3⁵+8 x 3⁴ x 4
Properties:	convex
Net:	Johnson solid 85 net.png

In geometry, the snub square antiprism is the Johnson solid that can be constructed by snubbing the square antiprism. 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 and properties

The snub is the process of constructing polyhedra by cutting loose the edge's faces, twisting them, and then attaching equilateral triangles to their edges. As the name suggested, the snub square antiprism is constructed by snubbing the square antiprism, and this construction results in 24 equilateral triangles and 2 squares as its faces. The Johnson solids are the convex polyhedra whose faces are regular, and the snub square antiprism is one of them, enumerated as

J₈₅

, the 85th Johnson solid.

Let

k ≈ 0.82354

be the positive root of the cubic polynomial

9x^3+3\sqrt\left(5-\sqrt\right)x^2-3\left(5-2\sqrt\right)x-17\sqrt+7\sqrt.

Furthermore, let

h ≈ 1.35374

be defined by

h = \frac.

Then, Cartesian coordinates of a snub square antiprism with edge length 2 are given by the union of the orbits of the points

(1,1,h),\,\left(1+\sqrtk,0,h-\sqrt\right)

under the action of the group generated by a rotation around the axis by 90° and by a rotation by 180° around a straight line perpendicular to the axis and making an angle of 22.5° with the axis. It has the three-dimensional symmetry of dihedral group

D_4d

of order 16.

The surface area and volume of a snub square antiprism with edge length

can be calculated as:

\begin A = \left(2+6\sqrt\right)a^2 &\approx 12.392a^2, \\ V &\approx 3.602 a^3.\end

This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Snub square antiprism".