Editing Snub square antiprism (section)

== Construction and properties ==
The [[Snub (geometry)|snub]] is the process of constructing polyhedra by cutting loose the edge's faces, twisting them, and then attaching [[Equilateral triangle|equilateral triangles]] to their edges.{{r|holme}} As the name suggested, the snub square antiprism is constructed by snubbing the [[square antiprism]],{{r|johnson}} and this construction results in 24 equilateral triangles and 2 squares as its faces.{{r|berman}} The [[Johnson solid]]s are the convex polyhedra whose faces are regular, and the snub square antiprism is one of them, enumerated as <math> J_{85} </math>, the 85th Johnson solid.{{r|francis}} <!--It can also be constructed as a square [[Johnson solid#Snub antiprisms|gyrobianticupolae]], connecting two [[anticupola]]e with gyrated orientations.-->

Let <math> k \approx 0.82354 </math> be the positive root of the [[cubic polynomial]]
<math display="block"> 9x^3+3\sqrt{3}\left(5-\sqrt{2}\right)x^2-3\left(5-2\sqrt{2}\right)x-17\sqrt{3}+7\sqrt{6}. </math>
Furthermore, let <math> h \approx 1.35374 </math> be defined by
<math display="block"> h = \frac{\sqrt{2}+8+2\sqrt{3}k-3\left(2+\sqrt{2}\right)k^2}{4\sqrt{3-3k^2}}. </math>
Then, [[Cartesian coordinate system|Cartesian coordinates]] of a snub square antiprism with edge length 2 are given by the union of the orbits of the points
<math display="block"> (1,1,h),\,\left(1+\sqrt{3}k,0,h-\sqrt{3-3k^2}\right) </math>
under the action of the [[Symmetry group|group]] generated by a rotation around the {{nowrap|1=<math> z </math>-}}axis by 90° and by a rotation by 180° around a straight line perpendicular to the {{nowrap|1=<math> z </math>-}}axis and making an angle of 22.5° with the {{nowrap|1=<math> x </math>-}}axis.{{r|timofeenko}} It has the [[Dihedral symmetry in three dimensions|three-dimensional symmetry]] of [[dihedral group]] <math> D_{4d} </math> of order 16.{{r|johnson}}

The surface area and volume of a snub square antiprism with edge length <math> a </math> can be calculated as:{{r|berman}}
<math display="block"> \begin{align}
 A = \left(2+6\sqrt{3}\right)a^2 &\approx 12.392a^2, \\
 V &\approx 3.602 a^3.
\end{align}
</math>