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.[1] As the name suggested, the snub square antiprism is constructed by snubbing the square antiprism,[2] and this construction results in 24 equilateral triangles and 2 squares as its faces.[3] The Johnson solids are the convex polyhedra whose faces are regular, and the snub square antiprism is one of them, enumerated as , the 85th Johnson solid.[4]
Let be the positive root of the cubic polynomial
Furthermore, let 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 -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.[5] It has the three-dimensional symmetry of dihedral group of order 8.[2]
The surface area and volume of a snub square antiprism with edge length can be calculated as:[3]