Square double gyroantiprismoid

The square double antiprismoid is a convex isogonal polychoron and the third member of the double antiprismoids that consists of 16 square antiprisms, 64 tetragonal disphenoids and 128 sphenoids obtained as the convex hull of two orthogonal square-square duoantiprisms. However, it cannot be made uniform.

Vertex coordinates
The vertices of a square double antiprismoid, assuming that the square antiprisms are regular of edge length 1, centered at the origin, are given by:
 * (0, ±$\sqrt{2}$/2, 0, ±$\sqrt{2+2√2}$/2)
 * (0, ±$\sqrt{2}$/2, ±$\sqrt{2+2√2}$/2, 0)
 * (±$\sqrt{2}$/2, 0, 0, ±$\sqrt{2+2√2}$/2)
 * (±$\sqrt{2}$/2, 0, ±$\sqrt{2+2√2}$/2, 0)
 * (0, ±$\sqrt{2+2√2}$/2, 0, ±$\sqrt{2}$/2)
 * (0, ±$\sqrt{2+2√2}$/2, ±$\sqrt{2}$/2, 0)
 * (±$\sqrt{2+2√2}$/2 0, 0, ±$\sqrt{2}$/2)
 * (±$\sqrt{2+2√2}$/2, 0, ±$\sqrt{2}$/2, 0)
 * (±1/2, ±1/2, ±$\sqrt{1+√2}$/2, ±$\sqrt{1+√2}$/2)
 * (±$\sqrt{1+√2}$/2, ±$\sqrt{1+√2}$/2, ±1/2, ±1/2)