Square double gyroantiprismoid

The square double gyroantiprismoid is a convex isogonal polychoron and the third member of the double gyroantiprismoid family. It consists of 16 square antiprisms, 32 tetragonal disphenoids, 64 rhombic disphenoids, and 128 sphenoids. 2 square antiprisms, 2 tetragonal disphenoids, 4 rhombic disphenoids, and 8 sphenoids join at each vertex. However, it cannot be made uniform. It is the second in an infinite family of isogonal square prismatic swirlchora.

Using the ratio method, the lowest possible ratio between the longest and shortest edges is 1:$$\sqrt{5-2\sqrt2}$$ ≈ 1:1.47363.

Vertex coordinates
Coordinates for the vertices of a square double gyroantiprismoid, assuming that the square antiprisms are uniform of edge length 1, centered at the origin, are given by:
 * $$\left(0,\,±\sqrt2,\,0,\,±\sqrt{\frac{1+\sqrt2}{2}}\right),$$
 * $$\left(0,\,±\frac{\sqrt2}{2},\,±\sqrt{\frac{1+\sqrt2}{2}},\,0\right),$$
 * $$\left(±\frac{\sqrt2}{2},\,0,\,0,\,±\sqrt{\frac{1+\sqrt2}{2}}\right),$$
 * $$\left(±\frac{\sqrt2}{2},\,0,\,±\sqrt{\frac{1+\sqrt2}{2}},\,0\right),$$
 * $$\left(0,\,±\sqrt{\frac{1+\sqrt2}{2}},\,0,,±\frac{\sqrt2}{2}\right),$$
 * $$\left(0,\,±\sqrt{\frac{1+\sqrt2}{2}},\,±\frac{\sqrt2}{2},\,0\right),$$
 * $$\left(±\sqrt{\frac{1+\sqrt2}{2}},\,0,\,0,\,±\frac{\sqrt2}{2}\right),$$
 * $$\left(±\sqrt{\frac{1+\sqrt2}{2}},\,0,\,±\frac{\sqrt2}{2},\,0\right),$$
 * $$\left(±\frac12,\,±\frac12,\,±\frac{\sqrt{1+\sqrt2}}{2},\,±\frac{\sqrt{1+\sqrt2}}{2}\right),$$
 * $$\left(±\frac{\sqrt{1+\sqrt2}}{2},\,±\frac{\sqrt{1+\sqrt2}}{2},\,±\frac12,\,±\frac12\right).$$