Square double prismantiprismoid

The square double prismantiprismoid is a convex isogonal polychoron and the third member of the double prismantiprismoid family. It consists of 16 square antiprisms, 16 square prisms, 32 rectangular trapezoprisms, 128 isosceles trapezoidal pyramids, 32 tetragonal disphenoids, and 64 digonal disphenoids. 1 square antiprism, 1 square prism, 2 rectangular trapezoprisms, 5 isosceles trapezoidal pyramids, 1 tetragonal disphenoid, and 2 didgonal disphenoids join at each vertex. It can be obtained as the convex hull of two orthogonal square-octagonal prismantiprismoids. However, it cannot be made scaliform.

A variant with uniform square antiprisms and regular cubes can be vertex-inscribed into a bitruncatotetracontoctachoron. Another variant can be vertex-inscribed into a biambotetracontoctachoron.

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