Square double antiprismoid

The square double antiprismoid is a convex isogonal polychoron and the third member of the double antiprismoid family. It consists of 16 square antiprisms, 64 tetragonal disphenoids, and 128 sphenoids. 2 square antiprisms, 4 tetragonal disphenoids, and 8 sphenoids join at each vertex. It can be obtained as the convex hull of two orthogonal square-square duoantiprisms or by alternating the octagonal ditetragoltriate. However, it cannot be made uniform. It is the first in an infinite family of isogonal square antiprismatic swirlchora.

Using the ratio method, the lowest possible ratio between the longest and shortest edges is 1:$$\frac{\sqrt{6-\sqrt2}}{2}$$ ≈ 1:1.07072. For this variant the edges of the squares of the inscribed duoantiprisms have ratio 1:$$\frac{2+\sqrt2}{2}$$ ≈ 1:1.70711. A variant with uniform square antiprisms also exists; this variant is based on a duoantiprism based on squares with edge length ratio 1:$$\sqrt{1+\sqrt2}$$ ≈ 1:1.55377.

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

An alternate set of coordinates, assuming that the edge length differences are minimized, centered at the origin, are given by:
 * $$\left(0,\,±\frac{\sqrt2}{2},\,0,\,±\frac{1+\sqrt2}{2}\right),$$
 * $$\left(0,\,±\frac{\sqrt2}{2},\,±\frac{1+\sqrt2}{2},\,0\right),$$
 * $$\left(±\frac{\sqrt2}{2},\,0,\,0,\,±\frac{1+\sqrt2}{2}\right),$$
 * $$\left(±\frac{\sqrt2}{2},\,0,\,±\frac{1+\sqrt2}{2},\,0\right),$$
 * $$\left(±\frac12,\,±\frac12,\,±\frac{2+\sqrt2}{4},\,±\frac{2+\sqrt2}{4}\right),$$
 * $$\left(0,\,±\frac{1+\sqrt2}{2},\,±\frac12,\,±\frac12\right),$$
 * $$\left(±\frac{1+\sqrt2}{2},\,0,\,±\frac12,\,±\frac12\right),$$
 * $$\left(±\frac{2+\sqrt2}{4},\,±\frac{2+\sqrt2}{4},\,0,\,±\frac{\sqrt2}{2}\right),$$
 * $$\left(±\frac{2+\sqrt2}{4},\,±\frac{2+\sqrt2}{4],\,±\frac{\sqrt2}{2},\,0\right).$$