Square double antiprismoid

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. 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:$\sqrt{6-√2}$/2 ≈ 1:1.07072.

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),
 * (±1/2, ±1/2, ±$\sqrt{1+√2}$/2, ±$\sqrt{1+√2}$/2),
 * (0, ±$\sqrt{2+2√2}$/2, ±1/2, ±1/2),
 * (±$\sqrt{2+2√2}$/2, 0, ±1/2, ±1/2),
 * (±$\sqrt{1+√2}$/2, ±$\sqrt{1+√2}$/2, 0, ±$\sqrt{2}$/2),
 * (±$\sqrt{1+√2}$/2, ±$\sqrt{1+√2}$/2, ±$\sqrt{2}$/2, 0).

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