Square-snub cubic duoantiprism

The square-snub cubic duoantiprism, or squasnicdap, is a convex isogonal polyteron that consists of 8 snub cubic antiprisms, 6 square-square duoantiprisms, 8 triangular-square duoantiprisms, 12 digonal-square duoantiprisms, and 192 sphenoidal pyramids. 2 snub cubic antiprisms, 1 square-square duoantiprisms, 1 triangular-square duoantiprism, 1 digonal-square duoantiprism, and 5 sphenoidal pyramids join at each vertex. It can be obtained through the process of alternating the octagonal-great rhombicuboctahedral duoprism. However, it cannot be made uniform.

Using the ratio method, the lowest possible ratio between the longest and shortest edges is 1:$$\sqrt{\frac{2+\sqrt2}{2}}$$ ≈ 1:1.30656.

Vertex coordinates
The vertices of a square-snub cubic duoantiprism, assuming that the edge length differences are minimized, centered at the origin, are given by all even permutations with an even number of sign changes, plus all odd permutations with an odd amount of sign changes of the first three coordinates of:
 * $$\left(c_1,\,c_2,\,c_3,\,0,\,±\frac{\sqrt{2+\sqrt2}}{2}\right),$$
 * $$\left(c_1,\,c_2,\,c_3,\,±\frac{\sqrt{2+\sqrt2}}{2},\,0\right),$$
 * $$\left(c_2,\,c_1,\,c_3,\,±\sqrt{\frac{2+\sqrt2}{8}},\,±\sqrt{\frac{2+\sqrt2}{8}}\right),$$

where

via the absolute value method, or
 * $$c_1=\text{root}(32x^3+16x^2-6x-1, 3) ≈ 0.3357307706942925520137148,$$
 * $$c_2=\text{root}(32x^3-14x+1, 3) ≈ 0.6223221429525196906982341,$$
 * $$c_3=\text{root}(16x^3-24x^2+5x+2, 3) ≈ 1.1223221429525196906982341,$$

where the ratio of the largest edge length to the smallest edge length is lowest (via the ratio method).
 * \left(\sqrt{\frac{2-\sqrt2}{8}},\,\sqrt{\frac{2+\sqrt2}{8}},\,\sqrt{\frac{10-\sqrt2}{8}},\,0,\,±\frac{\sqrt2}{2}\right),
 * $$\left(\sqrt{\frac{2-\sqrt2}{8}},\,\sqrt{\frac{2+\sqrt2}{8}},\,\sqrt{\frac{10-\sqrt2}{8}},\,±\frac{\sqrt2}{2},\,0\right),$$
 * $$\left(\sqrt{\frac{2+\sqrt2}{8}},\,\sqrt{\frac{2-\sqrt2}{8}},\,\sqrt{\frac{10-\sqrt2}{8}},\,±\frac12,\,±\frac12\right),$$