Square-pyritohedral icosahedral duoantiprism

The square-pyritohedral icosahedral duoantiprism, or squapidap, is a convex isogonal polyteron that consists of 8 pyritohedral icosahedral antiprisms, 8 triangular-square duoantiprisms, 6 digonal-square duoantiprisms, and 96 digonal disphenoidal pyramids. 2 pyritohedral icosahedral antiprisms, 1 digona-square duoantiprism, 2 triangular-square duoantiprisms, and 5 digonal disphenoidal pyramids join at each vertex. It can be obtained through the process of alternating the octagonal-truncated octahedral duoprism. However, it cannot be made uniform.

Using the ratio method, the lowest possible ratio between the longest and shortest edges is 1:$$\sqrt{\frac{24+9\sqrt2}{23}}$$ ≈ 1:1.26367.

Vertex coordinates
The vertices of a square-pyritohedral icosahedral duoantiprism, assuming that the edge length differences are minimized, centered at the origin, are given by: with all even permutations of the first three coordinates, and with all odd permutations of the first three coordinates.
 * $$\left(0,\,±\frac{\sqrt6}{6},\,±\frac{\sqrt6}{3},\,0,\,±\frac{\sqrt2}{2}\right),$$
 * $$\left(0,\,±\frac{\sqrt6}{6},\,±\frac{\sqrt6}{3},\,±\frac{\sqrt2}{2},\,0\right),$$
 * $$\left(0,\,±\frac{\sqrt6}{6},\,±\frac{\sqrt6}{3},\,±\frac12,\,±\frac12\right),$$
 * $$\left(0,\,±\frac{\sqrt6}{6},\,±\frac{\sqrt6}{3},\,±\frac12,\,±\frac12\right),$$