Tetrahedral-square prismantiprismoid

The tetrahedral-square prismantiprismoid is a convex isogonal polyteron that consists of 4 tetrahedral prisms, 4 tetrahedral antiprisms, 6 digonal-square prismantiprismoids and 16 tetrahedral wedges obtained through the process of edge-alternating the octagonal-cubic duoprism. However, it cannot be made uniform.

Using the ratio method, the lowest possible ratio between the longest and shortest edges is 1:$$\frac{1+\sqrt3}{2}$$ ≈ 1:1.36603.

Vertex coordinates
The vertices of a tetrahedral-square prismantiprismoid, assuming that the edge length differences are minimized, centered at the origin, are given by all even sign changes of the first three coordinates of:
 * ($\sqrt{2}$/4, $\sqrt{2}$/4, $\sqrt{2}$/4, ±($\sqrt{3}$-1)/2, ±1/2),
 * ($\sqrt{2}$/4, $\sqrt{2}$/4, -$\sqrt{2}$/4, ±1/2, ±($\sqrt{3}$-1)/2).