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. 1 tetrahedral prism, 1 tetrahedral antiprism, 3 digonal-square prismantiprismoids, and 4 tetrahedral wedges join at each vertex. It can be obtained through the process of edge-alternating the octagonal-cubic duoprism so that the octagons become long rectangles. 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:
 * $$\left(\frac{\sqrt2}{4},\,\frac{\sqrt2}{4},\,\frac{\sqrt2}{4},\,±\frac{\sqrt3-1}{2},\,±\frac12\right),$$
 * $$\left(\frac{\sqrt2}{4}, \frac{\sqrt2}{4},\,-\frac{\sqrt2}{4},\,±\frac12,\,±\frac{\sqrt3-1}{2}\right).$$

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