Digonal-octagonal tetraprismantiprismoid

The digonal-octagonal tetraprismantiprismoid is a convex isogonal polychoron that consists of 8 rectangular antiprisms, 16 digonal-rectangular gyrowedges, 8 rhombic disphenoids, and 32 phyllic disphenoids of two kinds. 1 rhombic disphenoid, 4 phyllic disphenoids, 2 rectangular gyroprisms, and 4 digonal-rectangular gyrowedges join at each vertex. It can be obtained as a subsymmetrical faceting of the octagonal-dioctagonal duoprism. However, it cannot be made scaliform.

Using the ratio method, the lowest possible ratio between the longest and shortest edges is 1:$$\frac{1+\sqrt2+\sqrt{5+2\sqrt{2}}}{2}$$ ≈ 1:2.60607.

Vertex coordinates
The vertices of a digonal-octagonal tetraprismantiprismoid, assuming that the edge length differences are minimized, using the ratio method, are given by all even permutations of the first two coordinates of:
 * $$±\left(\frac12,\,\frac{1+\sqrt2+\sqrt{5+2\sqrt2}}{4},\,0,\,\frac{1+\sqrt2+\sqrt{5+2\sqrt2}}{4}\right),$$
 * $$±\left(\frac12,\,-\frac{1+\sqrt2+\sqrt{5+2\sqrt2}}{4},\,0,\,\frac{1+\sqrt2+\sqrt{5+2\sqrt2}}{4}\right),$$
 * $$±\left(\frac{2-\sqrt2+\sqrt{10+4\sqrt2}}{8},\,\frac{2+3\sqrt2+\sqrt{10+4\sqrt2}}{8},\,\frac{2+\sqrt2+\sqrt{10+4\sqrt2}}{8},\,\frac{2+\sqrt2+\sqrt{10+4\sqrt2}}{8}\right),$$
 * $$±\left(\frac{2+3\sqrt2+\sqrt{10+4\sqrt2}}{8},\,\frac{2-\sqrt2+\sqrt{10+4\sqrt2}}{8},\,\frac{2+\sqrt2+\sqrt{10+4\sqrt2}}{8},\,\frac{2+\sqrt2+\sqrt{10+4\sqrt2}}{8}\right),$$
 * $$±\left(\frac{1+\sqrt2+\sqrt{5+2\sqrt2}}{4},\,\frac12,\,\frac{1+\sqrt2+\sqrt{5+2\sqrt2}}{4},\,0\right),$$
 * $$±\left(\frac{1+\sqrt2+\sqrt{5+2\sqrt2}}{4},\,-\frac12,\,\frac{1+\sqrt2+\sqrt{5+2\sqrt2}}{4},\,0\right),$$
 * $$±\left(\frac{2+3\sqrt2+\sqrt{10+4\sqrt5}}{8},\,-\frac{2-\sqrt2+\sqrt{10+4\sqrt2}}{8},\,\frac{2+\sqrt2+\sqrt{10+4\sqrt2}}{8},\,-\frac{2+\sqrt2+\sqrt{10+4\sqrt2}}{8}\right),$$
 * $$±\left(\frac{2-\sqrt2+\sqrt{10+4\sqrt2}}{8},\,-\frac{2+3\sqrt2+\sqrt{10+4\sqrt2}}{8},\,\frac{2+\sqrt2+\sqrt{10+4\sqrt2}}{8},\,-\frac{2+\sqrt2+\sqrt{10+4\sqrt2}}{8}\right).$$