Digonal-octagonal tetraprismantiprismoid

The digonal-octagonal tetraprismantiprismoid is a convex isogonal polychoron that consists of 8 rectangular antiprisms, 8 rhombic disphenoids, 16 digonal-rectangular gyrowedges, 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-hexadecagonal 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:
 * ±(1/2, (1+$\sqrt{2}$+$\sqrt{5+2√2}$)/4, 0, (1+$\sqrt{2}$+$\sqrt{5+2√2}$)/4),
 * ±(1/2, -(1+$\sqrt{2}$+$\sqrt{5+2√2}$)/4, 0, (1+$\sqrt{2}$+$\sqrt{5+2√2}$)/4),
 * ±((2-$\sqrt{2}$+$\sqrt{10+4√2}$)/8, (2+3$\sqrt{2}$+$\sqrt{10+4√2}$)/8, (2+$\sqrt{2}$+$\sqrt{10+4√2}$)/8, (2+$\sqrt{2}$+$\sqrt{10+4√2}$)/8),
 * ±((2+3$\sqrt{2}$+$\sqrt{10+4√2}$)/8, (2-$\sqrt{2}$+$\sqrt{10+4√2}$)/8, (2+$\sqrt{2}$+$\sqrt{10+4√2}$)/8, (2+$\sqrt{2}$+$\sqrt{10+4√2}$)/8),
 * ±((1+$\sqrt{2}$+$\sqrt{5+2√2}$)/4, 1/2, (1+$\sqrt{2}$+$\sqrt{5+2√2}$)/4, 0),
 * ±((1+$\sqrt{2}$+$\sqrt{5+2√2}$)/4, -1/2, (1+$\sqrt{2}$+$\sqrt{5+2√2}$)/4, 0),
 * ±((2+3$\sqrt{2}$+$\sqrt{10+4√2}$)/8, -(2-$\sqrt{2}$+$\sqrt{10+4√2}$)/8, (2+$\sqrt{2}$+$\sqrt{10+4√2}$)/8, -(2+$\sqrt{2}$+$\sqrt{10+4√2}$)/8),
 * ±((2-$\sqrt{2}$+$\sqrt{10+4√2}$)/8, -(2+3$\sqrt{2}$+$\sqrt{10+4√2}$)/8, (2+$\sqrt{2}$+$\sqrt{10+4√2}$)/8, -(2+$\sqrt{2}$+$\sqrt{10+4√2}$)/8).