Small triangular double gyroprismantiprismoid

The small triangular double prismantiprismoid is a convex isogonal polychoron and the second member of the double prismantiprismoid family. It consists of 12 triangular antiprisms, 12 triangular prisms, 18 rectangular trapezoprisms, 72 isosceles trapezoidal pyramids, 18 tetragonal disphenoids, and 36 rhombic disphenoids. 1 triangular antiprism, 1 triangular prism, 2 rectangular trapezoprisms, 5 isosceles trapezoidal pyramids, 1 tetragonal disphenoid, and 2 rhombic disphenoids join at each vertex. It can be obtained as the convex hull of two orthogonal triangular-hexagonal prismantiprismoids. However, it cannot be made scaliform.

A variant with regular octahedra and uniform triangular prisms can be vertex-inscribed into a small prismatotetracontoctachoron.

This polychoron cannot be optimized using the ratio method, because the solution (with intended minimal ratio 1:$$\frac{6}{3+3\sqrt3-\sqrt{6+6\sqrt3}}$$ ≈ 1:1.44686) would yield a great triangular double prismantiprismoid instead.

Vertex coordinates
The vertices of a triangular double prismantiprismoid, assuming that the triangular antiprisms and triangular prisms are uniform of edge length 1, centered at the origin, are given by:
 * $$\left(0,\,\frac{\sqrt3}{3},\,±\frac12,\,\frac{3\sqrt3+2\sqrt6}{6}\right),$$
 * $$\left(0,\,\frac{\sqrt3}{3},\,±\frac{2+\sqrt2}{2},\,-\frac{\sqrt6}{6}\right),$$
 * $$\left(0,\,\frac{\sqrt3}{3},\,±\frac{1+\sqrt2}{2},\,-\frac{3\sqrt3+\sqrt6}{6}\right),$$
 * $$\left(0,\,-\frac{\sqrt3}{3},\,±\frac12,\,-\frac{3\sqrt3+2\sqrt6}{6}\right),$$
 * $$\left(0,\,-\frac{\sqrt3}{3},\,±\frac{2+\sqrt2}{2},\,\frac{\sqrt6}{6}\right),$$
 * $$\left(0,\,-\frac{\sqrt3}{3},\,±\frac{1+\sqrt2}{2},\,\frac{3\sqrt3+\sqrt6}{6}\right),$$
 * $$\left(±\frac12,\,-\frac{\sqrt3}{6},\,±\frac12,\,\frac{3\sqrt3+2\sqrt6}{6}\right),$$
 * $$\left(±\frac12,\,-\frac{\sqrt3}{6},\,±\frac{2+\sqrt2}{2},\,-\frac{\sqrt6}{6}\right),$$
 * $$\left(±\frac12,\,-\frac{\sqrt3}{6},\,±\frac{1+\sqrt2}{2},\,-\frac{3\sqrt3+\sqrt6}{6}\right),$$
 * $$\left(±\frac12,\,\frac{\sqrt3}{6},\,±\frac12,\,-\frac{3\sqrt3+2\sqrt6}{6}\right),$$
 * $$\left(±\frac12,\,\frac{\sqrt3}{6},\,±\frac{2+\sqrt2}{2},\,\frac{\sqrt6}{6}\right),$$
 * $$\left(±\frac12,\,\frac{\sqrt3}{6},\,±\frac{1+\sqrt2}{2},\,\frac{3\sqrt3+\sqrt6}{6}\right),$$
 * $$\left(±\frac12,\,\frac{3\sqrt3+2\sqrt6}{6},\,0,\,\frac{\sqrt3}{3}\right),$$
 * $$\left(±\frac12,\,-\frac{3\sqrt3+2\sqrt6}{6},\,0,\,-\frac{\sqrt3}{3}\right),$$
 * $$\left(±\frac12,\,\frac{3\sqrt3+2\sqrt6}{6},\,±\frac12,,\,-\frac{\sqrt3}{6}\right),$$
 * $$\left(±\frac12,\,-\frac{3\sqrt3+2\sqrt6}{6},\,±\frac12,\,\frac{\sqrt3}{6}\right),$$
 * $$\left(±\frac{1+\sqrt2}{2},\,-\frac{3\sqrt3+\sqrt6}{6},\,0,\,\frac{\sqrt3}{3}\right),$$
 * $$\left(±\frac{1+\sqrt2}{2},\,\frac{3\sqrt3+\sqrt6}{6},\,0,\,-\frac{\sqrt3}{3}\right),$$
 * $$\left(±\frac{1+\sqrt2}{2},\,-\frac{3\sqrt3+\sqrt6}{6},\,±\frac12,\,-\frac{\sqrt3}{6}\right),$$
 * $$\left(±\frac{1+\sqrt2}{2},\,\frac{3\sqrt3+\sqrt6}{6},\,±\frac12,\,\frac{\sqrt3}{6}\right),$$
 * $$\left(±\frac{2+\sqrt2}{2},\,-\frac{\sqrt6}{6},\,0,\,\frac{\sqrt3}{3}\right),$$
 * $$\left(±\frac{2+\sqrt2}{2},\,\frac{\sqrt6}{6},\,0,\,-\frac{\sqrt3}{3}\right),$$
 * $$\left(±\frac{2+\sqrt2}{2},\,-\frac{\sqrt6}{6},\,±\frac12,,\,-\frac{\sqrt3}{6}\right),$$
 * $$\left(±\frac{2+\sqrt2}{2},\,\frac{\sqrt6}{6},\,±\frac12,\,\frac{\sqrt3}{6}\right).$$