Great disnub dodecahedron

The great disnub dodecahedron, gadsid, or compound of twelve pentagonal antiprisms is a uniform polyhedron compound. It consists of 120 triangles and 24 pentagons (whcih fall in pairs in the same plane and combine into 12 stellated decagons), with one pentagon and three triangles joining at a vertex.

This compound has rotational freedom, represented by an angle θ. we start at θ = 0° with all the pentagonal antiprisms inscribed in an icosahedron, and rotate pairs of antiprisms in opposite directions. At θ = 36° the antiprisms coincide by pairs, resulting in a double cover of the great snub dodecahedron.

Vertex coordinates
The vertices of a great disnub dodecahedron of edge length 1 and rotation angle θ are given by all even permutations of:
 * $$\left(±\frac{\sqrt5-(5+\sqrt5)\cos(\theta))}{10},\,±\sqrt{\frac{5+\sqrt5}{10}}\sin(\theta),\,±\frac{5+\sqrt5+4\sqrt5\cos(\theta)}{20}\right),$$
 * $$\left(±\frac{2\sqrt5-2\sqrt5\cos(\theta)-(1+\sqrt5)\sqrt{5\frac{5+\sqrt5}{2}}\sin(\theta)}{20},\,±\frac{-10(1+\sqrt5)\cos(\theta)+(\sqrt5-1)\sqrt{5\frac{5+\sqrt5}{2}}\sin(\theta)}{20},\,±\frac{5+\sqrt5+(5-\sqrt5)\cos(\theta)+2\sqrt{\5\frac{5+\sqrt5}{2}}\sin(\theta)}{20}\right),$$
 * $$\left(±\frac{2\sqrt5+(5+3\sqrt5)\cos(\theta)-2\sqrt{5\frac{5+\sqrt5}{2}}\sin(\theta)}{20},\,±\frac{-10\cos(\theta)-(1+\sqrt5)\sqrt{5\frac{5+\sqrt5}{2}}\sin(\theta)}{20},\,±\frac{5+\sqrt5-(5+\sqrt5)\cos(\theta)+(\sqrt5-1)\sqrt{5\frac{5+\sqrt5}{2}}\sin(\theta)}{20}\right),$$
 * $$\left(±\frac{2\sqrt5+(5+3\sqrt5)\cos(\theta)+2\sqrt{5\frac{5+\sqrt5}{2}}\sin(\theta)}{20},\,±\frac{10\cost(\theta)-(1+\sqrt5)\sqrt{5\frac{5+\sqrt5}{2}}\sin(\theta)}{20},\,±\frac{5+\sqrt5-(5+\sqrt5)\cos(\theta)-(\sqrt5-1)\sqrt{5\frac{5+\sqrt5}{2}}\sin(\theta)}{20}\right),$$
 * $$\left(±\frac{2\sqrt5-2\sqrt5\cos(\theta)+(1+\sqrt5)\sqrt{5\frac{5+\sqrt5}{2}}\sin(\theta)}{20},\,±\frac{10(1+\sqrt5)\cos(\theta)+(\sqrt5-1)\sqrt{5\frac{5+\sqrt5}{2}}\sin(\theta)}{20},\,±\frac{5+\sqrt5+(5-\sqrt5)\cos(\theta)-2\sqrt{5\frac{5+\sqrt5}{2}}\sin(\theta)}{20}\right).$$