Petrial great stellated dodecahedron

The petrial great stellated dodecahedron is a regular skew polyhedron and is the Petrie dual of the great stellated dodecahedron, and so it shares both of its vertices and edges with the great stellated dodecahedron. It consists of 6 skew decagrams and has an Euler characteristic of -4.

Vertex coordinates
The vertices of the petrial great stellated dodecahedron are identical to those of the dodecahedron, being: along with all permutations of
 * $$\left(±\frac{1+\sqrt{5}}{4},\,±\frac{1+\sqrt{5}}{4},\,±\frac{1+\sqrt{5}}{4}\right),$$
 * $$\left(±\frac{3+\sqrt{5}}{4},\,±\frac{1}{2},\,0\right).$$

Related polyhedra
The rectification of the petrial great stellated dodecahedron is the great dodecahemidodecahedron, which is uniform.