Petrial great dodecahedron

The petrial great icosahedron is a regular skew polyhedron and the Petrie dual of the great dodecahedron. It consists of 10 skew hexagons, has an Euler characteristic of -8, and it shares the vertices and edges of the icosahedron.

Vertex coordinates
The vertices of a petrial great dodecahedron of edge length 1 centered at the origin are the same as the icosahedron, being all cyclic permutations of:
 * $$\left(0,\,±\frac{1}{2},\,±\frac{1+\sqrt{5}}{4}\right).$$

Related polyhedra
The rectification of the petrial great icosahedron is the small dodecahemicosahedron, which is uniform.