Petrial icosahedron

The petrial icosahedron is a regular skew polyhedron and is the Petrie dual of the icosahedron, and so it shares both of its vertices and edges with the icosahedron. It consists of 6 skew decagons and has an Euler characteristic of -12.

Vertex coordinates
The vertices of a petrial icosahedron 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 icosahedron is the small dodecahemidodecahedron, which is uniform.