Petrial octahedron

The Petrial octahedron is a regular skew polyhedron consisting of 4 skew hexagons. It is the Petrie dual of the octahedron, therefore shares all of its edges and points with the octahedron. The Petrial octahedron has an Euler characteristic of -2.

Vertex coordinates
The vertex coordinates of the Petrial octahedron are shared with the regular octahedron, being all permutations of:
 * $$\left(±\frac{\sqrt2}{2},\,0,\,0\right).$$