Petrial dodecahedron

The petrial dodecahedron is a regular skew polyhedron consisting of 6 skew decagons. The petrial dodecahedron is the Petrie dual of the dodecahedron, so therefore it shares its edges and vertices with the dodecahedron. It has an Euler characteristic of -4.

Vertex coordinates
The vertices of the petrial dodecahedron are identical to those of the dodecahedron, being:
 * $$\left(\pm\frac{1+\sqrt{5}}{4},\,\pm\frac{1+\sqrt{5}}{4},\,\pm\frac{1+\sqrt{5}}{4}\right),$$

along with all permutations of
 * $$\left(\pm\frac{3+\sqrt{5}}{4},\,\pm\frac{1}{2},\,0\right).$$

Related polyhedra
The rectification of the Petrial dodecahedron is the small icosihemidodecahedron, which is uniform.