Dodecadodecahedron

The dodecadodecahedron, or did, is a quasiregular uniform polyhedron. It consists of 12 pentagons and 12 pentagrams, with two of each joining at a vertex. It can be derived as a rectified small stellated dodecahedron or great dodecahedron.

All its flags are isomorphic, leading some to consider it regular. In fact it is an abstractly regular polyhedron.

Vertex coordinates
A dodecadodecahedron of side length 1 has vertex coordinates given by all permutations of and even permutations of
 * $$\left(±1,\,0,\,0\right),$$
 * $$\left(±\frac{1+\sqrt5}{4},\,±\frac{\sqrt5-1}{4},\,±\frac12\right).$$

The first set of vertices corresponds to a scaled octahedron which can be inscribed into the dodecadodecahedron.

Related polyhedra
The dodecadodecahedron is the colonel of a three-member regiment that also includes the small dodecahemicosahedron and the great dodecahemicosahedron.