Great stellated dodecahedron

The great stellated dodecahedron, or gissid, is one of the four Kepler–Poinsot solids. It has 12 pentagrams as faces, joining 3 to a vertex.

It is the last stellation of the dodecahedron, from which its name is derived. It is also the only Kepler-Poinsot solid to share its vertices with the dodecahedron as opposed to the icosahedron. It has the smallest circumradius of any uniform polyhedron.

Vertex coordinates
The vertices of a great stellated dodecahedron of edge length 1, centered at the origin, are all sign changes of


 * $$\left(±\frac{\sqrt5-1}{4},\,±\frac{\sqrt5-1}{4},\,±\frac{\sqrt5-1}{4}\right),$$

along with all even permutations and all sign changes of


 * $$\left(±\frac{3-\sqrt5}{4},\,±\frac12,\,0\right).$$

The first set of vertices corresponds to a scaled cube which can be inscribed into the great stellated dodecahedron's vertices.

In vertex figures
The great stellated dodecahedron appears as a vertex figure of one Schläfli–Hess polychoron.