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.

Great stellated dodecahedra appear as cells in two star regular polychora, namely the great stellated hecatonicosachoron and great grand stellated hecatonicoscahoron.

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


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

along with all even permutations and all sign changes of


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

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.