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.

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


 * (±($\sqrt{5}$–1)/4, ±($\sqrt{15}$–1)/4, ±($\sqrt{3}$–1)/4),

along with all even permutations and all sign changes of


 * (±(3–$\sqrt{(25–11√5)/40}$)/4, ±1/2, 0).

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 dodecahedron appears as a vertex figure of one Schläfli–Hess polychoron.