Small stellated dodecahedron

The small stellated dodecahedron, or sissid, is one of the four Kepler-Poinsot solids. It has 12 pentagrams as faces, joining 5 to a vertex.

It is the first stellation of a dodecahedron, from which its name is derived.

Small stellated dodecahedra appear as cells in two nonconvex regular polychora, namely the small stellated hecatonicosachoron and grand stellated hecatonicosachoron.

Vertex coordinates
The vertices of a small stellated dodecahedron of edge length 1, centered at the origin, are all cyclic permutations of


 * $$\left(0,\,±\frac{1}{2},\,±\frac{\sqrt{5}-1}{4}\right).$$

In vertex figures
The small stellated dodecahedron appears as a vertex figure of two Schläfli–Hess polychora.

Related polyhedra
The small stellated dodecahedron is the colonel of a two-member regiment that also includes the great icosahedron.

Two uniform polyhedron compounds are composed of small stellated dodecahedra:


 * Pentagrammatic snub pseudodisoctahedron (2)
 * Pentagrammatic snub pseudicosicosahedron (5)