Retrograde pentagrammic cupola

The retrograde pentagrammic cupola or rastacu, also called the crossed pentagrammic cupola or sometimes just the pentagrammic cupola, is a cupola based on the pentagram, seen as a 5/3-gon rather than 5/2. It consists of 5 triangles, 5 squares, 1 pentagram, and 1 decagram.

It can be obtained as a segment of the quasirhombicosidodecahedron, just as its conjugate, the convex pentagonal cupola, can be obtained from the small rhombicosidodecahedron.

It also appears in some scaliform polychora, namely the retroprismatorhombisnub hecatonicosachoron and retroprismatorhombiretrosnub hecatonicosachoron.

Vertex coordinates
A retrograde pentagrammic cupola of edge length 1 has vertices given by the following coordinates:


 * $$\left(±\frac12,\,-\sqrt{\frac{5-2\sqrt5}{20}},\,\sqrt{\frac{5+\sqrt5}{10}}\right),$$
 * $$\left(±\frac{\sqrt5-1}{4},\,\sqrt{\frac{5-\sqrt5}{40}},\,\sqrt{\frac{5+\sqrt5}{10}}\right),$$
 * $$\left(0,\,\sqrt{\frac{5-\sqrt5}{10}},\,\sqrt{\frac{5+\sqrt5}{10}}\right),$$
 * $$\left(±\frac12,\,±\frac{\sqrt{5-2\sqrt5}}{2},\,0\right),$$
 * $$\left(±\frac{3-\sqrt5}{4},\,±\sqrt{\frac{5-\sqrt5}{8}},\,0\right),$$
 * $$\left(±\frac{\sqrt5-1}{2},\,0,\,0\right).$$

Related polyhedra
The retrograde pentagrammic cupola is the pentagram-first cap of the quasirhombicosidodecahedron.