Pentagrammic retroprism

The pentagrammic retroprism, or starp, also called the pentagrammic crossed antiprism, is a prismatic uniform polyhedron. It consists of 10 triangles and 2 pentagrams. Each vertex joins one pentagram and three triangles. As the name suggests, it is a crossed antiprism based on a pentagram, seen as a 5/3-gon rather than 5/2. This makes it the simplest possible crossed antiprism.

Similar to how the pentagonal antiprism can be vertex-inscribed into the regular icosahedron, the pentagrammic retroprism can be vertex-inscribed into a great icosahedron.

Vertex coordinates
A pentagrammic retroprism of edge length 1 has vertex coordinates given by:
 * $$\left(±\frac{\sqrt5-1}2,\,±\frac12,\,0\right),$$
 * $$\left(0,\,±\frac{\sqrt5-1}2,\,±\frac12\right),$$
 * $$\left(\frac12,\,0,\,\frac{\sqrt5-1}2\right),$$
 * $$\left(-\frac12,\,0,\,-\frac{\sqrt5-1}2\right).$$

These coordinates are obtained by removing two opposite vertices from a great icosahedron.

An alternative set of coordinates can be constructed in a similar way to other polygonal antiprisms, giving the vertices as the following points along with their central inversions:


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

Related polyhedra
Two non-prismatic uniform polyhedron compounds are composed of pentagrammic retroprisms:


 * Great inverted snub dodecahedron (6)
 * Great inverted disnub dodecahedron (12)

There are also an infinite amount of prismatic uniform compounds that are the crossed antiprisms of compounds of pentagrams.