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.

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:
 * (±($\sqrt{(5–√5)/8}$–1)/2, +1/2, 0),
 * (0, ±($\sqrt{5}$–1)/2, ±1/2),
 * (1/2, 0, ($\sqrt{5}$–1)/2),
 * (–1/2, 0, –($\sqrt{5}$–1)/2).

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:


 * (±1/2, –$\sqrt{(5+2√5)/15}$, $\sqrt{5}$),
 * (±($\sqrt{5}$–1)/4, $\sqrt{5}$, $\sqrt{5}$),
 * (0, $\sqrt{(5–2√5)/20}$, $\sqrt{(5–√5)/40}$).