Retrograde pentagonal cuploid

The retrograde pentagonal cuploid, also called the retrograde pentagonal semicupola or rapescu, is an orbiform polyhedron. It consists of 5 triangles, 5 squares, and 1 pentagon. It is a cuploid based on the pentagon seen as {5/4}, with a pseudo {10/4} base.

Vertex coordinates
A retrograde pentagonal cuploid of edge length 1 has vertices given by the following coordinates:


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

Related polyhedra
The retrograde pentagonal cuploid can be edge-inscribed into the small ditrigonary icosidodecahedron; it uses triangles and pentagons of the great ditrigonary icosidodecahedron as well as squares of the rhombihedron, the inscribed compound of 5 cubes.