Pentagonal cupofastegium

The pentagonal cupofastegium, or pecuf, also sometimes called the pentagonal orthobicupolic ring, is a CRF segmentochoron (designated K-4.154 on Richard Klitzing's list). It consists of 1 pentagonal prism, 5 tetrahedra, 5 triangular prisms, and 2 pentagonal cupolas.

It has two representations as a segmentochoron: pentagon atop pentagonal cupola or pentagonal prism atop decagon.

The pentagonal cupofastegium can be obtained as a cap of the small disprismatohexacosihecatonicosachoron in pentagonal prism first orientation.

Vertex coordinates
The vertices of a pentagonal cupofastegium with edge length 1 are given by:
 * $$\left(±\frac12,\,-\sqrt{\frac{5+2\sqrt5}{20}},\,±\frac12,\,\sqrt{\frac{5-2\sqrt5}{20}}\right),$$
 * $$\left(±\frac{1+\sqrt5}{4},\,\sqrt{\frac{5-\sqrt5}{40}},\,±\frac12,\,\sqrt{\frac{5-2\sqrt5}{20}}\right),$$
 * $$\left(0,\,±\sqrt{\frac{5+\sqrt5}{10}},\,±\frac12,\,\sqrt{\frac{5-2\sqrt5}{20}}\right),$$
 * $$\left(±\frac12,\,±\frac{\sqrt{5+2\sqrt5}}{2},\,0,\,0\right),$$
 * $$\left(±\frac{3+\sqrt5}{4},\,±\sqrt{\frac{5+\sqrt5}{8}},\,0,\,0\right),$$
 * $$\left(±\frac{1+\sqrt5}{2},\,0,\,0,\,0\right).$$

Representations
A pentagonal cupofastegium has the following Coxeter diagrams:


 * ox xx5xo&#x (full symmetry)
 * xxx5oxo&#x (H2 symmetry only, seen with pentagon atop pentagonal cupola)