Pentagonal prismatic pyramid

The pentagonal prismatic pyramid, or pippy, is a CRF segmentochoron (designated K-4.141 on Richard Klitzing's list). It has 5 square pyramids, 2 pentagonal pyramids, and 1 pentagonal prism as cells. As the name suggests, it is a pyramid based on the pentagonal prism.

The pentagonal prismatic pyramid is the vertex pyramid of the rectified hexacosichoron.

The pentagonal prism is the largest CRF polyhedron with radius less than 1. As a result, this pyramid is extremely flat, which means that it can be used to form a large number of CRF augmentations from other polychora.

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

Representations
A pentagonal prismatic pyramid has the following Coxeter diagrams:


 * ox ox5oo&#x (full symmetry)
 * oxx5ooo&#x (H2 symmetry, prism seen as frustum symmetry)