Pentagonal pyramidal prism

The pentagonal pyramidal prism, or peppyp, is a CRF segmentochoron (designated K-4.38 on Richard Klitzing's list). It consists of 2 pentagonal pyramids, 1 pentagonal prism, and 5 triangular prisms.

As the name suggests, it is a prism based on the pentagonal pyramid. As such, it is a segmentochoron between two pentagonal pyramids. It can also be viewed as a segmentochoron between a pentagonal prism and a dyad.

Two pentagonal pyramidal prisms can be attached to the pentagonal prismatic cells of the pentagonal antiprismatic prism to produce the uniform icosahedral prism.

Vertex coordinates
Coordinates of the vertices of a pentagonal pyramidal prism of edge length 1 centered at the origin are given by:


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

Representations
A pentagonal pyramidal prism has the following Coxeter diagrams:


 * xx o54oo&#x (full symmetry)
 * oxxo5oooo&#xr (H2 axial only, pentagonal pyramid atop pentagonal pyramid)