Pentagonal antifastegium

The pentagonal antifastegium, or paf, is a CRF segmentochoron (designated K-4.22 on Richard Klitzing's list). It consists of 1 pentagonal prism, 2 pentagonal antiprisms, 5 tetrahedra, and 5 square pyramids. It is a member of the infinite family of polygonal antifastegiums.

It is a segmentochoron between a pentagon and a pentagonal antiprism or between a pentagon and a gyro pentagonal prism.

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

Representations
The pentagonal antifastegium can be represented by the following Coxeter diagrams:
 * ox xo5ox&#x
 * xoo5oxx&#x