Pentagonal antiprismatic pyramid

The pentagonal antiprismatic pyramid, or pappy, is a CRF segmentochoron (designated K-4.80 on Richard Klitzing's list). It consists of 10 regular tetrahedra, 2 pentagonal pyramids, and 1 pentagonal antiprism. As the name suggests, it is a pyramid based on the pentagonal antiprism.

It can be obtained as the middle piece of an icosahedral pyramid, with the remainder formed by augmenting two pentagonal scalenes onto the pentagonal pyramids. It also occurs as a part of icosahedron atop dodecahedron, as the vertex pyramid of the icosahedral vertices.

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