Biaugmented pentagonal prism

The biaugmented pentagonal prism, or baupip, is one of the 92 Johnson solids (J53). It consists of 2+2+4 triangles, 1+2 squares, and 2 pentagons. It can be constructed by attaching square pyramids to two non-adjacent square faces of the pentagonal prism.

Vertex coordinates
A biaugmented pentagonal prism of edge length 1 has the following vertices:
 * $$\left(±\frac12,\,-\sqrt{\frac{5+2\sqrt5}{20}},\,±\frac12\right),$$
 * $$\left(±\frac{1+\sqrt5}{4},\,\sqrt{\frac{5-\sqrt5}{40}},\,±\frac12\right),$$
 * $$\left(0,\,±\sqrt{\frac{5+\sqrt5}{10}},\,±\frac12\right),$$
 * $$\left(±\frac{3+\sqrt5+2\sqrt{5+\sqrt5}}{8},\,-\sqrt{\frac{35-9\sqrt5+4\sqrt{25-5\sqrt5}}{160}},\,0\right).$$