Pentagonal-octagonal duoprism

The pentagonal-octagonal duoprism or podip, also known as the 5-8 duoprism, is a uniform duoprism that consists of 5 octagonal prisms and 8 pentagonal prisms, with two of each joining at each vertex.

Vertex coordinates
Coordinates for the vertices of a pentagonal-octagonal duoprism with edge length 1 are given by:
 * $$\left(0,\,\sqrt{\frac{5+\sqrt5}{10}},\,±\frac12,\,±\frac{1+\sqrt2}{2}\right),$$
 * $$\left(0,\,\sqrt{\frac{5+\sqrt5}{10}},\,±\frac{1+\sqrt2}{2},\,±\frac12\right),$$
 * $$\left(±\frac{1+\sqrt5}{4},\,\sqrt{\frac{5-\sqrt5}{40}},\,±\frac12,\,±\frac{1+\sqrt2}{2}\right),$$
 * $$\left(±\frac{1+\sqrt5}{4},\,\sqrt{\frac{5-\sqrt5}{40}},\,±\frac{1\sqrt2}{2},\,±\frac12\right),$$
 * $$\left(±\frac12,\,-\sqrt{\frac{5+2\sqrt5}{20}},\,±\frac12,\,±\frac{1+\sqrt2}{2}\right),$$
 * $$\left(±frac12,\,-\sqrt{\frac{5+2\sqrt5}{20}},\,±\frac{1+\sqrt2}{2},\,±\frac12\right).$$

Representations
A pentagonal-octagonal duoprism has the following Coxeter diagrams:


 * x5o x8o (full symmetry)
 * x4x x5o (octagons as ditetragons)
 * ofx xxx8ooo&#xt (octagonal axial)
 * ofx xxx4xxx&#xt (ditetragonal axial)