Pentagonal duoprism

The pentagonal duoprism or pedip, also known as the pentagonal-pentagonal duoprism, the 5 duoprism or the 5-5 duoprism, is a noble uniform duoprism that consists of 10 pentagonal prisms, with 4 meeting at each vertex. It is also the 10-4 gyrochoron and the square funk prism. It is the first in an infinite family of isogonal pentagonal dihedral swirlchora and also the first in an infinite family of isochoric pentagonal hosohedral swirlchora.

A pentagonal duoprism of edge length 1 contains the vertices of a regular pentachoron of edge length $$\sqrt{\frac{5+\sqrt5}{2}}$$, due to the fact the pentachoron is also the 5-2 step prism.

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

Representations
A pentagonal duoprism has the following Coxeter diagrams:


 * x5o x5o (full symmetry)
 * ofx xxx5ooo&#xt (pentagonal axial)