Rectified pentagonal duoprism

The rectified pentagonal duoprism or repdip is a convex isogonal polychoron that consists of 10 rectified pentagonal prisms and 25 tetragonal disphenoids. 3 rectified pentagonal prisms and 2 tetragonal disphenoids join at each vertex. It can be formed by rectifying the pentagonal duoprism.

It can also be formed as the convex hull of 2 oppositely oriented semi-uniform pentagonal duoprisms, where the edges of one pentagon are $$\sqrt5-1 ≈ 1.23607$$ times as long as the edges of the other.

The ratio between the longest and shortest edges is 1:$$\frac{\sqrt{3+\sqrt5}}{2}$$ ≈ 1:1.14412.

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