Pentagonal-icosidodecahedral duoprism

The pentagonal-icosidodecahedral duoprism or pid is a convex uniform duoprism that consists of 5 icosidodecahedral prisms, 12 pentagonal duoprisms, and 20 triangular-pentagonal duoprisms. Each vertex joins 2 icosidodecahedral prisms, 2 triangular-pentagonal duoprisms, and 2 pentagonal duoprisms.

Vertex coordinates
The vertices of a pentagonal-icosidodecahedral duoprism of edge length 1 are given by all permutations of the last three coordinates of: as well as all even permutations of the last three coordinates of:
 * $$\left(0,\, \sqrt{\frac{5+\sqrt5}{10}},\,0,\,0,\,±\frac{1+\sqrt5}2\right),$$
 * $$\left(±\frac{1+\sqrt5}{4},\, \sqrt{\frac{5-\sqrt5}{40}},\,0,\,0,\,±\frac{1+\sqrt5}2\right),$$
 * $$\left(±\frac12,\, -\sqrt{\frac{5+2\sqrt5}{20}},\,0,\,0,\,±\frac{1+\sqrt5}2\right),$$
 * $$\left(0,\, \sqrt{\frac{5+\sqrt5}{10}},\,±\frac12,\,±\frac{1+\sqrt5}4,\,±\frac{3+\sqrt5}4\right),$$
 * $$\left(±\frac{1+\sqrt5}{4},\, \sqrt{\frac{5-\sqrt5}{40}},\,±\frac12,\,±\frac{1+\sqrt5}4,\,±\frac{3+\sqrt5}4\right),$$
 * $$\left(±\frac12,\, -\sqrt{\frac{5+2\sqrt5}{20}},\,±\frac12,\,±\frac{1+\sqrt5}4,\,±\frac{3+\sqrt5}4\right).$$