Pentagonal-truncated dodecahedral duoprism

The pentagonal-truncated dodecahedral duoprism or petid is a convex uniform duoprism that consists of 5 truncated dodecahedral prisms, 12 pentagonal-decagonal duoprisms, and 20 triangular-pentagonal duoprisms. Each vertex joins 2 truncated dodecahedral prisms, 1 triangular-pentagonal duoprism, and 2 pentagonal-decagonal duoprisms.

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