Pentagonal-icosahedral duoprism

The pentagonal-icosahedral duoprism or pike is a convex uniform duoprism that consists of 5 icosahedral prisms and 20 triangular-pentagonal duoprisms. Each vertex joins 2 icosahedral prisms and 5 triangular-pentagonal duoprisms.

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