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.

Vertex coordinates
The vertices of a pentagonal-icosahedral duoprism of edge length 1 are given by all even permutations and all sign changes of the last three coordinates of:
 * (0, $\sqrt{450+90√5}$/10, 0, 1/2, (1+$\sqrt{50+10√5}$)/4)
 * (±(1+$\sqrt{5}$)/4, $\sqrt{5}$/20, 0, 1/2, (1+$\sqrt{50–10√5}$)/4)
 * (±1/2, -$\sqrt{5}$/10, 0, 1/2, (1+$\sqrt{25+10√5}$)/4)