Icosahedron atop icosidodecahedron

Icosahedron atop icosidodecahedron, or ikaid, is a CRF segmentochoron (designated K-4.137 on Richard Klitzing's list). As the name suggests, it consists of an icosahedron and an icosidodecahedron as bases, connected by 20 octahedra and 12 pentagonal pyramids.

It can also be seen as a rectification of the CRF icosahedral pyramid.

It is also the cap of the rectified hexacosichoron in icosahedron-first orientation.

Vertex coordinates
The vertices of an icosahedron atop icosidodecahedron segmentochoron of edge length 1 are given by:
 * $$\left(±\frac{1+\sqrt5}{4},\,±\frac12,\,0,\,\frac{\sqrt5-1}{4}\right)$$ and all even permutations of first three coordinates
 * $$\left(±\frac{1+\sqrt5}{2},\,0,\,0,\,0\right)$$ and all permutations of first three coordinates
 * $$\left(±\frac{3+\sqrt5}{4},\,±\frac{1+\sqrt5}{4},\,±\frac12,\,0\right)$$ and all even permutations of first there coordinates