Icosahedron atop dodecahedron

The icosahedron atop dodecahedron, or ikadoe, is a CRF segmentochoron (designated K-4.78 on Richard Klitzing's list). As the name suggests, it consists of a dodecahedron and an icosahedron as bases, connected by 12 pentagonal pyramids and 20+30 tetrahedra.

It is also commonly referred to as a dodecahedral or icosahedral antiprism, as the two bases are a pair of dual polyhedra.

The icosahedron atop dodecahedron can also be obtained from the hexacosichoron as a monostratic stack. This is more readily seen from the hexacosichoron's vertex-first projection (where the two bases are concentric) or its edge-first projection (where the two bases are flattened).

Vertex coordinates
Coordinates for the vertices of an icosahedron atop dodecahedron of edge length 1 are given by:
 * $$\left(±\frac{1+\sqrt5}{4},\,±\frac12,\,0,\,±\frac12\right)$$ and all permutations of the first three coordinates,
 * $$\left(±\frac{3+\sqrt5}{4},\,±\frac12,\,0,\,0\right)$$ and all permutations of the first three coordinates,
 * $$\left(±\frac{1+\sqrt5}{4},\,±\frac{1+\sqrt5}{4},\,±\frac{1+\sqrt5}{4},\,0\right).$$