Cube atop icosahedron

The cube atop icosahedron, or cubaike, is a CRF segmentochoron (designated K-4.21 on Richard Klitzing's list). It consists of a cube and an icosahedron located on parallel hyperplanes, connected by 6 triangular prisms (attached to the cube's faces), 12 square pyramids (attached to the icosahedron and the remaining square faces of the triangular prisms), and 8 tetrahedra that fill in the remaining gaps.

The drastically different symmetries of the two "bases" set the cube atop icosahedron apart from other segmentochora. In fact the common symmetry here is only pyritohedral.

This segmentochoron may at first not seem to be related to any other polychora, but it can be thought of as a subset of the vertices of the hexacosichoron, when seen vertex-first. The top cube forms some of the vertices of the dodecahedral layer, while the bottom icosahedron is the larger icosahedral layer from the hexacosichoron.

Vertex coordinates
Coordinates for the vertices of a cube atop icosahedron of edge length 1 are given by:

and all cyclic permutations in the first 3 coordinates of
 * $$\left(±\frac12,\,±\frac12,\,±\frac12,\,\frac12\right),$$
 * $$\left(±\frac{1+\sqrt5}{4},\,±\frac12,\,0,\,\frac{1-\sqrt5}{4}\right).$$

The first set of coordinates defines the cube, and the second set defines the icosahedron.

Representations
A cube atop icosahedron can be represented by the following Coxeter diagrams:


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