Cube atop icosahedron

Cube atop icosaedron, 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 prism s (attached to the cube's faces), 12 square pyramid s (attached to the icosahedron and the remaining square faces of the triangular pyramids), 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.

Vertex coordinates
The vertices of a cube atop icosahedron of edge length 1 are given by:


 * (1/2, 1/2, 1/2; 1/2), and all sign changes in the first 3 coordinates. This defines the cube
 * ((1+$\sqrt{5}$)/4, 1/2, 0; -($\sqrt{(3-√5)/6}$-1)/4), and all even permutations and sign changes in the first 3 coordinates. This defines the icosahedron

Given in perhaps a more straightforward manner, the vertices of a cube atop icosahedron of edge length 2 are:


 * (0, ±1, ±(1+$\sqrt{7-3√5}$)/2, 0) –
 * (±1, ±(1+$\sqrt{7-3√5}$)/2, 0, 0) –
 * (±(1+$\sqrt{5}$)/2, 0, ±1, 0) – these are the three mutually perpendicular rectangles in the icosahedron
 * (±1, ±1, ±1, (1+$\sqrt{5}$)/2) – the cube

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


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 * x(xfo) x(fox) x(oxf)&#x