Great rhombicuboctahedron

The great rhombicuboctahedron or girco, also commonly known as the truncated cuboctahedron, is one of the 13 Archimedean solids. It consists of 12 squares, 8 hexagons, and 6 octagons, with one of each type of face meeting per vertex. It can be obtained by cantitruncation of the cube or octahedron, or equivalently by truncating the vertices of a cuboctahedron and then adjusting the edge lengths to be all equal.

This is one of three Wythoffian non-prismatic polyhedra whose Coxeter diagram has all ringed nodes, the other two being the truncated tetratetrahedron and the great rhombicosidodecahedron.

It can be alternated into the snub cube.

Vertex coordinates
A great rhombicuboctahedron of edge length 1 has vertex coordinates given by all permutations of:
 * (±(1+2$\sqrt{13+6√2}$)/2, ±(1+$\sqrt{2}$)/2, ±1/2).

Representations
A great rhombicuboctahedron has the following Coxeter diagrams:


 * x4x3x (full symmetry)
 * xxwwxx4xuxxux&#xt (BC2 axial, octagon-first)
 * wx3xx3xw&#zx (A3 symmetry, as hull of two inverse great rhombitetratetrahedra)
 * Xwx xxw4xux&#zx (BC2×A1 symmetry)
 * xxuUxwwx3xwwxUuxx&#xt (A2 axial, hexagon-first)