Quasirhombicuboctahedron

The quasirhombicuboctahedron is the name given by Wenninger (1989, p.  132) to the uniform polyhedron with Maeder index 17 (Maeder 1997), Wenninger index 85 (Wenninger 1989), Coxeter index 59 (Coxeter et al. 1954), Har'El index 22 (Har'El 1993), faces , Schläfli symbol r', and Wythoff symbol .

Unfortunately, other authors (e.g., Maeder 1997) use the term "great rhombicuboctahedron" to refer to this solid, despite the fact that "'great rhombicuboctahedron" is commonly used to refer to a distinct (and more common) Archimedean solid (Cundy and Rowlett 1989, p. 106).

The quasirhombicuboctahedron is implemented in the Wolfram Language as UniformPolyhedron[85], UniformPolyhedron["GreatRhombicuboctahedron"], UniformPolyhedron["Coxeter", 59], UniformPolyhedron["Kaleido", 22], UniformPolyhedron["Uniform", 17], or UniformPolyhedron["Wenninger", 85]. It is also implemented in the Wolfram Language as PolyhedronData["Quasirhombicuboctahedron"].

The skeleton of the quasirhombicuboctahedron is the small rhombicuboctahedral graph, illustrated above.

The dual of the quasirhombicuboctahedron is the great deltoidal icositetrahedron.

Its circumradius for unit edge length is

The convex hull of the quasirhombicuboctahedron is the Archimedean truncated cube, whose dual is the small triakis octahedron, so the dual of the quasirhombicuboctahedron (i.e., the great deltoidal icositetrahedron) is one of the stellations of the small triakis octahedron (Wenninger 1983, p. 57).