degrees creates the pseudo­rhombi­cubocta­hedron. Both of these polyhedra have the same vertex figure: 3.4.4.4. There are three pairs of parallel planes that each intersect the rhombicuboctahedron in a regular octagon. The rhombicuboctahedron may be divided along any of these to obtain an octagonal prism with regular faces and two additional polyhedra called square cupolae, which count among the Johnson solids; it is thus an elongated square orthobicupola. These pieces can be reassembled to give a new solid called the elongated square gyrobicupola or pseudorhombicuboctahedron, with the symmetry of a square antiprism. In this the vertices are all locally the same as those of a rhombicuboctahedron, with one triangle and three squares meeting at each one, but are not all identical with respect to the entire polyhedron, since some are closer to the symmetry axis than others. Orthogonal projections The rhombicuboctahedron has six special orthogonal projections, centered, on a vertex, on two types of edges, and three types of faces: triangles, and two squares. The last two correspond to the B2 and A2 Coxeter planes. Spherical tiling The rhombicuboctahedron can also be represented as a spherical tiling, and projected onto the plane via a stereographic projection. This projection is conformal, preserving angles but not areas or lengths. Straight lines on the sphere are projected as circular arcs on the plane. Pyritohedral symmetry A half symmetry form of the rhombicuboctahedron, , exists with pyritohedral symmetry, [4,3+], (3*2) as Coxeter diagram , Schläfli symbol s2{3,4}, and can be called a cantic snub octahedron. This form can be visualized by alternatingly coloring the edges of the 6 squares. These squares can then be distorted into rectangles, while the 8 triangles remain equilateral. The 12 diagonal square faces will become isosceles trapezoids. In the limit, the rectangles can be reduced to edges, and the trapezoids become triangles, and an icosahedron is formed, by a snub octahedron construction, , s{3,4}. (The compound of two icosahedra is constructed from both alternated positions.) Algebraic properties Cartesian coordinates Cartesian coordinates for the vertices of a rhombicuboctahedron centred at the origin, with edge length 2 units, are all the even permutations of (±1, ±1, ±(1 + )). If the original rhombicuboctahedron has unit edge length, its dual strombic icositetrahedron has edge lengths Area and volume The area A and the volume V of the rhombicuboctahedron of edge length a are: Close-packing density The optimal packing fraction of rhombicuboctahedra is given