by . It was noticed that this optimal value is obtained in a Bravais lattice by . Since the rhombicuboctahedron is contained in a rhombic dodecahedron whose inscribed sphere is identical to its own inscribed sphere, the value of the optimal packing fraction is a corollary of the Kepler conjecture: it can be achieved by putting a rhombicuboctahedron in each cell of the rhombic dodecahedral honeycomb, and it cannot be surpassed, since otherwise the optimal packing density of spheres could be surpassed by putting a sphere in each rhombicuboctahedron of the hypothetical packing which surpasses it. In the arts The 1495 Portrait of Luca Pacioli, traditionally attributed to Jacopo de' Barbari, includes a glass rhombicuboctahedron half-filled with water, which may have been painted by Leonardo da Vinci. The first printed version of the rhombicuboctahedron was by Leonardo and appeared in Pacioli's Divina proportione (1509). A spherical 180° × 360° panorama can be projected onto any polyhedron; but the rhombicuboctahedron provides a good enough approximation of a sphere while being easy to build. This type of projection, called Philosphere, is possible from some panorama assembly software. It consists of two images that are printed separately and cut with scissors while leaving some flaps for assembly with glue. Objects The Freescape games Driller and Dark Side both had a game map in the form of a rhombicuboctahedron. The "Hurry-Scurry Galaxy" and "Sea Slide Galaxy" in the videogame Super Mario Galaxy have planets in the similar shape of a rhombicuboctahedron. Sonic the Hedgehog 3s Icecap Zone features pillars topped with rhombicuboctahedra. During the Rubik's Cube craze of the 1980s, at least two twisty puzzles sold had the form of a rhombicuboctahedron (the mechanism was similar to that of a Rubik's Cube). Related polyhedra The rhombicuboctahedron is one of a family of uniform polyhedra related to the cube and regular octahedron. Symmetry mutations This polyhedron is topologically related as a part of sequence of cantellated polyhedra with vertex figure (3.4.n.4), and continues as tilings of the hyperbolic plane. These vertex-transitive figures have (*n32) reflectional symmetry. Vertex arrangement It shares its vertex arrangement with three nonconvex uniform polyhedra: the stellated truncated hexahedron, the small rhombihexahedron (having the triangular faces and six square faces in common), and the small cubicuboctahedron (having twelve square faces in common). Rhombicuboctahedral graph The rhombicuboctahedral graph''' is the graph of vertices and edges of the rhombicuboctahedron. It has 24 vertices and