Cuboctahedron

The cuboctahedron is a convex polyhedron with eight triangular faces and six square faces, twelve identical vertices and twenty-four edges, and it can be formed by cutting the corners off a cube exactly at the midpoints of its edges, a process called rectification, or equivalently by rectifying a regular octahedron, reflecting the fact that the cube and octahedron are dual to one another. It is one of the thirteen Archimedean solids, the convex polyhedra built from more than one type of regular polygon meeting in an identical arrangement at every vertex, a class of solids credited to Archimedes in a work that has not survived and is known only through later accounts, and cataloged completely for the first time in a surviving written source by the German astronomer Johannes Kepler in his 1619 book Harmonices Mundi. The cuboctahedron's twelve vertices, when connected to a common center, give the closest possible packing arrangement of twelve spheres around a single central sphere, so that the shape appears naturally in the geometry of cubic close packing and hexagonal close packing of equal spheres.