The process of stellation takes a given polyhedron and extends its surface planes until they meet other planes, resulting in a larger solid, a stellation of the interior solid. This page describes the stellations of the cuboctahedron, which is an interesting example because it is quasi-regular. The simplest stellation (other than the "zeroth stellation" which is the cuboctahedron itself) is the compound of the cube and octahedron, illustrated at right. The intersection of the cube and the octahedron is the inner cuboctahedron. (Go inside to see it.)
Here are some other stellations of the cuboctahedron. As with other stellated polyhedra, there is no essential order for listing them, so they appear here in a no particular order:
Three of these appear in Magnus Wenninger's Polyhedron Models (see the references) along with patterns for their construction in paper. His figure numbers are given here [in square brackets].