Depending on how the face planes of a given polyhedron divide space, there may be several ways to extend them and choose regions which they bound. Starting with an octahedron, stellation produces only one possibility: the compound of two tetrahedra (stella octangula). This model shows two tetrahedra in outline; their face planes are generated by extending the faces of the inner octahedron.
Observe that two planes which meet at an edge of the stellation will
not share an edge of the inner polyhedron. So there are no stellations
of the cube --- nonadjacent faces are parallel,
and so never meet to form a finite solid.
Starting with the dodecahedron, the three stellations possible: the small stellated dodecahedron, the great dodecahedron, and the great stellated dodecahedron are shown above. In that order, each is a continuation of the face planes of the previous one. These three stellations fall within the class of Kepler-Poinsot polyhedra. Try flying inside them to find the inner dodecahedron in each.
The remaining Kepler-Poinsot solid, the great icosahedron, is a stellation of the icosahedron. Travel inside this great icosahedron to find its inner icosahedron. Starting with the icosahedron, it turns out that there are 59 possible stellations, of which the great icosahedron is only one. To learn about the others, read about the 59 Stellations of the Icosahedron.
The first stellation of a convex quasi-regular solid is a compound of two dual regular solids. For example, the first stellation of the cuboctahedron is the compound of the cube and octahedron. The first stellation of the icosidodecahedron is the compound of the icosahedron and the dodecahedron.
There are three stellations of the rhombic dodecahedron: