Stellation is the process of constructing polyhedra by extending the facial planes past the polyhedron edges of a given polyhedron until they intersect (Wenninger 1989). The set of all possible polyhedron edges of the stellations can be obtained by finding all intersections on the facial planes. Since the number and variety of intersections can become unmanageable for complicated polyhedra, additional rules (e.g., Miller's rules) are sometimes added to constrain allowable stellations.
There are a number of subtlties and ambiguities about the stellation process. As noted by Cromwell (1997, pp. 263-264), "The stellation process may seem clear enough, but there is some ambiguity about how we should interpret the result. For example, is the great dodecahedron composed of twelve regular pentagons, or 60 isosceles triangles.... This freedom on interpretation means that there are complementary ways to think about the process of face-stellation."
There are no stellations of the cube or tetrahedron (Wenninger 1989, p. 35), although the stella octangula is sometimes improperly called the "stellated tetrahedron." The only stellated form of the octahedron is the stella octangula, which is a compound of two tetrahedra (Wenninger 1989, pp. 35 and 37). There are three dodecahedron stellations: the (non-convex hulls of the) small stellated dodecahedron, great dodecahedron, and great stellated dodecahedron (Wenninger 1989, pp. 35 and 38-40). Coxeter et al. (1999) shows that 58 icosahedron stellations exist (although Coxeter et al. include the icosahedron itself in the count giving a total of 59 "stellations"), subject to certain restrictions.
The (non-convex hulls of the) Kepler-Poinsot polyhedra consist of the three dodecahedron stellations and one of the icosahedron stellations, and these are the only stellations of Platonic solids which are uniform polyhedra.
Archimedean stellations have received much less attention than Platonic stellations. However, there are three rhombic dodecahedron stellations (Wells 1991, pp. 216-217).
