Stellated Polyhedron

A convex regular Polyhedron. Stellated polyhedra include the Kepler-Poinsot Solids, which consist of three Dodecahedron Stellations and one of the Icosahedron Stellations. Coxeter (1982) shows that 59 Icosahedron Stellations exist. The Cube and the Tetrahedron cannot be stellated. The Octahedron has only one stellation, the Stella Octangula which is a compound of two Tetrahedra.

There are therefore a total of $3+1+(59-1)+1=63$ stellated Polyhedra, although some are Compound Polyhedra and therefore not Uniform Polyhedra. The set of all possible Edges of the stellations can be obtained by finding all intersections on the facial planes.

See also Archimedean Solid Stellation, Dodecahedron Stellations, Icosahedron Stellations, Kepler-Poinsot Solid, Polyhedron, Stella Octangula, Stellated Truncated Hexahedron, Stellation, Uniform Polyhedron

References

Coxeter, H. S. M. The Fifty-Nine Icosahedra. New York: Springer-Verlag, 1982.

Cundy, H. and Rollett, A. Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Publications, 1989.

Wenninger, M. J. Polyhedron Models. Cambridge, England: University Press, 1974.