Stellate polyhedrons

Introduction:

Stellation is the process of constructing polyhedra by extending the facial planes past the polyhedron edges of a given polyhedron until they intersect . 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 are sometimes added to constrain allowable stellations.

There are no stellations of the cube or tetrahedron . The only stellated form of the octahedron is the stella octangula, which is a compound of two tetrahedra. There are three dodecahedron stellations: the small stellated dodecahedron, great dodecahedron, and great stellated dodecahedron . Coxeter et al. (1982) shows that 58 icosahedron stellations exist (although Coxeter et al. include the icosahedron itself in their count, for a total of 59), subject to certain restrictions.

The Kepler-Poinsot solids consist of the three dodecahedron stellations and one of the icosahedron stellations

1. Octahedron stellation

2. Kepler-Poinsot solids

3. Icosahedron stellations

4. Rhombic triacontahedron stellations

5. References