Star polyhedron

There are two general kinds of star polyhedron:

• Polyhedra which self-intersect in a repetitive way.
• Concave polyhedra of a particular kind which alternate convex and concave or saddle vertices in a repetitive way. Mathematically these figures are examples of star domains.

Mathematical studies of star polyhedra are usually concerned with regular, uniform polyhedra, or the duals of the uniform polyhedra. All these stars are of the self-intersecting kind.

Self-intersecting star polyhedra

Regular star polyhedra

The regular star polyhedra are self-intersecting polyhedra. They may either have self-intersecting faces, or self-intersecting vertex figures.

There are four regular star polyhedra, known as the Kepler–Poinsot polyhedra. The Schläfli symbol {p,q} implies faces with p sides, and vertex figures with q sides. Two of them have pentagrammic {5/2} faces and two have pentagrammic vertex figures.

There are also an infinite number of regular star dihedra and hosohedra {2,p/q} and {p/q,2} for any star polygon {p/q}. While degenerate in Euclidean space, they can be realised spherically in nondegenerate form.

Uniform and uniform dual star polyhedra

There are many uniform star polyhedra including two infinite series, of prisms and of antiprisms, and their duals.

The uniform and dual uniform star polyhedra are also self-intersecting polyhedra. They may either have self-intersecting faces, or self-intersecting vertex figures or both.

The uniform star polyhedra have regular faces or regular star polygon faces. The dual uniform star polyhedra have regular faces or regular star polygon vertex figures.

Stellations and facettings

Beyond the forms above, there are unlimited classes of self-intersecting (star) polyhedra.

Two important classes are the stellations of convex polyhedra and their duals, the facettings of the dual polyhedra.

For example, the complete stellation of the icosahedron (illustrated) can be interpreted as a self-intersecting polyhedron composed of 20 identical faces, each a (9/4) wound polygon. Below is an illustration of this polyhedron with one face drawn in yellow.

Star polytopes

A similarly self-intersecting polytope in any number of dimensions is called a star polytope.

A regular polytope {p,q,r,...,s,t} is a star polytope if either its facet {p,q,...s} or its vertex figure {q,r,...,s,t} is a star polytope.

In four dimensions, the 10 regular star polychora are called the Schläfli–Hess polychora. Analogous to the regular star polyhedra, these 10 are all composed of facets which are either one of the five regular Platonic solids or one of the four regular star Kepler–Poinsot polyhedra.

For example, the great grand stellated 120-cell, projected orthogonally into 3-space, looks like this:

There are no regular star polytopes in dimensions higher than 4.