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. == Star-domain star polyhedra ==