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Concept
Star polyhedron
Summary
In geometry, a star polyhedron is a polyhedron which has some repetitive quality of nonconvexity giving it a star-like visual quality.
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.
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.
These images show each form with a single face colored yellow to show the visible portion of that face.
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.
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.
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.
Official source
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