Star polyhedron
From Wikipedia, the free encyclopedia
In geometry, the term star polyhedron does not seem to have been propely defined, even though it is in common use. We can say that 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.
Studies of star polyhedra are usually concerned with uniform and regular polyhedra. All these stars are of the self-intersecting kind. So some authorities might argue that the concave kind are not proper stars. But the latter usage seems so common that it cannot be ignored. The important thing is to be clear which kind you mean.
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[edit] Uniform and regular star polyhedra
There are many uniform star polyhedra including two infinite series, of prisms and of antiprisms.
There are four regular star polyhedra, known as the Kepler-Poinsot polyhedra.
All these are self-intersecting. There are no uniform or regular star polyhedra of the concave kind.
[edit] Star polytopes
Higher dimensional intersecting polytopes are called star polytopes; for example, the 10 regular star polychora, called the Schläfli-Hess polychora.
[edit] See also
[edit] References
- Coxeter, H.S.M., M.S. Longuet-Higgins and J.C.P Miller, Uniform Polyhedra, Phil. Trans. 246 A (1954) pp. 401-450.
- Coxeter, H.S.M., Regular Polytopes, 3rd. ed., Dover Publications, 1973. ISBN 0-486-61480-8.
