Star polyhedron

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.

Studies of star polyhedra are usually concerned with regular, uniform polyhedra, or the duals to the uniform polyhedra. All these stars are of the self-intersecting kind. So some authorities^[who?] might argue that the concave kind are not proper stars. But the latter usage is sufficiently widespread that it cannot be ignored. The important thing is to be clear which kind you mean.

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.

These images show each form with a single face colored yellow to show the visible portion of that face.

Uniform and dual uniform 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.

Examples

Uniform polyhedron	Dual polyhedron
The pentagrammic prism is a prismatic star polyhedron. It is composed of two pentagram faces connected by five intersecting square faces.	The pentagrammic dipyramid is also a star polyhedron, representing the dual to the pentagrammic prism. It is face-transitive, composed of ten intersecting isosceles triangles.
The great dodecicosahedron is a star polyhedron, constructed from a single vertex figure of intersecting hexagonal and decagrammic, {10/3}, faces.	The great dodecicosacron is the dual to the great dodecicosahedron. It is face-transitive, composed of 60 intersecting bow-tie-shaped quadrilateral faces.

Other star polyhedra

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

One class is the isohedral figures, which are like the uniform figures, but don't require regular faces.

For example, the complete stellation of the icosahedron can be interpreted as a self-intersecting polyhedron composed of 12 identical faces, each a (9/4) wound polygon. Below is an illustration of this polyhedron with one face drawn in yellow.

Star polytopes

Self-intersecting polytopes in any dimension are called star polytopes.

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.

Notes

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. (VI. Star-polyhedra, XIV. Star-polytopes) (p. 263) [1]
John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 26, Regular star-polytopes, pp. 404–408)

External links