- Star polyhedron
In

geometry , a**star polyhedron**is apolyhedron 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 authoritiesWho|date=August 2008 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 figure s.There are four regular star polyhedra, known as the

Kepler-Poinsot polyhedra . TheSchläfli symbol {p,q} implies faces with "p" sides, and vertex figures with "q" sides. Two of them havepentagram mic {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 figure s or both.The uniform star polyhedra have regular faces or regular

star polygon faces. The dual uniform star polyhedra have regular faces or regularstar polygon vertex figures.**Examples****Other star polyhedra**Beyond the forms above, there are unlimited classes of self-intersecting (star) polyhedra.

One class is the

isohedral figure s, which are like the uniform figures, but don't require regular faces.For example, the

complete 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 to help visualize the face within the complex figure.**Star polytopes**Higher dimensional intersecting

polytope s are called**star polytopes**.A regular polytope {p,q,r,...,s,t} is a star-polytope if either its facets {p,q,...s}, or its vertex figure {q,r,...,s,t} is a star polytope.

In four dimensions, the 10 regular star polychora, called the Schläfli-Hess polychora. Like the regular star polyhedra, these 10 are all composed of facets which are either one of the five regular

Platonic solid s or one of the four regular starKepler-Poinsot solid s.For example, the

great grand stellated 120-cell , projected orthogonally into 3-space looks like this::**See also***

Star polygon

*Stellation

*Polyhedral compound

*List of uniform polyhedra **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)

* John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, "The Symmetry of Things" 2008, ISBN 978-1-56881-220-5 (Chapter 26. pp. 404: Regular star-polytopes Dimension 3)**External links***Mathworld | urlname=StarPolyhedron | title=Star Polyhedron

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**Uniform star polyhedron**— A display of uniform polyhedra at the Science Museum in London … Wikipedia**Star polygon**— Set of regular star polygons {5/2} {7/2} … Wikipedia**Polyhedron**— Polyhedra redirects here. For the relational database system, see Polyhedra DBMS. For the game magazine, see Polyhedron (magazine). For the scientific journal, see Polyhedron (journal). Some Polyhedra Dodecahedron (Regular polyhedron) … Wikipedia**Nonconvex uniform polyhedron**— In geometry, a nonconvex uniform polyhedron, or uniform star polyhedron, is a self intersecting uniform polyhedron. Each can contain either star polygon faces, star polygon vertex figures or both.The complete set of 57 nonprismatic uniform star… … Wikipedia**Omnitruncated polyhedron**— In geometry, an omnitruncated polyhedron is a truncated quasiregular polyhedron. When they are alternated, they produce the snub polyhedra. All omnitruncated polyhedra are zonohedra. They have Wythoff symbol p q r | and vertex figures as 2p.2q.2r … Wikipedia**Regular polyhedron**— A regular polyhedron is a polyhedron whose faces are congruent (all alike) regular polygons which are assembled in the same way around each vertex. A regular polyhedron is highly symmetrical, being all of edge transitive, vertex transitive and… … Wikipedia**Kepler–Poinsot polyhedron**— In geometry, a Kepler–Poinsot polyhedron is any of four regular star polyhedra. They may be obtained by stellating the regular convex dodecahedron and icosahedron, and differ from these in having regular pentagrammic faces or vertex figures.… … Wikipedia**Uniform polyhedron**— A uniform polyhedron is a polyhedron which has regular polygons as faces and is transitive on its vertices (i.e. there is an isometry mapping any vertex onto any other). It follows that all vertices are congruent, and the polyhedron has a high… … Wikipedia**Kepler-Poinsot polyhedron**— The Kepler Poinsot polyhedra is a popular name for the regular star polyhedra. Each has faces which are congruent regular convex polygons or star polygons and has the same number of faces meeting at each vertex (compare to Platonic solids).There… … Wikipedia**List of Wenninger polyhedron models**— This table contains an indexed list of the Uniform and stellated polyhedra from the book Polyhedron Models , by Magnus Wenninger.The book was written as a guide book to building polyhedra as physical models. It includes templates of face elements … Wikipedia