geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

, a star polyhedron is a

polyhedron In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices. A convex polyhedron is the convex hull of finitely many points, not all on th ...

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 domain In geometry, a set S in the Euclidean space \R^n is called a star domain (or star-convex set, star-shaped set or radially convex set) if there exists an s_0 \in S such that for all s \in S, the line segment from s_0 to s lies in S. This defini ...

s. Mathematical studies of star polyhedra are usually concerned with regular,

uniform A uniform is a variety of clothing worn by members of an organization while participating in that organization's activity. Modern uniforms are most often worn by armed forces and paramilitary organizations such as police, emergency services, se ...

polyhedra, or the

duals ''Duals'' is a compilation album by the Irish rock band U2. It was released in April 2011 to u2.com subscribers. Track listing :* "Where the Streets Have No Name" and "Amazing Grace" are studio mix of U2's performance at the Rose Bowl, Pas ...

of the uniform polyhedra. All these stars are of the self-intersecting kind.

Self-intersecting star polyhedra

Regular star polyhedra

The regular star polyhedra are self-intersecting polyhedra. They may either have self-intersecting

faces The face is the front of an animal's head that features the eyes, nose and mouth, and through which animals express many of their emotions. The face is crucial for human identity, and damage such as scarring or developmental deformities may affe ...

, or self-intersecting

vertex figure In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off. Definitions Take some corner or Vertex (geometry), vertex of a polyhedron. Mark a point somewhere along each connect ...

s. There are four regular star polyhedra, known as the Kepler–Poinsot polyhedra. The

Schläfli symbol In geometry, the Schläfli symbol is a notation of the form \ that defines regular polytopes and tessellations. The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, who generalized Euclidean geometry to more ...

implies faces with ''p'' sides, and vertex figures with ''q'' sides. Two of them have

pentagram A pentagram (sometimes known as a pentalpha, pentangle, or star pentagon) is a regular five-pointed star polygon, formed from the diagonal line segments of a convex (or simple, or non-self-intersecting) regular pentagon. Drawing a circle aroun ...

mic 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 A dihedron is a type of polyhedron, made of two polygon faces which share the same set of ''n'' edges. In three-dimensional Euclidean space, it is degenerate if its faces are flat, while in three-dimensional spherical space, a dihedron with flat ...

and

hosohedra In spherical geometry, an -gonal hosohedron is a tessellation of lunes on a spherical surface, such that each lune shares the same two polar opposite vertices. A regular -gonal hosohedron has Schläfli symbol with each spherical lune havi ...

and for any star polygon . 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 In geometry, a uniform star polyhedron is a self-intersecting uniform polyhedron. They are also sometimes called nonconvex polyhedra to imply self-intersecting. Each polyhedron can contain either star polygon faces, star polygon vertex figures, ...

including two infinite series, of prisms and of antiprisms, and their

. The

and dual uniform star polyhedra are also self-intersecting polyhedra. They may either have self-intersecting

, or self-intersecting

s or both. The uniform star polyhedra have regular faces or regular

star polygon In geometry, a star polygon is a type of non-convex polygon. Regular star polygons have been studied in depth; while star polygons in general appear not to have been formally defined, certain notable ones can arise through truncation operations ...

faces. The dual uniform star polyhedra have regular faces or regular

vertex figures.

Stellations and facettings

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

stellation In geometry, stellation is the process of extending a polygon in two dimensions, polyhedron in three dimensions, or, in general, a polytope in ''n'' dimensions to form a new figure. Starting with an original figure, the process extends specific el ...

s of convex polyhedra and their duals, the

facetting Stella octangula as a faceting of the cube In geometry, faceting (also spelled facetting) is the process of removing parts of a polygon, polyhedron or polytope, without creating any new vertices. New edges of a faceted polyhedron may be cre ...

s 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 elementary geometry, a polytope is a geometric object with flat sides (''faces''). Polytopes are the generalization of three-dimensional polyhedra to any number of dimensions. Polytopes may exist in any general number of dimensions as an -d ...

in any number of dimensions is called a star polytope. A regular polytope is a star polytope if either its facet or its vertex figure 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 solid In geometry, a Platonic solid is a convex, regular polyhedron in three-dimensional Euclidean space. Being a regular polyhedron means that the faces are congruent (identical in shape and size) regular polygons (all angles congruent and all edges c ...

s or one of the four regular star Kepler–Poinsot polyhedra. For example, the

great grand stellated 120-cell In geometry, the great grand stellated 120-cell or great grand stellated polydodecahedron is a regular star 4-polytope with Schläfli symbol , one of 10 regular Schläfli-Hess 4-polytopes. It is unique among the 10 for having 600 vertices, and ...

, projected orthogonally into 3-space, looks like this: : There are no regular star polytopes in dimensions higher than 4.

Star-domain star polyhedra

A polyhedron which does not cross itself, such that all of the interior can be seen from one interior point, is an example of a

. The visible exterior portions of many self-intersecting star polyhedra form the boundaries of star domains, but despite their similar appearance, as abstract polyhedra these are different structures. For instance, the small stellated dodecahedron has 12 pentagram faces, but the corresponding star domain has 60 isosceles triangle faces, and correspondingly different numbers of vertices and edges. Polyhedral star domains appear in various types of architecture, usually religious in nature. For example, they are seen on many baroque churches as symbols of the

Pope The pope ( la, papa, from el, πάππας, translit=pappas, 'father'), also known as supreme pontiff ( or ), Roman pontiff () or sovereign pontiff, is the bishop of Rome (or historically the patriarch of Rome), head of the worldwide Cathol ...

who built the church, on Hungarian churches and on other religious buildings. These stars can also be used as decorations.

Moravian star A Moravian star (german: Herrnhuter Stern) is an illuminated Advent, Christmas, or Epiphany decoration popular in Germany and in places in Europe and America where there are Moravian congregations, notably the Lehigh Valley of Pennsylvania and ...

s are used for both purposes and can be constructed in various forms.

Notes

References

* Coxeter, H.S.M.,

M. S. Longuet-Higgins Michael Selwyn Longuet-Higgins FRS (8 December 1925 – 26 February 2016) was a mathematician and oceanographer at the Department of Applied Mathematics and Theoretical Physics (DAMTP), Cambridge University, England and Institute for Nonlinea ...

and J.C.P Miller, Uniform Polyhedra, ''Phil. Trans.'' 246 A (1954) pp. 401–450. *Coxeter, H.S.M., ''

Regular Polytopes In mathematics, a regular polytope is a polytope whose symmetry group acts transitively on its flags, thus giving it the highest degree of symmetry. All its elements or -faces (for all , where is the dimension of the polytope) — cells, ...

'', 3rd. ed., Dover Publications, 1973. . (VI. Star-polyhedra, XIV. Star-polytopes) (p. 263

* John Horton Conway, John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, ''The Symmetries of Things'' 2008, (Chapter 26, Regular star-polytopes, pp. 404–408) * Tarnai, T., Krähling, J. and Kabai, S.; "Star polyhedra: from St. Mark's Basilica in Venice to Hungarian Protestant churches", Paper ID209, ''Proc. of the IASS 2007, Shell and Spatial Structures: Structural Architecture-Towards the Future Looking to the Past'', University of IUAV, 2007

o

External links

*{{Mathworld , urlname=StarPolyhedron , title=Star Polyhedron Polyhedra