In
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 ...
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 defin ...
s.
Mathematical studies of star polyhedra are usually concerned with
regular
The term regular can mean normal or in accordance with rules. It may refer to:
People
* Moses Regular (born 1971), America football player
Arts, entertainment, and media Music
* "Regular" (Badfinger song)
* Regular tunings of stringed instrum ...
,
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, ...
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, 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 of a polyhedron. Mark a point somewhere along each connected edge. Draw lines ...
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 mor ...
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 aro ...
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 f ...
and
hosohedra 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 including two infinite series, of
prisms and of antiprisms, and their
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, ...
.
The
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 ...
and
dual
Dual or Duals may refer to:
Paired/two things
* Dual (mathematics), a notion of paired concepts that mirror one another
** Dual (category theory), a formalization of mathematical duality
*** see more cases in :Duality theories
* Dual (grammatical ...
uniform star polyhedra are also self-intersecting polyhedra. They may either have self-intersecting
faces, 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 of a polyhedron. Mark a point somewhere along each connected edge. Draw lines ...
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 operation ...
faces. The dual 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 operation ...
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
facettings of the dual polyhedra.
For example, the
complete stellation of the icosahedron
In geometry, the complete or final stellation of the icosahedron is the outermost stellation of the icosahedron, and is "complete" and "final" because it includes all of the cells in the icosahedron's stellation diagram. That is, every three inte ...
(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 ...
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 e ...
s 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.
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
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 defin ...
. The visible exterior portions of many self-intersecting star polyhedra form the boundaries of star domains,
but despite their similar appearance, as
abstract polyhedra
In mathematics, an abstract polytope is an algebraic partially ordered set which captures the dyadic property of a traditional polytope without specifying purely geometric properties such as points and lines.
A geometric polytope is said to be ...
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.
See also
*
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 operation ...
*
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 ...
*
Polyhedral compound
In geometry, a polyhedral compound is a figure that is composed of several polyhedra sharing a common centre. They are the three-dimensional analogs of polygonal compounds such as the hexagram.
The outer vertices of a compound can be connec ...
*
List of uniform polyhedra
*
List of uniform polyhedra by Schwarz triangle
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. . (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