 A {5/2} star polygon constructed in a pentagon. | In geometry, a star polygon is a complex, equilateral equiangular polygon, so named for its starlike appearance, created by connecting one vertex of a simple, regular, n-sided polygon to another, non-adjacent vertex and continuing the process until the original vertex is reached again. For instance, in a regular pentagon, a five-pointed star can be obtained by drawing a line from the first to the third vertex, from the third vertex to the fifth vertex, from the fifth vertex to the second vertex, from the second vertex to the fourth vertex, and from the fourth vertex to the first vertex. This involves repeated addition with a modulus of n, where n is the number of sides of the polygon and the number x to be repeatedly added is greater than 1 and less than n-1, or: 1 < x < n-1. The notation for such a polygon is {n/x} (see Schläfli symbol), which is equal to {n/n-x}. The polygon at right is {5/2}. Image File history File links Pentagram_in_pentagon. ...
Image File history File links Pentagram_in_pentagon. ...
Table of Geometry, from the 1728 Cyclopaedia. ...
Look up Polygon in Wiktionary, the free dictionary. ...
The Pleiades star cluster A star is a massive body of plasma in outer space that is currently producing or has produced energy through nuclear fusion. ...
In geometry, a vertex (Latin: whirl, whirlpool; plural vertices) is a corner of a polygon (where two sides meet) or of a polyhedron (where three or more faces and an equal number of edges meet). ...
Mathematical meanings Especially in British/European usage, the modulus of a number is its absolute value. ...
In mathematics, the Schläfli symbol is a simple notation that gives a summary of some important properties of a particular regular polytope. ...
Examples
A six pointed star can be created using a compas and a straight edge: Make a circle of any size with the compas. Move the point of the compas from the center to the outside edge, without changing the compas, and then walk the compas around the outside edge of the circle. Take a straight edge and make lines between every other mark and you can create a star. It has been stated that creating a five pointed star in a similar fashion was one of the esoteric teachings of the Pythagoreans. Divulging that secret was punishable by death. If the modulus n is evenly divisible by x, the star polygon obtained will be a regular polygon with n/x sides. A new figure is obtained by rotating these regular n/x-gons one vertex to the left on the original polygon until the number of vertices rotated equals n/x minus one, and combining these figures. An extreme case of this is where n is an even number and n/x is 2, producing a figure consisting of n/2 straight line segments; this is called a degenerate star polygon. Image File history File links Green_pentagram. ...
Image File history File links Obtuse_heptagram. ...
Image File history File links Acute_heptagram. ...
Image File history File links Octagram. ...
In mathematics, a degenerate case is a limiting case in which a class of object changes its nature so as to belong to another, usually simpler, class. ...
Star figures
 star figure Hexagram {6/2} In other cases where n and x have a common factor, a star polygon for a lower n is obtained, and rotated versions can be combined. These figures are called star figures or improper star polygons or polygon compounds. The same notation {n/x} is used for them. The non-degenerate example with the smallest n is the complex {10/4} consisting of two pentagrams, differing by a rotation of 36°. Image File history File links Hexagram. ...
Image File history File links Hexagram. ...
Geometric interiors Convex polygons divide space into two clear regions, inside and outside. In contrast star polygons leave an ambiguity of interpretations. The diagram below demonstrates three interpretations of a pentagram.
 Image File history File links Pentagram_interior. ...
- The first converts it to a concave decagon (10-pointed polygon).
- The middle interpretation recognizes space is still divided into two regions defined by following a directional path and saying everything left and right from each edge are opposite sides. This makes the most interior region actually "outside", and in general you can determine inside by an odd/even rule of counting how many edges are intersected from a point along a ray to infinity.
- The last interpretation considers multiple levels of interior regions. This interpretation, like the first must also consider geometric intersections of the edges. The resulting shape can no longer be considered a simple polygon but a network of edge-attached polygons. (This image, if regular, could also be considerd the net of a pentagonal pyramid polyhedron!)
Categories: Polyhedra | Stub ...
In geometry, the Pentagonal pyramid is one of the Johnson solids (J2). ...
Example star prisms with different face interior renderings {7/2} heptagrammic prism: In geometry, the {7/2} septagrammic prism is one of an infinite set of nonconvex prisms formed by square sides and two regular star polygon caps, in this case two {7/2} septagrams. ...
Simple (binary) face interior |
Complex face interior | Image File history File links Download high resolution version (640x639, 14 KB) Summary Uniform septagrammic prism 4. ...
Image File history File links Download high resolution version (1000x1000, 279 KB) Licensing Robert Webb produced this image for Wikipedia upon my request, and offered the credit statement below: File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ...
Symmetry Star polygons can be thought of as diagramming cosets of the subgroups  of the finite group  . In mathematics, if G is a group, H a subgroup of G, and g an element of G, then gH = { gh : h an element of H } is a left coset of H in G, and Hg = { hg : h an element of H } is a right coset of H in G...
In group theory, given a group G under a binary operation *, we say that some subset H of G is a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H is a group...
In mathematics, a finite group is a group which has finitely many elements. ...
The symmetry group of {n/k} is dihedral group Dn of order 2n, independent of k. The symmetry group of an object (e. ...
This article may be confusing for some readers, and should be edited to enhance clarity. ...
 An {8/3} star polygon (octagram) constructed in an octagon Certain star polygons feature prominently in art and culture. These include: Image File history File links Octagram. ...
Image File history File links Octagram. ...
- The {5/2} star polygon is known as a pentagram, pentalpha or pentangle, and is considered by some to have occult significance.
- The {8/3} star polygon (octagram), and the complex star polygon of two {16/6} polygons, are frequent geometrical motifs in Mughal Islamic art and architecture; the first is on the
coat of arms of Azerbaijan.
- The complex {8/2} star polygon (i.e. two squares) is known as the Star of Lakshmi, and figures in Hinduism;
- The simplest non-degenerate complex star polygon is two {6/2} polygons (i.e., triangles), the hexagram (Star of David, Seal of Solomon).
- The {7/3} and {7/2} star polygons are known as heptagrams. They also have occult significance, for example in the Kabbalah and in Wicca.
- An eleven pointed star is called a hendecagram.
The star polygons were first studied by Thomas Bradwardine. The pentangle otherwise known as the pentogram, first appeared in a 14th century poem Sir Gawain and the Green knight. It was used on Sir Gawains shield, it was a token of truth. ...
To meet Wikipedias quality standards, this article or section may require cleanup. ...
The Mughal Empire, (Persian: دولت٠مغل) was an empire that at its greatest territorial extent ruled most of the Indian Subcontinent, then known as Hindustan, and parts of Afghanistan and Persia, between 1526 and 1707. ...
Mediums of Islamic art Islamic art throughout history has been mainly abstract and decorative, portraying geometric, floral, Arabesque, and calligraphic designs. ...
Islamic architecture, a part of the Islamic studies, is the entire range of architecture that has evolved within Muslim culture in the course of the history of Islam. ...
Coat of Arms of Azerbaijan public Other version at Image:Azerbaijan coa large. ...
The complex {8/2} star polygon (i. ...
Hinduism {Sanskrit/Hindi - HindÅ« Dharma, also known as SanÄtana (eternal) Dharma and Vaidika (of the Vedas) Dharma} is the religion based on the Vedas as well as other traditional scriptures and beliefs. ...
A hexagram (also known as sexagram or a magicians/sorcerers star) is a six-pointed star, a type of complex star polygon. ...
The Star of David The Star of David (Hebrew: pronounced: maw-gÄn daw-vÄ“d, translit. ...
In Medieval Jewish, Islamic and Christian legends, the Seal of Solomon was a magical signet ring said to have been possessed by King Solomon (or Sulayman in the Islamic version), which variously gave him the power to command demons (or jinni), or to speak with animals. ...
Acute heptagram Obtuse heptagram Acute and obtuse heptagrams inscribed within a heptagon. ...
This article is about the overall Jewish mysticisms tradition. ...
This article or section is missing references or citation of sources. ...
A hendeceagram is a star polygon that has eleven points. ...
One simple form of the Seal of Solomon. ...
One simple form of the Seal of Solomon. ...
In Medieval Jewish, Islamic and Christian legends, the Seal of Solomon was a magical signet ring said to have been possessed by King Solomon (or Sulayman in the Islamic version), which variously gave him the power to command demons (or jinni), or to speak with animals. ...
A hexagram (also known as sexagram or a magicians/sorcerers star) is a six-pointed star, a type of complex star polygon. ...
Thomas Bradwardine (c. ...
Some symbols based on a star polygon have interlacing, by small gaps, and/or, in the case of a star figure, using different colors.
See also Categories: Stub | Polygons ...
This page lists the regular polytopes in Euclidean space. ...
In a standard magic star, or magic polygram, numbers are placed at each vertex and intersection to produce lines of four that each sum to the same magic total. ...
Stellation is a process of constructing new polygons (in two dimensions), new polyhedra in three dimensions, or in general new polytopes in n dimensions. ...
External links - Star Polygon -- from MathWorld
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