Star polygon
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Star polygons in general do not appear to have been formally defined. We can say only that they are polygons which look starlike. Only the regular ones have been studied in any depth.
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[edit] Regular star polygons
In geometry, a regular star polygon is a self-intersecting, equilateral equiangular polygon, 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}.
A regular star polygon can also be represented as a sequence of stellations of a convex regular core polygon.
[edit] Examples
 {5/2} |
 {7/2} |
 {7/3} |
 {8/3} |
 {9/2} |
 {9/4} |
[edit] Star figures
If the number of sides 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/x is 2, producing a figure consisting of n/2 straight line segments; this is called a degenerate star polygon.
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°.
A six-pointed star, like a hexagon, can be created using a compass and a straight edge:
- Make a circle of any size with the compass.
- Without changing the radius of the compass, set its pivot on the circle's circumference, and find one of the two points where a new circle would intersect the first circle.
- With the pivot on the last point found, similarly find a third point on the circumference, and repeat until six such points have been marked.
- With a straight edge, join alternate points on the circumference to form two overlapping equilateral triangles.
[edit] Symmetry
Star polygons can be thought of as diagramming cosets of the subgroups 
of the finite group 
.
The symmetry group of {n/k} is dihedral group Dn of order 2n, independent of k.
Certain star polygons feature prominently in art and culture. These include:
- The {5/2} star polygon known as a pentagram, pentalpha or pentangle, and considered by some to have occult significance.
- The simplest non-degenerate complex star polygon which is two {6/2} polygons (i.e., triangles), the hexagram (Star of David, Seal of Solomon).
- The Marian Star, the proper star polygon for Roman Catholic symbolic depictions of celestial objects.
- The {7/3} and {7/2} star polygons which are known as heptagrams and also have occult significance, particularly in the Kabbalah and in Wicca.
- The complex {8/2} star polygon (i.e. two squares), which is known as the Star of Lakshmi and figures in Hinduism;
- The {8/3} star polygon (octagram), and the complex star polygon of two {16/6} polygons, which are frequent geometrical motifs in Mughal Islamic art and architecture; the first is on the coat of arms of Azerbaijan.
- An eleven pointed star called the hendecagram, which apparently was used on the tomb of Shah Nemat Ollah Vali.
The star polygons were first studied by Thomas Bradwardine.
Some symbols based on a star polygon have interlacing, by small gaps, and/or, in the case of a star figure, using different colors.
[edit] Irregular cyclic star polygons
The edge lengths are defined by the distance between alternate vertices in the faces of the uniform polyhedron, with the last edge turning retrograde
Irregular cyclic star polygons are used in the definition of vertex figures for the uniform polyhedra, defined by the sequence of regular polygon faces around each vertex, allowing for both multiple turns, and retrograde directions. (See vertex figures at List of uniform polyhedra)
[edit] Interiors of star polygons
Star polygons leave an ambiguity of interpretation for vertices and interiors. This diagram demonstrates three interpretations of a pentagram.
- The left complex polygon interpretation has 5 vertices of a regular pentagon connected alternately on a cyclic path, skipping alternate vertices. The interior is considered everything immediately left (or right) from each edge (until the next intersection). This makes the most interior region actually "outside", and in general you can determine inside by a binary odd/even rule of counting how many edges are intersected from a point along a ray to infinity.
- The middle complex polygon interpretation also has 5 vertices of a regular pentagon connected alternately on a cyclic path, but the interior is considered as inside a simple polygon perimeter boundary.
- The right interpretation creates new vertices at the intersections of the edges (5 in this case) and defines a new concave decagon (10-pointed polygon) formed by perimeter path of the middle interpretation.
[edit] Example interpretations of a star prism
 Heptagrams with 2-sided interior |
 Heptagrams with a simple perimeter interior |
The heptagrammic prism above shows different interpretations can create very different appearances.
Builders of polyhedron models, like Magnus Wenninger, usually represent star polygon faces in the concave form, without internal edges shown.
[edit] See also
- Complex polygon
- List of regular polytopes - Nonconvex forms (2D)
- Magic star
- Star polyhedron
- Star polychoron (4-polytopes)
- Star domain — a similar-sounding but unrelated concept
[edit] References
- Cromwell, P.; Polyhedra, CUP, Hbk. 1997, ISBN 0-521-66432-2. Pbk. 1999, ISBN 0-521-66405-5.
- Grünbaum, Branko, and G. C. Shephard. Tilings and Patterns. New York: W. H. Freeman & Co., 1987. ISBN 0-7167-1193-1.
[edit] External links
- Eric W. Weisstein, Star polygon at MathWorld.
- Eric W. Weisstein, Polygram at MathWorld.
