Star polygon

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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}.

The star polygons can also be represented as a sequence of stellations of the regular polygons.

1 Examples
- 1.1 Star figures
2 Interpretations of a pentagram and its interior
- 2.1 Example star prism interpretations
3 Symmetry
4 See also
5 References
6 External links

[edit] Examples

{5/2}	{7/2}	{7/3}	{8/3}	{9/2}	{9/4}

star figure
Hexagram
{6/2}

star figure
Enneagram
{9/3}

[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 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.

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] Interpretations of a pentagram and its interior

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 star prism interpretations

{7/2} heptagrammic 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] Symmetry

Star polygons can be thought of as diagramming cosets of the subgroups $x\mathbb{Z}_n$ of the finite group $\mathbb{Z}_n$ .

The symmetry group of {n/k} is dihedral group D_n of order 2n, independent of k.

An {8/3} star polygon (octagram) constructed in an octagon

Certain star polygons feature prominently in art and culture. These include:

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.