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}.

Examples

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

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.

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.

Star figures

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°.

Geometric interiors

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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.

1. The first converts it to a concave decagon (10-pointed polygon).
2. 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.
3. 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!)

Example star prisms with different face interior renderings

{7/2} heptagrammic prism:

Simple (binary) face interior
Complex face interior

Symmetry

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

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

  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

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  • Complex polygon
  • List of regular polytopes#Nonconvex forms (2D)
  • Magic star
  • Stellation

  External links

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  • Star Polygon -- from MathWorld

  polygons

  Stern (Geometrie) | Stelo (figuro) | Wielokąt gwiaździsty | Звезда (геометрическая фигура)