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In geometry, a star polygon is a type of non-convex polygon. Only the regular star polygons have been studied in any depth; star polygons in general appear not to have been formally defined.
Contents
- Etymology
- Regular star polygon
- Degenerate regular star polygons
- Simple isotoxal star polygons
- Interiors of star polygons
- Star polygons in art and culture
- References
Branko Grünbaum identified two primary definitions used by Kepler, one being the regular star polygons with intersecting edges that don't generate new vertices, and the second being simple isotoxal concave polygons.
The first usage is included in polygrams which includes polygons like the pentagram but also compound figures like the hexagram.
Etymology
Star polygon names combine a numeral prefix, such as penta-, with the Greek suffix -gram (in this case generating the word pentagram). The prefix is normally a Greek cardinal, but synonyms using other prefixes exist. For example, a nine-pointed polygon or enneagram is also known as a nonagram, using the ordinal nona from Latin. The -gram suffix derives from γραμμή (grammḗ) meaning a line.
Regular star polygon
A "regular star polygon" is a self-intersecting, equilateral equiangular polygon, created by connecting one vertex of a simple, regular, p-sided polygon to another, non-adjacent vertex and continuing the process until the original vertex is reached again. Alternatively for integers p and q, it can be considered as being constructed by connecting every qth point out of p points regularly spaced in a circular placement. 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.
A regular star polygon is denoted by its Schläfli symbol {p/q}, where p and q are relatively prime (they share no factors) and q ≥ 2.
The symmetry group of {n/k} is dihedral group Dn of order 2n, independent of k.
A regular star polygon can also be obtained as a sequence of stellations of a convex regular core polygon.
Regular star polygons were first studied systematically by Thomas Bradwardine, and later Kepler.
Degenerate regular star polygons
If p and q are not coprime, a degenerate polygon will result with coinciding vertices and edges. For example {6/2} will appear as a triangle, but can be labeled with two sets of vertices 1-6. This should be seen not as two overlapping triangles, but a double-winding of a single unicursal hexagon.
Simple isotoxal star polygons
When the intersecting lines are removed, the star polygons are no longer regular, but can be seen as simple concave isotoxal 2n-gons, alternating vertices at two different radii, which do not necessarily have to match the regular star polygon angles. Branko Grünbaum in Tilings and Patterns represents these stars as |n/d| that match the geometry of polygram {n/d} with a notation {nα} more generally, representing an n-sided star with each internal angle α<180°(1-2/n) degrees. For |n/d|, the inner vertices have an exterior angle, β, as 360°(d-1)/n.
These polygons are often seen in tiling patterns. The parametric angle α (degrees or radians) can be chosen to match internal angles of neighboring polygons in a tessellation pattern. Johannes Kepler in his 1619 work Harmonices Mundi, including among other period tilings, nonperiodic tilings like that three regular pentagons, and a regular star pentagon (5.5.5.5/2) can fit around a vertex, and related to modern penrose tilings.
Interiors of star polygons
The interior of a star polygon may be treated in different ways. Three such treatments are illustrated for a pentagram. Branko Grunbaum and Geoffrey Shephard consider two of them, as regular star polygons and concave isogonal 2n-gons.
These include:
When the area of the polygon is calculated, each of these approaches yields a different answer.
Star polygons in art and culture
Star polygons feature prominently in art and culture. Such polygons may or may not be regular but they are always highly symmetrical. Examples include: