 {5/2} |  |5/2| |
| A regular starpentagon, {5/2}, has five vertices (its corner tips) and five intersecting edges, while a concavedecagon, |5/2|, has ten edges and two sets of five vertices. The first is used in definitions ofstar polyhedra and staruniform tilings, while the second is sometimes used in planar tilings. | |
 Small stellated dodecahedron |  Tessellation |
Ingeometry, astar polygon is a type of non-convex polygon.Regular star polygons have been studied in depth; while star polygons in general appear not to have been formally defined,certain notable ones can arise through truncation operations on regular simple or star polygons.
Branko Grünbaum identified two primary usages of this terminology byJohannes Kepler, one corresponding to theregular star polygons withintersecting edges that do not generate new vertices, and the other one to theisotoxalconcavesimple polygons.[1]
Polygrams include polygons like thepentagram, but also compound figures like thehexagram.
One definition of astar polygon, used inturtle graphics, is a polygon havingq ≥ 2turns (q is called theturning number ordensity), like inspirolaterals.[2]
Star polygon names combine anumeral prefix, such aspenta-, with theGreek suffix-gram (in this case generating the wordpentagram). The prefix is normally a Greekcardinal, but synonyms using other prefixes exist. For example, a nine-pointed polygon orenneagram is also known as anonagram, using theordinalnona fromLatin.[citation needed] The-gram suffix derives fromγραμμή (grammḗ), meaning a line.[3] The namestar polygon reflects the resemblance of these shapes to thediffraction spikes of real stars.
 {5/2} |  {7/2} |  {7/3} | ... |
Aregular star polygon is a self-intersecting, equilateral, and equiangularpolygon.
A regular star polygon is denoted by itsSchläfli symbol {p/q}, wherep (the number of vertices) andq (thedensity) arerelatively prime (they share no factors) and whereq ≥ 2. The density of a polygon can also be called itsturning number: the sum of theturn angles of all the vertices, divided by 360°.
Thesymmetry group of {p/q} is thedihedral group Dp, of order 2p, independent ofq.
Regular star polygons were first studied systematically byThomas Bradwardine, and laterJohannes Kepler.[4]
Regular star polygons can be created by connecting onevertex of a regularp-sided simple polygon to another vertex, non-adjacent to the first one, and continuing the process until the original vertex is reached again.[5] Alternatively, for integersp andq, it can be considered as being constructed by connecting everyqth point out ofp points regularly spaced in a circular placement.[6] For instance, in a regular pentagon, a five-pointed star can be obtained by drawing a line from the 1st to the 3rd vertex, from the 3rd to the 5th vertex, from the 5th to the 2nd vertex, from the 2nd to the 4th vertex, and from the 4th to the 1st vertex.
Ifq ≥p/2, then the construction of {p/q} will result in the same polygon as {p/(p −q)}; connecting every third vertex of the pentagon will yield an identical result to that of connecting every second vertex. However, the vertices will be reached in the opposite direction, which makes a difference when retrograde polygons are incorporated in higher-dimensional polytopes. For example, anantiprism formed from a prograde pentagram {5/2} results in apentagrammic antiprism; the analogous construction from a retrograde "crossed pentagram" {5/3} results in apentagrammic crossed-antiprism. Another example is thetetrahemihexahedron, which can be seen as a "crossed triangle" {3/2}cuploid.
Ifp andq 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: 1-3 and 4-6. This should be seen not as two overlapping triangles, but as a double-winding single unicursal hexagon.[7][8]
Alternatively, a regular star polygon can also be obtained as a sequence ofstellations of a convex regularcore polygon. Constructions based on stellation also allow regular polygonal compounds to be obtained in cases where the densityq and amountp of vertices are not coprime. When constructing star polygons from stellation, however, ifq >p/2, the lines will instead diverge infinitely, and ifq =p/2, the lines will be parallel, with both resulting in no further intersection in Euclidean space. However, it may be possible to construct some such polygons in spherical space, similarly to themonogon anddigon; such polygons do not yet appear to have been studied in detail.
When the intersecting line segments are removed from a regular starn-gon, the resulting figure is no longer regular, but can be seen as anisotoxalconcavesimple2n-gon, alternating vertices at two different radii.Branko Grünbaum, inTilings and patterns, represents such a star that matches the outline of a regularpolygram {n/d} as |n/d|, or more generally with {n𝛼}, which denotes anisotoxal concaveor convex simple2n-gon with outerinternal angle 𝛼.
| |n/d| {n𝛼} | |9/4| {920°} | {330°} | {630°} | |5/2| {536°} | {445°} | |8/3| {845°} | |6/2| {660°} | {572°} |
|---|---|---|---|---|---|---|---|---|
| 𝛼 | 20° | 30° | 36° | 45° | 60° | 72° | ||
| βext | 60° | 150° | 90° | 108° | 135° | 90° | 120° | 144° |
| Isotoxal simple n-pointed star |  |  |  | |  |  |  |  |
| Related regular polygram {n/d} |  {9/4} |  {12/5} | {5/2} |  {8/3} |  2{3} Star figure |  {10/3} | ||
These polygons are often seen in tiling patterns. The parametric angle 𝛼 (in degrees or radians) can be chosen to matchinternal angles of neighboring polygons in a tessellation pattern. In his 1619 workHarmonices Mundi, among periodic tilings,Johannes Kepler includes nonperiodic tilings, like that with three regular pentagons and one regular star pentagon fitting around certain vertices, 5.5.5.5/2, and related to modernPenrose tilings.[9]
| Isotoxal simple n-pointed stars | "Triangular" stars (n = 3) | "Square" stars (n = 4) | "Hexagonal" stars (n = 6) | "Octagonal" stars (n = 8) | ||
|---|---|---|---|---|---|---|
| Image of tiling |  |  |  |  |  |  |
| Vertex config. | 3.3* 𝛼.3.3** 𝛼 | 8.4* π/4.8.4* π/4 | 6.6* π/3.6.6* π/3 | 3.6* π/3.6** π/3 | 3.6.6* π/3.6 | not edge-to-edge |
The interior of a star polygon may be treated in different ways. Three such treatments are illustrated for a pentagram.Branko Grünbaum and Geoffrey Shephard consider two of them, as regular starn-gons and as isotoxal concave simple2n-gons.[9]
These three treatments are:
When the area of the polygon is calculated, each of these approaches yields a different result.
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Star polygons feature prominently in art and culture. Such polygons may or may not beregular, but they are always highlysymmetrical. Examples include:
 An {8/3}octagram constructed in a regularoctagon | .svg) Seal of Solomon with circle and dots (star figure) |
Hart, Vi (2010)."Doodling in Math Class: Stars".YouTube.