{5/2} |
|5/2| |
A regular star pentagon, {5/2}, has five corner vertices and intersecting edges, while a concave decagon, |5/2|, has ten edges and two sets of five vertices. The first are used in definitions of star polyhedra and star uniform tilings, while the second are sometimes used in planar tilings. | |
Small stellated dodecahedron |
Tessellation |
In geometry, a star 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 and star polygons.
Branko Grünbaum identified two primary definitions used by Johannes Kepler, one being the regular star polygons with intersecting edges that don't generate new vertices, and the second being simple isotoxal concave polygons.^{[1]}
The first usage is included in polygrams which includes polygons like the pentagram but also compound figures like the hexagram.
One definition of a star polygon, used in turtle graphics, is a polygon having 2 or more turns (turning number and density), like in spirolaterals.^{[2]}
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.^{[citation needed]} The -gram suffix derives from γραμμή (grammḗ) meaning a line.^{[3]}
{5/2} |
{7/2} |
{7/3}... |
A "regular star polygon" is a self-intersecting, equilateral equiangular polygon.
A regular star polygon is denoted by its Schläfli symbol {p/q}, where p (the number of vertices) and q (the density) are relatively prime (they share no factors) and q ≥ 2. The density of a polygon can also be called its turning number, the sum of the turn angles of all the vertices divided by 360°.
The symmetry group of {n/k} is dihedral group D_{n} of order 2n, independent of k.
Regular star polygons were first studied systematically by Thomas Bradwardine, and later Johannes Kepler.^{[4]}
Regular star polygons can be 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.^{[5]} 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.^{[6]} 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.
If q is greater than half of p, then the construction will result in the same polygon as 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, an antiprism formed from a prograde pentagram {5/2} results in a pentagrammic antiprism; the analogous construction from a retrograde "crossed pentagram" {5/3} results in a pentagrammic crossed-antiprism. Another example is the tetrahemihexahedron, which can be seen as a "crossed triangle" {3/2} cuploid.
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.^{[7]}^{[8]}
Alternatively, a regular star polygon can also be obtained as a sequence of stellations of a convex regular core polygon. Constructions based on stellation also allow for regular polygonal compounds to be obtained in cases where the density and amount of vertices are not coprime. When constructing star polygons from stellation, however, if q is greater than p/2, the lines will instead diverge infinitely, and if q is equal to 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 the monogon and digon; such polygons do not yet appear to have been studied in detail.
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 {n_{α}} more generally, representing an n-sided star with each internal angle α<180°(1-2/n) degrees.^{[1]} For |n/d|, the inner vertices have an exterior angle, β, as 360°(d-1)/n.
|n/d| {n_{α}} |
{3_{30°}} |
{6_{30°}} |
|5/2| {5_{36°}} |
{4_{45°}} |
|8/3| {8_{45°}} |
|6/2| {6_{60°}} |
{5_{72°}} |
---|---|---|---|---|---|---|---|
α | 30° | 36° | 45° | 60° | 72° | ||
β | 150° | 90° | 72° | 135° | 90° | 120° | 144° |
Isotoxal star |
|||||||
Related polygram {n/d} |
{12/5} |
{5/2} |
{8/3} |
2{3} Star figure |
{10/3} |
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.^{[9]}
Star triangles | Star squares | Star hexagons | Star octagons | ||
---|---|---|---|---|---|
(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 star polygons and concave isogonal 2n-gons.^{[9]}
These include:
When the area of the polygon is calculated, each of these approaches yields a different answer.
Star polygons feature prominently in art and culture. Such polygons may or may not be regular but they are always highly symmetrical. Examples include:
An {8/3} octagram constructed in a regular octagon |
Seal of Solomon with circle and dots (star figure) |