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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 Simple or SIMPLE may refer to: *Simplicity, the state or quality of being simple Arts and entertainment * ''Simple'' (album), by Andy Yorke, 2008, and its title track * "Simple" (Florida Georgia Line song), 2018 * "Simple", a song by Johnn ...
isotoxal In geometry, a polytope (for example, a polygon or a polyhedron) or a tiling is isotoxal () or edge-transitive if its symmetries act transitively on its edges. Informally, this means that there is only one type of edge to the object: given ...
concave polygon A simple polygon that is not convex is called concave, non-convex or reentrant. A concave polygon will always have at least one reflex interior angle—that is, an angle with a measure that is between 180 degrees and 360 degrees exclusive. Polyg ...
s. 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
spirolateral In Euclidean geometry, a spirolateral is a polygon created by a sequence of fixed vertex internal angles and sequential edge lengths 1,2,3,…,''n'' which repeat until the figure closes. The number of repeats needed is called its cycles. Gardner, M ...
s.Abelson, Harold, diSessa, Andera, 1980, ''Turtle Geometry'', MIT Press, p.24


Etymology

Star polygon names combine a
numeral prefix Numeral or number prefixes are prefixes derived from numerals or occasionally other numbers. In English and many other languages, they are used to coin numerous series of words. For example: * unicycle, bicycle, tricycle (1-cycle, 2-cycle, 3-c ...
, such as ''
penta- Numeral or number prefixes are prefixes derived from numerals or occasionally other numbers. In English and many other languages, they are used to coin numerous series of words. For example: * unicycle, bicycle, tricycle (1-cycle, 2-cycle, 3-c ...
'', with the Greek suffix '' -gram'' (in this case generating the word '' pentagram''). The prefix is normally a Greek
cardinal Cardinal or The Cardinal may refer to: Animals * Cardinal (bird) or Cardinalidae, a family of North and South American birds **'' Cardinalis'', genus of cardinal in the family Cardinalidae **'' Cardinalis cardinalis'', or northern cardinal, t ...
, but synonyms using other prefixes exist. For example, a nine-pointed polygon or ''
enneagram Enneagram is a compound word derived from the Greek neoclassical stems for "nine" (''ennea'') and something "written" or "drawn" (''gramma''). Enneagram may refer to: * Enneagram (geometry), a nine-sided star polygon with various configurations ...
'' is also known as a ''nonagram'', using the ordinal ''nona'' from
Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally a dialect spoken in the lower Tiber area (then known as Latium) around present-day Rome, but through ...
. The ''-gram'' suffix derives from '' γραμμή'' (''grammḗ'') meaning a line.


Regular star polygon

A "regular star polygon" is a self-intersecting, equilateral equiangular polygon. A regular star polygon is denoted by its Schläfli symbol , 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 angle In geometry, an angle of a polygon is formed by two sides of the polygon that share an endpoint. For a simple (non-self-intersecting) polygon, regardless of whether it is convex or non-convex, this angle is called an interior angle (or ) if ...
s of all the vertices divided by 360°. The symmetry group of is dihedral group ''D''n of order 2''n'', independent of ''k''. Regular star polygons were first studied systematically by Thomas Bradwardine, and later Johannes Kepler.


Construction via vertex connection

Regular star polygons can be created by connecting one
vertex Vertex, vertices or vertexes may refer to: Science and technology Mathematics and computer science *Vertex (geometry), a point where two or more curves, lines, or edges meet *Vertex (computer graphics), a data structure that describes the position ...
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 ''q''th 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. 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 results in a pentagrammic antiprism; the analogous construction from a retrograde "crossed pentagram" results in a pentagrammic crossed-antiprism. Another example is the
tetrahemihexahedron In geometry, the tetrahemihexahedron or hemicuboctahedron is a uniform star polyhedron, indexed as U4. It has 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices. Its vertex figure is a crossed quadrilateral. Its Coxeter–Dynkin dia ...
, which can be seen as a "crossed triangle" cuploid.


Degenerate regular star polygons

If ''p'' and ''q'' are not coprime, a degenerate polygon will result with coinciding vertices and edges. For example 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.Coxeter, The Densities of the Regular polytopes I, p.43: If d is odd, the truncation of the polygon is naturally . But if not, it consists of two coincident 's; two, because each side arises from an original side and once from an original vertex. Thus the density of a polygon is unaltered by truncation. :


Construction via stellation

Alternatively, a regular star polygon can also be obtained as a sequence of
stellation In geometry, stellation is the process of extending a polygon in two dimensions, polyhedron in three dimensions, or, in general, a polytope in ''n'' dimensions to form a new figure. Starting with an original figure, the process extends specif ...
s 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.


Simple isotoxal star polygons

When the intersecting lines are removed, the star polygons are no longer regular, but can be seen as
simple Simple or SIMPLE may refer to: *Simplicity, the state or quality of being simple Arts and entertainment * ''Simple'' (album), by Andy Yorke, 2008, and its title track * "Simple" (Florida Georgia Line song), 2018 * "Simple", a song by Johnn ...
concave Concave or concavity may refer to: Science and technology * Concave lens * Concave mirror Mathematics * Concave function, the negative of a convex function * Concave polygon, a polygon which is not convex * Concave set In geometry, a subset o ...
isotoxal In geometry, a polytope (for example, a polygon or a polyhedron) or a tiling is isotoxal () or edge-transitive if its symmetries act transitively on its edges. Informally, this means that there is only one type of edge to the object: given ...
2''n''-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 with a notation 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''.


Examples in tilings

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.Branko Grunbaum and Geoffrey C. Shephard, Tilings by Regular Polygons
Mathematics Magazine 50 (1977), 227–247 and 51 (1978), 205–206]


Interiors

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 2''n''-gons. These include: * Where a side occurs, one side is treated as outside and the other as inside. This is shown in the left hand illustration and commonly occurs in computer vector graphics rendering. *The number of times that the polygonal curve winds around a given region determines its '' density''. The exterior is given a density of 0, and any region of density > 0 is treated as internal. This is shown in the central illustration and commonly occurs in the mathematical treatment of polyhedra. (However, for non-orientable polyhedra density can only be considered modulo 2 and hence the first treatment is sometimes used instead in those cases for consistency.) * Where a line may be drawn between two sides, the region in which the line lies is treated as inside the figure. This is shown in the right hand illustration and commonly occurs when making a physical model. When the area of the polygon is calculated, each of these approaches yields a different answer.


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: *The star pentagon ( pentagram) is also known as a pentalpha or pentangle, and historically has been considered by many magical and religious cults to have occult significance. *The and star polygons ( heptagrams) also have occult significance, particularly in the
Kabbalah Kabbalah ( he, קַבָּלָה ''Qabbālā'', literally "reception, tradition") is an esoteric method, discipline and Jewish theology, school of thought in Jewish mysticism. A traditional Kabbalist is called a Mekubbal ( ''Məqūbbāl'' "rece ...
and in Wicca. *The star polygon ( octagram) is a frequent geometrical motif in Mughal Islamic art and architecture; the first is on the emblem of Azerbaijan. *An eleven pointed star called the
hendecagram In geometry, a hendecagram (also endecagram or endekagram) is a star polygon that has eleven vertices. The name ''hendecagram'' combines a Greek numeral prefix, '' hendeca-'', with the Greek suffix ''-gram''. The ''hendeca-'' prefix derives fr ...
was used on the tomb of Shah Nemat Ollah Vali.


See also

* List of regular polytopes and compounds#Stars * Five-pointed star * Magic star * Moravian star * Pentagramma mirificum *
Regular star 4-polytope In mathematics, a regular 4-polytope is a regular four-dimensional polytope. They are the four-dimensional analogues of the regular polyhedra in three dimensions and the regular polygons in two dimensions. There are six convex and ten star reg ...
* Rub el Hizb * Star (glyph) * Star polyhedron, Kepler–Poinsot polyhedron, and
uniform star polyhedron In geometry, a uniform star polyhedron is a self-intersecting uniform polyhedron. They are also sometimes called nonconvex polyhedra to imply self-intersecting. Each polyhedron can contain either star polygon faces, star polygon vertex figures, ...
* Starfish


References

*Cromwell, P.; ''Polyhedra'', CUP, Hbk. 1997, . Pbk. (1999), . p. 175 * Grünbaum, B. and G.C. Shephard; ''Tilings and Patterns'', New York: W. H. Freeman & Co., (1987), . * Grünbaum, B.; Polyhedra with Hollow Faces, ''Proc of NATO-ASI Conference on Polytopes ... etc. (Toronto 1993)'', ed T. Bisztriczky et al., Kluwer Academic (1994) pp. 43–70. *
John H. Conway John Horton Conway (26 December 1937 – 11 April 2020) was an English mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. He also made contributions to many branches o ...
, Heidi Burgiel, Chaim Goodman-Strass, ''The Symmetries of Things'' 2008, (Chapter 26. pp. 404: Regular star-polytopes Dimension 2) * Branko Grünbaum, ''Metamorphoses of polygons'', published in ''The Lighter Side of Mathematics: Proceedings of the Eugène Strens Memorial Conference on Recreational Mathematics and its History'', (1994) {{DEFAULTSORT:Star Polygon Star symbols