Editing Star polygon

{{Short description|Regular non-convex polygon}}
{{Distinguish|Star-shaped polygon}}
{| class=wikitable align=right width=320 style="margin: 0px 0px 10px 10px"
|+ Two types of star pentagons
|- align=center
|[[File:Alfkors.svg|160px]]<BR>{5/2}
|[[File:Stjärna.svg|160px]]<BR>{{pipe}}5/2{{pipe}}
|-
|colspan=2|A regular star [[pentagon]], {5/2}, has five vertices (its corner tips) and five intersecting edges, while a concave [[decagon]], {{pipe}}5/2{{pipe}}, has ten edges and two sets of five vertices. The first is used in definitions of [[star polyhedra]] and star [[uniform tiling]]s, while the second is sometimes used in planar tilings.
|- align=center
|[[File:Small stellated dodecahedron.png|160px]]<BR>[[Small stellated dodecahedron]]
|[[File:Kepler decagon pentagon pentagram tiling.svg|160px]]<BR>[[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, [[Decagram (geometry)#Related figures|certain notable ones]] can arise through truncation operations on regular simple or star polygons.

[[Branko Grünbaum]] identified two primary usages of this terminology by [[Johannes Kepler]], one corresponding to the [[regular star polygon]]s with [[List of self-intersecting polygons|intersecting edges]] that do not generate new vertices, and the other one to the [[isotoxal]] [[Concave polygon|concave]] [[simple polygon]]s.<ref name=tilingsandpatterns>Grünbaum & Shephard (1987). Tilings and Patterns. Section 2.5</ref>

[[Polygram (geometry)|Polygrams]] include 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 ''q'' ≥ 2 [[Turn (geometry)|turns]] (''q'' is called the [[turning number]] or [[Density (polygon)|density]]), like in [[spirolateral]]s.<ref name="turtle">Abelson, Harold, diSessa, Andera, 1980, ''Turtle Geometry'', MIT Press, p. 24</ref>

==Names==
Star polygon names combine a [[numeral prefix]], such as ''[[wikt:penta-|penta-]]'', with the [[Greek language|Greek]] suffix ''[[wikt:-gram|-gram]]'' (in this case generating the word ''[[pentagram]]''). The prefix is normally a Greek [[Cardinal number (linguistics)|cardinal]], but synonyms using other prefixes exist. For example, a nine-pointed polygon or ''[[Enneagram (geometry)|enneagram]]'' is also known as a ''nonagram'', using the [[Ordinal number (linguistics)|ordinal]] ''nona'' from [[Latin language|Latin]].{{citation needed|date=February 2015|reason=Do authoritative sources use both prefixes?}} The ''-gram'' suffix derives from ''[[wikt:γραμμή|γραμμή]]'' (''grammḗ''), meaning a line.<ref>[https://www.perseus.tufts.edu/hopper/text?doc=Perseus%3Atext%3A1999.04.0057%3Aentry%3Dgrammh%2F γραμμή], Henry George Liddell, Robert Scott, ''A Greek-English Lexicon'', on Perseus</ref> The name ''star polygon'' reflects the resemblance of these shapes to the [[diffraction spike]]s of real stars.

==Regular star polygon==
{{Further|Regular polygon#Regular star polygons}}
{| class="wikitable" align=right style="margin: 0px 0px 10px 10px"
|[[File:Regular star polygon 5-2.svg|80px]]<BR>[[Pentagram|{5/2}]]
|[[File:Regular star polygon 7-2.svg|80px]]<BR>[[Heptagram|{7/2}]]
|[[File:Regular star polygon 7-3.svg|80px]]<BR>[[Heptagram|{7/3}]]
|...
|}
[[File:regular star polygons.svg|thumb|300px|Regular convex and star polygons with 3 to 12 vertices, labeled with their Schläfli symbols]]
A ''regular star polygon'' is a self-intersecting, equilateral, and equiangular [[polygon]].

A regular star polygon is denoted by its [[Schläfli symbol]] {''p''/''q''}, where ''p'' (the number of vertices) and ''q'' (the [[Density_(polytope)|density]]) are [[Coprime|relatively prime]] (they share no factors) and where ''q'' ≥ 2. The density of a polygon can also be called its [[turning number]]: the sum of the [[turn angle]]s of all the vertices, divided by 360°.

The [[symmetry group]] of {''p''/''q''} is the [[dihedral group]] D<sub>''p''</sub>, of order 2''p'', independent of ''q''.

Regular star polygons were first studied systematically by [[Thomas Bradwardine]], and later [[Johannes Kepler]].<ref>Coxeter, Introduction to Geometry, second edition, 2.8 ''Star polygons'', pp. 36–38</ref>

===Construction via vertex connection===
Regular star polygons can be created by connecting one [[Vertex (geometry)|vertex]] of a regular ''p''-sided simple polygon to another vertex, non-adjacent to the first one, and continuing the process until the original vertex is reached again.<ref>{{cite book |last=Coxeter |first=Harold Scott Macdonald |title=[[Regular polytopes (book)|Regular polytopes]] |publisher=Courier Dover Publications |page=[https://archive.org/details/regularpolytopes0000coxe/page/93 93] |date=1973 |isbn=978-0-486-61480-9 }}</ref> 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.<ref>{{MathWorld |urlname=StarPolygon |title=Star Polygon}}</ref> 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.

If ''q'' ≥ ''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, 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]].

====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: 1–3 and 4–6. This should be seen not as two overlapping triangles, but as a double-winding single unicursal hexagon.<ref>[http://www.math.washington.edu/~grunbaum/Your%20polyhedra-my%20polyhedra.pdf Are Your Polyhedra the Same as My Polyhedra?] {{Webarchive|url=https://web.archive.org/web/20160803160413/http://www.math.washington.edu/~grunbaum/Your%20polyhedra-my%20polyhedra.pdf |date=2016-08-03 }}, Branko Grünbaum</ref><ref>Coxeter, The Densities of the Regular Polytopes I, p. 43:<BR>If ''q'' is odd, the truncation of {''p''/''q''} is naturally {2''p''/''q''}. But if ''q'' is even, the truncation of {''p''/''q''} consists of two coincident {{nowrap|{''p''/(''q''/2)}'s;}} two, because each side arises once from an original side and once from an original vertex. Since {{nowrap|2(''q''/2) {{=}} ''q'',}} the density of a polygon is never altered by truncation.</ref>
:[[File:Doubly wound hexagon.svg|160px]]

===Construction via stellation===
Alternatively, a regular star polygon can also be obtained as a sequence of [[Stellation#Stellating polygons|stellation]]s of a convex regular ''core'' polygon. Constructions based on stellation also allow regular polygonal compounds to be obtained in cases where the density ''q'' and amount ''p'' of vertices are not coprime. When constructing star polygons from stellation, however, if ''q'' > ''p''/2, the lines will instead diverge infinitely, and if ''q'' = ''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.

==Isotoxal star simple polygons==
When the intersecting line segments are removed from a regular star ''n''-gon, the resulting figure is no longer regular, but can be seen as an [[isotoxal]] [[Concave polygon|concave]] [[Simple polygon|simple]] '''2'''''n''-gon, alternating vertices at two different radii. [[Branko Grünbaum]], in ''[[Tilings and patterns]]'', represents such a star that matches the outline of a regular [[Polygram (geometry)|polygram]] {''n''/''d''} as |''n''/''d''|, or more generally with {''n''<sub>𝛼</sub>}, which denotes an [[Isotoxal figure#Isotoxal polygons|isotoxal concave ''or convex'']] simple '''2'''''n''-gon with outer [[internal angle]] 𝛼.
* For |''n''/''d''|, the outer internal angle {{nowrap|𝛼 {{=}} 180(1 − 2''d''/''n'')}} degrees, necessarily, and the inner (new) vertices have an [[external angle]] {{nowrap|''β''{{sub|ext}} {{=}} 180[1 − 2(''d'' − 1)/''n'']}} degrees, necessarily.
* For {''n''<sub>𝛼</sub>}, the outer internal and inner external angles, also denoted by 𝛼 and ''β''{{sub|ext}}, do not have to match those of any regular polygram {''n''/''d''}; however, {{nowrap|𝛼 < 180(1 − 2/''n'')}} degrees and {{nowrap|''β''{{sub|ext}} < 180°,}} necessarily (here, {''n''<sub>𝛼</sub>} is concave).<ref name=tilingsandpatterns/>
{| class=wikitable
|+ Examples of isotoxal star simple polygons
|- valign=top align=center
!{{pipe}}''n''/''d''{{pipe}}<BR>{''n''<sub>𝛼</sub>}
!{{pipe}}9/4{{pipe}}<BR>{9<sub>20°</sub>}
!&nbsp;<BR>{3<sub>30°</sub>}
!&nbsp;<BR>{6<sub>30°</sub>}
!{{pipe}}5/2{{pipe}}<BR>{5<sub>36°</sub>}
!&nbsp;<BR>{4<sub>45°</sub>}
!{{pipe}}8/3{{pipe}}<BR>{8<sub>45°</sub>}
!{{pipe}}6/2{{pipe}}<BR>{6<sub>60°</sub>}
!&nbsp;<BR>{5<sub>72°</sub>}
|-
!𝛼
!20°
!colspan=2|30°
!36°
!colspan=2|45°
!60°
!72°
|-
!''β''{{sub|ext}}
!60°
!150°
!90°
!108°
!135°
!90°
!120°
!144°
|- valign=top align=center
!valign=center|Isotoxal<BR>simple<BR>''n''-pointed<BR>star
|[[File:Isotoxal star simple 18-gon fitting regular star 9÷4-gon outline.png|75px]]
|[[File:Isotoxal star triangle 12-5.svg|75px]]
|[[File:Isotoxal star hexagon 12-5.png|75px]]
|[[File:Stjärna.svg|75px]]
|[[File:Isotoxal square star 8-3.svg|75px]]
|[[File:Octagonal star.png|75px]]
|[[File:Roundel of Israel – Low Visibility – Type 2.svg|60px]]
|[[File:Wide pentagram.png|75px]]
|- valign=top align=center
!valign=center|Related<BR>regular<BR>polygram<BR>{''n''/''d''}
|[[File:Star polygon 9-4.svg|75px]]<BR>{9/4}
|colspan=2|[[File:Regular star polygon 12-5.svg|75px]]<BR>{12/5}
|[[File:Alfkors.svg|75px]]<BR>{5/2}
|colspan=2|[[File:Regular star polygon 8-3.svg|75px]]<BR>{8/3}
|[[File:Hexagram.svg|65px]]<BR>2{3}<BR>[[Star figure]]
|[[File:Decagram 10 3.png|75px]]<BR>{10/3}
|}

===Examples in tilings===
{{See|Uniform tiling#Uniform tilings using star polygons as concave alternating faces}}
These polygons are often seen in tiling patterns. The parametric angle 𝛼 (in degrees or radians) can be chosen to match [[internal angle]]s of neighboring polygons in a tessellation pattern. In his 1619 work ''[[Harmonice 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 modern [[Penrose tiling]]s.<ref name="maa.org">[https://www.maa.org/sites/default/files/pdf/upload_library/22/Allendoerfer/1978/0025570x.di021102.02p0230f.pdf Branko Grunbaum and Geoffrey C. Shephard, Tilings by Regular Polygons, Mathematics Magazine #50 (1977), pp. 227–247, and #51 (1978), pp. 205–206]</ref>
{| class="wikitable nowrap"
|+ Examples of isogonal tilings with isotoxal simple stars<ref>[http://www.polyomino.org.uk/publications/2004/star-polygon-tiling.pdf Tiling with Regular Star Polygons], Joseph Myers</ref>
!valign=center|Isotoxal simple<BR>''n''-pointed stars
!"Triangular" stars<BR>(''n'' = 3)
!"Square" stars<BR>(''n'' = 4)
!colspan=3|"Hexagonal" stars<BR>(''n'' = 6)
!"Octagonal" stars<BR>(''n'' = 8)
|- align=center valign=center
!Image of tiling
|[[File:Triangle and triangular star tiling.svg|140px]]
|[[File:Octagon star square tiling.svg|150px]]
|[[File:Hexagon hexagram tiling.png|140px]]
|[[File:Gyrated truncated hexagonal tiling2.png|140px]]
|[[File:Trihexagonal tiling stars.svg|150px]]
|[[File:Hexagon hexagram tiling2.png|140px]]
|- align=center valign=center
!Vertex config.
|3.3{{supsub|*|𝛼}}.3.3{{supsub|**|𝛼}}
|8.4{{supsub|*|π/4}}.8.4{{supsub|*|π/4}}
|6.6{{supsub|*|π/3}}.6.6{{supsub|*|π/3}}
|3.6{{supsub|*|π/3}}.6{{supsub|**|π/3}}
|3.6.6{{supsub|*|π/3}}.6
|not edge-to-edge
|}

==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 ''n''-gons and as isotoxal concave simple '''2'''''n''-gons.<ref name="maa.org"/>

[[File:Pentagram interpretations.svg|400px]]

These three treatments are:
* Where a line segment 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 (polytope)|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, in those cases, for consistency, the first treatment is sometimes used instead.)
* Wherever a line segment may be drawn between two sides, the region in which the line segment lies is treated as inside the figure.<!--This definition of the interior is false for a concave figure, isn't it? Perhaps mentioning the outline of a polygram would help?--> 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 result.

==In art and culture==
{{Main|Star polygons in art and culture}}
{{refimprove section|date=March 2024}}

Star polygons feature prominently in art and culture. Such polygons may or may not be [[Regular polygon|regular]], but they are always highly [[symmetrical]]. Examples include:
* The {5/2} star pentagon ([[pentagram]]) is also known as a pentalpha or pentangle, and historically has been considered by many [[Magic (paranormal)|magic]]al and [[religious]] cults to have [[occult]] significance.
* The {7/2} and {7/3} star polygons ([[heptagram]]s) also have occult significance, particularly in the [[Kabbalah]] and in [[Wicca]].
* The {8/3} star polygon ([[octagram]]) is a frequent geometrical motif in [[Mughal Empire|Mughal]] [[Islamic art history|Islamic art]] and [[Islamic architecture|architecture]]; the first is on the [[emblem of Azerbaijan]].
* An eleven pointed star called the [[hendecagram]] was used on the [[Shah Nematollah Vali Shrine|tomb of Shah Nematollah Vali]].<ref>{{Cite book |last=Broug |first=Eric |title=Islamic Geometric Patterns |date=2008-05-27 |publisher=Thames and Hudson |pages=183–185, 193 |isbn=978-0-500-28721-7 |location=London |language=English}}</ref>

{| class=wikitable width=300
|- valign=top
|[[File:Octagram.svg|150px]]<BR>An {8/3} [[octagram]] constructed in a regular [[octagon]]
|[[File:Seal of Solomon (Simple Version).svg|150px]]<BR>[[Seal of Solomon]] with circle and dots (star figure)
|}

==See also==
* [[List of regular polytopes and compounds#Stars]]
* [[Five-pointed star]]
* [[Magic star]]
* [[Moravian star]]
* [[Pentagramma mirificum]]
* [[Regular star 4-polytope]]
* [[Rub el Hizb]]
* [[Star (glyph)]]
* [[Star polyhedron]], [[Kepler–Poinsot polyhedron]], and [[uniform star polyhedron]]
* [[Starfish]]

==References==
{{reflist}}
* Cromwell, P.; ''Polyhedra'', CUP, Hbk. 1997, {{isbn|0-521-66432-2}}. Pbk. 1999, {{isbn|0-521-66405-5}}. p. 175
* [[Branko Grünbaum|Grünbaum, B.]] and G. C. Shephard; ''Tilings and Patterns'', New York: W. H. Freeman & Co. (1987), {{isbn|0-7167-1193-1}}.
* [[Branko Grünbaum|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 Horton Conway|John H. Conway]], Heidi Burgiel, Chaim Goodman-Strauss, ''The Symmetries of Things'', 2008, {{isbn|978-1-56881-220-5}} (Chapter 26, p. 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)

==External Links==
{{Cite web|last=Hart|first=Vi|url=http://www.youtube.com/watch?v=CfJzrmS9UfY|title=Doodling in Math Class: Stars|website=[[YouTube]] |year=2010}}

{{Polygons}}

{{DEFAULTSORT:Star polygon}}

[[Category:Star symbols]]
[[Category:Star polygons| ]]