Editing Regular polygon (section)

==Regular star polygons==
<div class="skin-invert-image">
{{Infobox
| title            = Regular star polygons
| above            = 2 < 2q < p, [[greatest common divisor|gcd]](p, q) = 1
| abovestyle       = font-size: 14px;
| subheader        = {{image array|perrow=3||width=80
   | image1 = Regular star polygon 5-2.svg | caption1 = [[Pentagram|{5/2}]]
   | image2 = Regular star polygon 7-2.svg | caption2 = [[Heptagram|{7/2}]]
   | image3 = Regular star polygon 7-3.svg | caption3 = [[Heptagram|{7/3}]]
}}
| label1           = [[Schläfli symbol]]
| data1            = {p/q}
| label2           = [[Vertex (geometry)|Vertices]] and [[Edge (geometry)|Edges]]
| data2            = p
| label3           = [[Density (polygon)|Density]]
| data3            = q
| label4           = [[Coxeter diagram]]
| data4            = {{CDD|node_1|p|rat|dq|node}}
| label5           = [[Symmetry group]]
| data5            = [[Dihedral symmetry|Dihedral]] (D<sub>p</sub>)
| label6           = [[Dual polygon]]
| data6            = Self-dual
| label7           = [[Internal angle]]<br>([[degree (angle)|degree]]s)
| data7            = <math>180-\frac{360q}{p}</math><ref>{{cite book |last=Kappraff |first=Jay |title=Beyond measure: a guided tour through nature, myth, and number |publisher=World Scientific |year=2002 |page=258 |isbn= 978-981-02-4702-7 |url=https://books.google.com/books?id=vAfBrK678_kC&q=star+polygon&pg=PA256}}</ref>
}}</div>

A non-convex regular polygon is a regular [[star polygon]]. The most common example is the [[pentagram]], which has the same vertices as a [[pentagon]], but connects alternating vertices.

For an ''n''-sided star polygon, the [[Schläfli symbol]] is modified to indicate the ''density'' or "starriness" ''m'' of the polygon, as {''n''/''m''}. If ''m'' is 2, for example, then every second point is joined. If ''m'' is 3, then every third point is joined. The boundary of the polygon winds around the center ''m'' times.

The (non-degenerate) regular stars of up to 12 sides are:
*[[Pentagram]] – {5/2}
*[[Heptagram]] – {7/2} and {7/3}
*[[Octagram]] – {8/3}
*[[Enneagram (geometry)|Enneagram]] – {9/2} and {9/4}
*[[Decagram (geometry)|Decagram]] – {10/3}
*[[Hendecagram]] – {11/2}, {11/3}, {11/4} and {11/5}
*[[Dodecagram]] – {12/5}

''m'' and ''n'' must be [[coprime]], or the figure will degenerate.

The degenerate regular stars of up to 12 sides are:
*Tetragon – {4/2}
*Hexagons – {6/2}, {6/3}
*Octagons – {8/2}, {8/4}
*Enneagon – {9/3}
*Decagons – {10/2}, {10/4}, and {10/5}
*Dodecagons – {12/2}, {12/3}, {12/4}, and {12/6}

{| class="wikitable floatright skin-invert-image"
|+ Two interpretations of {6/2}
! Grünbaum<br>{6/2} or 2{3}<ref name=Branko>[http://www.math.washington.edu/~grunbaum/Your%20polyhedra-my%20polyhedra.pdf Are Your Polyhedra the Same as My Polyhedra?] [[Branko Grünbaum]] (2003), Fig. 3</ref>
! Coxeter<br>''2''{3} or {6}[2{3}]{6}
|-
| [[File:Doubly wound hexagon.svg|160px]]
| [[File:Regular star figure 2(3,1).svg|160px]]
|-
! Doubly-wound hexagon
! Hexagram as a compound<br>of two triangles
|}
Depending on the precise derivation of the Schläfli symbol, opinions differ as to the nature of the degenerate figure. For example, {6/2} may be treated in either of two ways:
* For much of the 20th century (see for example {{harvtxt|Coxeter|1948}}), we have commonly taken the /2 to indicate joining each vertex of a convex {6} to its near neighbors two steps away, to obtain the regular [[compound polygon|compound]] of two triangles, or [[hexagram]]. {{paragraph}}Coxeter clarifies this regular compound with a notation {kp}[k{p}]{kp} for the compound {p/k}, so the [[hexagram]] is represented as {6}[2{3}]{6}.<ref>Regular polytopes, p.95</ref> More compactly Coxeter also writes ''2''{n/2}, like ''2''{3} for a hexagram as compound as [[Alternation (geometry)|alternations]] of regular even-sided polygons, with italics on the leading factor to differentiate it from the coinciding interpretation.<ref>Coxeter, The Densities of the Regular Polytopes II, 1932, p.53</ref>
* Many modern geometers, such as Grünbaum (2003),<ref name=Branko/> regard this as incorrect. They take the /2 to indicate moving two places around the {6} at each step, obtaining a "double-wound" triangle that has two vertices superimposed at each corner point and two edges along each line segment. Not only does this fit in better with modern theories of [[abstract polytope]]s, but it also more closely copies the way in which Poinsot (1809) created his star polygons – by taking a single length of wire and bending it at successive points through the same angle until the figure closed.