View source for Regular polygon

{{short description|Equiangular and equilateral polygon}}
{| class=wikitable align="right" width="255"
!bgcolor=#e7dcc3 colspan=2|Set of convex regular n-gons
|-
|align=center colspan=2|
[[Image:Regular polygon 3 annotated.svg|60px]]60px60px[[Image:Regular polygon 6 annotated.svg|60px]]<br>
[[Image:Regular polygon 7 annotated.svg|60px]]60px60px[[Image:Regular polygon 10 annotated.svg|60px]]<br>
[[Image:Regular polygon 11 annotated.svg|60px]]60px60px[[Image:Regular polygon 14 annotated.svg|60px]]<br>
[[Image:Regular polygon 15 annotated.svg|60px]]60px60px60px<br>60px[[Image:Regular polygon 20 annotated.svg|60px]]<br>
Regular polygons
|-
|bgcolor=#e7dcc3|[[Edge (geometry)|Edge]]s and [[Vertex (geometry)|vertices]]||''n''
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]||{''n''}
|-
|bgcolor=#e7dcc3|Coxeter–Dynkin diagram||{{CDD|node_1|n|node}}
|-
|bgcolor=#e7dcc3|[[Point group|Symmetry group]]||D<sub>n</sub>, order 2n
|-
|bgcolor=#e7dcc3|[[Dual polygon]]||Self-dual
|-
| bgcolor=#e7dcc3 | [[Area]]<br /> (with side length, ''s'') || <math>A = \tfrac{1}{4}ns^2 \cot\left(\frac{\pi}{n}\right)</math>
|-
| bgcolor=#e7dcc3 | Internal angle || <math>(n - 2) \times \frac{180^\circ}{n}</math>
|-
| bgcolor=#e7dcc3 | Internal angle sum || <math>\left(n - 2\right)\times 180^\circ</math>
|-
| bgcolor=#e7dcc3 | Inscribed circle diameter || <math>d_\text{IC} = s\cot\left(\frac{\pi}{n}\right)</math>
|-
| bgcolor=#e7dcc3 | Circumscribed circle diameter || <math>d_\text{OC} = s\csc\left(\frac{\pi}{n}\right)</math>
|-
|bgcolor=#e7dcc3|Properties || [[Convex polygon|Convex]], cyclic, equilateral, [[Isogonal figure|isogonal]], [[Isotoxal figure|isotoxal]]
|}
In [[Euclidean geometry]], a '''regular polygon''' is a [[Polygon|polygon]] that is [[Equiangular polygon|equiangular]] (all angles are equal in measure) and [[Equilateral polygon|equilateral]] (all sides have the same length). Regular polygons may be either '''[[Convex polygon|convex]]''' or '''[[Star polygon|star]]'''. In the [[Limit (mathematics)|limit]], a sequence of regular polygons with an increasing number of sides approximates a [[Circle|circle]], if the [[Perimeter|perimeter]] or [[Area|area]] is fixed, or a regular [[Apeirogon|apeirogon]] (effectively a [[Line (geometry)|straight line]]), if the edge length is fixed.

==General properties==
[[File:regular star polygons.svg|thumb|300px|Regular convex and star polygons with 3 to 12 vertices labelled with their Schläfli symbols]]
''These properties apply to all regular polygons, whether convex or [[Star polygon|star]].''

A regular ''n''-sided polygon has [[Rotational symmetry|rotational symmetry]] of order ''n''.

All vertices of a regular polygon lie on a common circle (the [[Circumscribed circle|circumscribed circle]]); i.e., they are [[Concyclic points|concyclic points]]. That is, a regular polygon is a cyclic polygon.

Together with the property of equal-length sides, this implies that every regular polygon also has an inscribed circle or incircle that is tangent to every side at the midpoint. Thus a regular polygon is a [[Tangential polygon|tangential polygon]].

A regular ''n''-sided polygon can be constructed with compass and straightedge if and only if the [[Odd number|odd]] [[Prime number|prime]] factors of ''n'' are distinct Fermat primes. See [[Constructible polygon|constructible polygon]].

===Symmetry===
The [[Symmetry group|symmetry group]] of an ''n''-sided regular polygon is [[Dihedral group|dihedral group]] ''D<sub>n</sub>'' (of order 2''n''): ''D''<sub>2</sub>, [[Dihedral group of order 6|''D''<sub>3</sub>]], [[Examples of groups#A symmetry group|''D''<sub>4</sub>]], ... It consists of the rotations in ''C<sub>n</sub>'', together with [[Reflection symmetry|reflection symmetry]] in ''n'' axes that pass through the center. If ''n'' is even then half of these axes pass through two opposite vertices, and the other half through the midpoint of opposite sides. If ''n'' is odd then all axes pass through a vertex and the midpoint of the opposite side.

==Regular convex polygons==
All regular [[Simple polygon|simple polygon]]s (a simple polygon is one that does not intersect itself anywhere) are convex. Those having the same number of sides are also [[Similarity (geometry)|similar]].

An ''n''-sided convex regular polygon is denoted by its [[Schläfli symbol]] {''n''}. For ''n'' &lt; 3, we have two [[Degeneracy (mathematics)|degenerate]] cases:
; [[Monogon]] {1}: Degenerate in [[Euclidean geometry|ordinary space]]. (Most authorities do not regard the monogon as a true polygon, partly because of this, and also because the formulae below do not work, and its structure is not that of any [[Abstract polytope|abstract polygon]].)
; [[Digon]] {2}; a "double line segment": Degenerate in [[Euclidean geometry|ordinary space]]. (Some authorities do not regard the digon as a true polygon because of this.)

In certain contexts all the polygons considered will be regular. In such circumstances it is customary to drop the prefix regular. For instance, all the faces of uniform polyhedra must be regular and the faces will be described simply as triangle, square, pentagon, etc.

===Angles===
For a regular convex ''n''-gon, each interior angle has a measure of:
: <math>\left(1 - \frac{2}{n}\right)\times 180\,\,\,</math> degrees, or equivalently <math>\frac{180(n - 2)}{n}</math> degrees;
: <math>\frac{(n - 2)\pi}{n}</math> radians; or
: <math>\frac{(n - 2)}{2n}</math> full [[Turn (geometry)|turns]],

and each exterior angle (i.e., supplementary to the interior angle) has a measure of <math>\tfrac{360}{n}</math> degrees, with the sum of the exterior angles equal to 360 degrees or 2π radians or one full turn.

As the number of sides, n approaches infinity, the internal angle approaches 180 degrees. For a regular polygon with 10,000 sides (a [[Myriagon|myriagon]]) the internal angle is 179.964°. As the number of sides increase, the internal angle can come very close to 180°, and the shape of the polygon approaches that of a circle. However the polygon can never become a circle. The value of the internal angle can never become exactly equal to 180°, as the circumference would effectively become a straight line. For this reason, a circle is not a polygon with an infinite number of sides.

===Diagonals===
For ''n'' &gt; 2, the number of [[Diagonal|diagonal]]s is <math>\tfrac{1}{2}n(n - 3)</math>; i.e., 0, 2, 5, 9, ..., for a triangle, square, pentagon, hexagon, ... . The diagonals divide the polygon into 1, 4, 11, 24, ... pieces {{oeis|A007678}}.

For a regular ''n''-gon inscribed in a unit-radius circle, the product of the distances from a given vertex to all other vertices (including adjacent vertices and vertices connected by a diagonal) equals ''n''.

===Points in the plane===

For a regular simple ''n''-gon with circumradius ''R'' and distances ''d<sub>i</sub>'' from an arbitrary point in the plane to the vertices, we have<ref>Park, Poo-Sung. "Regular polytope distances", [[Organization:Forum Geometricorum|Forum Geometricorum]] 16, 2016, 227-232. http://forumgeom.fau.edu/FG2016volume16/FG201627.pdf</ref>

:<math>\frac{\sum_{i=1}^n d_i^4}{n} + 3R^4 = \left(\frac{\sum_{i=1}^n d_i^2}{n} + R^2\right)^2.</math>

====Interior points====
For a regular ''n''-gon, the sum of the perpendicular distances from any interior point to the ''n'' sides is ''n'' times the [[Apothem|apothem]]<ref name=Johnson>Johnson, Roger A., ''Advanced Euclidean Geometry'', Dover Publ., 2007 (orig. 1929).</ref>{{rp|p. 72}} (the apothem being the distance from the center to any side). This is a generalization of [[Viviani's theorem]] for the ''n''=3 case.<ref>Pickover, Clifford A, ''The Math Book'', Sterling, 2009: p. 150</ref><ref>Chen, Zhibo, and Liang, Tian. "The converse of Viviani's theorem", ''[[Organization:The College Mathematics Journal|The College Mathematics Journal]]'' 37(5), 2006, pp. 390–391.</ref>

===Circumradius===
[[Image:PolygonParameters.png|thumb|left|180px|Regular [[Pentagon|pentagon]] (''n'' = 5) with [[Edge (geometry)|side]] ''s'', circumradius ''R'' and [[Apothem|apothem]] ''a'']]
The circumradius ''R'' from the center of a regular polygon to one of the vertices is related to the side length ''s'' or to the [[Apothem|apothem]] ''a'' by
:<math>R = \frac{s}{2 \sin\left(\frac{\pi}{n}\right)} = \frac{a}{\cos\left(\frac{\pi}{n} \right)}</math>

For [[Constructible polygon|constructible polygon]]s, [[Algebraic expression|algebraic expression]]s for these relationships exist; see [[Bicentric polygon]].

The sum of the perpendiculars from a regular ''n''-gon's vertices to any line tangent to the circumcircle equals ''n'' times the circumradius.<ref name=Johnson/>{{rp|p. 73}}

The sum of the squared distances from the vertices of a regular ''n''-gon to any point on its circumcircle equals 2''nR''<sup>2</sup> where ''R'' is the circumradius.<ref name=Johnson/>{{rp|p.73}}

The sum of the squared distances from the midpoints of the sides of a regular ''n''-gon to any point on the circumcircle is 2''nR''<sup>2</sup> − {{sfrac|''ns''<sup>2</sup>|4}}, where ''s'' is the side length and ''R'' is the circumradius.<ref name=Johnson/>{{rp|p. 73}}
{{Clear|left}}
=== Dissections===
Coxeter states that every [[Zonogon|zonogon]] (a 2''m''-gon whose opposite sides are parallel and of equal length) can be dissected into <math>\tbinom{n}{2}</math> or ''m''(''m''-1)/2 parallelograms.
These tilings are contained as subsets of vertices, edges and faces in orthogonal projections [[Hypercube|''m''-cubes]].<ref>Coxeter, Mathematical recreations and Essays, Thirteenth edition, p.141</ref>
In particular this is true for regular polygons with evenly many sides, in which case the parallelograms are all rhombi.
The list {{OEIS2C|1=A006245}} gives the number of solutions for smaller polygons.
{| class=wikitable
|+ Example dissections for select even-sided regular polygons
!2''m''
![[Hexagon#Dissection|6]]
![[Octagon#Dissection|8]]
![[Decagon#Dissection|10]]
![[Dodecagon#Dissection|12]]
![[Tetradecagon#Dissection|14]]
![[Hexadecagon#Dissection|16]]
![[Octadecagon#Dissection|18]]
![[Icosagon#Dissection|20]]
![[Icositetragon#Dissection|24]]
!30
![[Tetracontagon#Dissection|40]]
![[Pentacontagon#Dissection|50]]
|- align=center valign=top
!Image
|[[File:6-gon rhombic dissection.svg|60px]]
|[[File:8-gon rhombic dissection.svg|60px]]
|[[File:Sun_decagon.svg|60px]]
|[[File:12-gon rhombic dissection.svg|60px]]
|[[File:14-gon-dissection-star.svg|60px]]
|[[File:16-gon rhombic dissection.svg|60px]]
|[[File:18-gon-dissection-star.svg|60px]]
|[[File:20-gon rhombic dissection.svg|60px]]
|[[File:24-gon rhombic dissection.svg|60px]]
|[[File:30-gon-dissection-star.svg|60px]]
|[[File:40-gon rhombic dissection.svg|60px]]
|[[File:50-gon-dissection-star.svg|60px]]
|- align=center valign=top
!Rhombs
|3
|6
|10
|15
|21
|28
|36
|45
|66
|105
|190
|300
|}

===Area===<!--This section is linked from [[Truncated icosahedron]]-->
The area ''A'' of a convex regular ''n''-sided polygon having [[Edge (geometry)|side]] ''s'', [[Circumscribed circle|circumradius]] ''R'', [[Apothem|apothem]] ''a'', and [[Perimeter|perimeter]] ''p'' is given by<ref>{{cite web |url=http://www.mathopenref.com/polygonregulararea.html |title=Math Open Reference |accessdate=4 Feb 2014}}</ref><ref>{{cite web |url=http://www.mathwords.com/a/area_regular_polygon.htm |title=Mathwords}}</ref>
:<math>A = \tfrac{1}{2}nsa = \tfrac{1}{2}pa = \tfrac{1}{4}ns^2\cot\left(\tfrac{\pi}{n}\right) = na^2\tan\left(\tfrac{\pi}{n}\right) = \tfrac{1}{2}nR^2\sin\left(\tfrac{2\pi}{n}\right)</math>

For regular polygons with side ''s'' = 1, circumradius ''R'' = 1, or apothem ''a'' = 1, this produces the following table:<ref>Results for ''R'' = 1 and ''a'' = 1 obtained with [[Software:Maple|Maple]], using function definition:
<source lang="mupad">
f := proc (n)
options operator, arrow;
[
 [convert(1/4*n*cot(Pi/n), radical), convert(1/4*n*cot(Pi/n), float)],
 [convert(1/2*n*sin(2*Pi/n), radical), convert(1/2*n*sin(2*Pi/n), float), convert(1/2*n*sin(2*Pi/n)/Pi, float)],
 [convert(n*tan(Pi/n), radical), convert(n*tan(Pi/n), float), convert(n*tan(Pi/n)/Pi, float)]
]
end proc
</source><p>The expressions for ''n''=16 are obtained by twice applying the [[Tangent half-angle formula|tangent half-angle formula]] to tan(π/4)</ref> (Note that since <math>\cot x \rightarrow 1/x</math> as <math>x \rightarrow 0</math><ref>[[Trigonometric functions]]</ref>, the area when <math>s=1</math> is tending to <math>n^2/4\pi</math> as <math>n</math> grows large.)
{{Clear}}
<div style="overflow:auto">
{| class=wikitable style="text-align:center;"
|-
!                            rowspan="2" | Number <br/>of sides
! style="background:#FF8080" colspan="2" | Area when side ''s'' = 1
! style="background:#80FF80" colspan="3" | Area when circumradius ''R'' = 1
! style="background:#8080FF" colspan="3" | Area when apothem ''a'' = 1
|-
! Exact
! Approximation
! Exact
! Approximation
! As an (approximate)<br/>fraction of<br/>circumcircle{{nbsp}}area
! Exact
! Approximation
! As an (approximate)<br/>multiple of<br/>{{nbsp}}incircle{{nbsp}}area
|-
| ''n''
| <math>\tfrac{n}{4}\cot\left(\tfrac{\pi}{n}\right)</math>  ||
| <math>\tfrac{n}{2}\sin\left(\tfrac{2\pi}{n}\right)</math> ||  || <math>\tfrac{n}{2\pi}\sin\left(\tfrac{2\pi}{n}\right)</math>
| <math>n \tan\left(\tfrac{\pi}{n}\right)</math>            ||  || <math>\tfrac{n}{\pi}\tan\left(\tfrac{\pi}{n}\right)</math>
|-
| [[Equilateral triangle|3]]
| {{tmath|\tfrac{ \sqrt{3} }{4} }} || 0.433012702
| {{tmath|\tfrac{3\sqrt{3} }{4} }} || 1.299038105 || 0.4134966714
| {{tmath|3\sqrt{3}             }} || 5.196152424 || 1.653986686
|-
| [[Square|4]]
| 1 || 1.000000000
| 2 || 2.000000000 || 0.6366197722
| 4 || 4.000000000 || 1.273239544
|-
| 5
| {{tmath|\tfrac{1}{4}\sqrt{25 + 10\sqrt{5} } }}                      || 1.720477401
| {{tmath|\tfrac{5}{4}\sqrt{\tfrac{1}{2}\left(5 + \sqrt{5}\right)} }} || 2.377641291 || 0.7568267288
| {{tmath|5\sqrt{5 - 2\sqrt{5} } }}                                   || 3.632712640 || 1.156328347
|-
| 6
| {{tmath|\tfrac{3\sqrt{3} }{2} }} || 2.598076211
| {{tmath|\tfrac{3\sqrt{3} }{2} }} || 2.598076211 || 0.8269933428
| {{tmath|       2\sqrt{3}      }} || 3.464101616 || 1.102657791
|-
| 7
|  || 3.633912444
|  || 2.736410189 || 0.8710264157
|  || 3.371022333 || 1.073029735
|-
| 8
| {{tmath|2 + 2\sqrt{2}             }} || 4.828427125
| {{tmath|    2\sqrt{2}             }} || 2.828427125 || 0.9003163160
| {{tmath|8\left(\sqrt{2} - 1\right)}} || 3.313708500 || 1.054786175
|-
| 9
|  || 6.181824194
|  || 2.892544244 || 0.9207254290
|  || 3.275732109 || 1.042697914
|-
| 10
| {{tmath|\tfrac{5}{2}\sqrt{5 + 2\sqrt{5} } }}                        || 7.694208843
| {{tmath|\tfrac{5}{2}\sqrt{\tfrac{1}{2}\left(5 - \sqrt{5}\right)} }} || 2.938926262 || 0.9354892840
| {{tmath|2\sqrt{25 - 10\sqrt{5} } }}                                 || 3.249196963 || 1.034251515
|-
| 11
|  || 9.365639907
|  || 2.973524496 || 0.9465022440
|  || 3.229891423 || 1.028106371
|-
| 12
| {{tmath|6 + 3\sqrt{3} }}               || 11.19615242
| 3                                      || 3.000000000 || 0.9549296586
| {{tmath|12\left(2 - \sqrt{3} \right)}} || 3.215390309 || 1.023490523
|-
| 13
|  || 13.18576833
|  || 3.020700617 || 0.9615188694
|  || 3.204212220 || 1.019932427
|-
| 14
|  || 15.33450194
|  || 3.037186175 || 0.9667663859
|  || 3.195408642 || 1.017130161
|-
| 15
| <ref>{{tmath| \tfrac{15}{8}\left(\sqrt{15} + \sqrt{3} + \sqrt{2\left(5 + \sqrt{5} \right)} \right)}}</ref> || 17.64236291
| <ref>{{tmath| \tfrac{15}{16}\left(\sqrt{15} + \sqrt{3} - \sqrt{10 - 2\sqrt{5} } \right)}}</ref> || 3.050524822 || 0.9710122088
| <ref>{{tmath| \tfrac{15}{2}\left(3\sqrt{3} - \sqrt{15} - \sqrt{2\left(25 - 11\sqrt{5} \right)} \right)}}</ref> || 3.188348426 || 1.014882824
|-
| 16
| <ref>{{tmath| 4 \left(1 + \sqrt{2} + \sqrt{2 \left(2 + \sqrt{2} \right)} \right)}}</ref>     || 20.10935797
| {{tmath| 4\sqrt{2 - \sqrt{2} } }}                                                 || 3.061467460 || 0.9744953584
| <ref>{{tmath| 16 \left(1 + \sqrt{2}\right)\left(\sqrt{2 \left(2 - \sqrt{2} \right)} - 1\right)}}</ref> || 3.182597878 || 1.013052368
|-
| 17
|  || 22.73549190
|  || 3.070554163 || 0.9773877456
|  || 3.177850752 || 1.011541311
|-
| 18
|  || 25.52076819
|  || 3.078181290 || 0.9798155361
|  || 3.173885653 || 1.010279181
|-
| 19
|  || 28.46518943
|  || 3.084644958 || 0.9818729854
|  || 3.170539238 || 1.009213984
|-
| 20
| <ref>{{tmath| 5 \left(1 + \sqrt{5} + \sqrt{5 + 2\sqrt{5} } \right) }}</ref> || 31.56875757
| <ref>{{tmath| \tfrac{5}{2}\left(\sqrt{5} - 1\right)}}</ref>                 || 3.090169944 || 0.9836316430
| <ref>{{tmath| 20 \left(1 + \sqrt{5} -\sqrt{5 + 2\sqrt{5} } \right) }}</ref> || 3.167688806 || 1.008306663
|-
| 100
|  || 795.5128988
|  || 3.139525977 || 0.9993421565
|  || 3.142626605 || 1.000329117
|-
| 1000
|  || 79577.20975
|  || 3.141571983 || 0.9999934200
|  || 3.141602989 || 1.000003290
|-
| 10,000
|  || 7957746.893
|  || 3.141592448 || 0.9999999345
|  || 3.141592757 || 1.000000033
|-
| 1,000,000
|  || 79577471545
|  || 3.141592654 || 1.000000000
|  || 3.141592654 || 1.000000000
|}
</div>
[[File:Polygons comparison.png|thumb|right|400px|Comparison of sizes of regular polygons with the same edge length, from [[Equilateral triangle|three]] to [[Hexacontagon|sixty]] sides. The size increases without bound as the number of sides approaches infinity.]]
Of all ''n''-gons with a given perimeter, the one with the largest area is regular.<ref>Chakerian, G.D. "A Distorted View of Geometry." Ch. 7 in ''Mathematical Plums'' (R. Honsberger, editor). Washington, DC: Mathematical Association of America, 1979: 147.</ref>

== Constructible polygon==
{{main|Constructible polygon}}

Some regular polygons are easy to construct with compass and straightedge; other regular polygons are not constructible at all.
The ancient Greek mathematicians knew how to construct a regular polygon with 3, 4, or 5 sides,<ref name=Bold/>{{rp|p. xi}} and they knew how to construct a regular polygon with double the number of sides of a given regular polygon.<ref name=Bold>Bold, Benjamin. ''Famous Problems of Geometry and How to Solve Them'', Dover Publications, 1982 (orig. 1969).</ref>{{rp|pp. 49–50}} This led to the question being posed: is it possible to construct ''all'' regular ''n''-gons with compass and straightedge? If not, which ''n''-gons are constructible and which are not?

[[Biography:Carl Friedrich Gauss|Carl Friedrich Gauss]] proved the constructibility of the regular [[Heptadecagon|17-gon]] in 1796. Five years later, he developed the theory of [[Gaussian period]]s in his ''[[Disquisitiones Arithmeticae]]''. This theory allowed him to formulate a sufficient condition for the constructibility of regular polygons:

:A regular ''n''-gon can be constructed with compass and straightedge if ''n'' is the product of a power of 2 and any number of distinct Fermat primes (including none).

(A Fermat prime is a [[Prime number|prime number]] of the form <math>2^{(2^n)}+1.</math>) Gauss stated without proof that this condition was also necessary, but never published his proof.  A full proof of necessity was given by [[Biography:Pierre Wantzel|Pierre Wantzel]] in 1837. The result is known as the '''Gauss–Wantzel theorem'''.

Equivalently, a regular ''n''-gon is constructible if and only if the [[Cosine|cosine]] of its common angle is a [[Constructible number|constructible number]]—that is, can be written in terms of the four basic arithmetic operations and the extraction of square roots.

=={{anchor|Skew regular polygons}}Regular skew polygons==
{|class=wikitable align=right width=400
|- valign=top
|[[File:Cube petrie polygon sideview.png|160px]]<br>The [[Cube|cube]] contains a skew regular [[Hexagon|hexagon]], seen as 6 red edges zig-zagging between two planes perpendicular to the cube's diagonal axis.
|[[File:Antiprism17.jpg|240px]]<br>The zig-zagging side edges of a ''n''-[[Antiprism|antiprism]] represent a regular skew 2''n''-gon, as shown in this 17-gonal antiprism.
|}

A ''regular [[Skew polygon|skew polygon]]'' in 3-space can be seen as nonplanar paths zig-zagging between two parallel planes, defined as the side-edges of a uniform [[Antiprism|antiprism]]. All edges and internal angles are equal.
{|class=wikitable width=480
|[[File:Petrie polygons.png|480px]]<br>The [[Platonic solid|Platonic solid]]s (the [[Tetrahedron|tetrahedron]], [[Cube|cube]], [[Octahedron|octahedron]], [[Dodecahedron|dodecahedron]], and [[Icosahedron|icosahedron]]) have Petrie polygons, seen in red here, with sides 4, 6, 6, 10, and 10 respectively.
|}

More generally ''regular skew polygons'' can be defined in ''n''-space. Examples include the [[Petrie polygon]]s, polygonal paths of edges that divide a [[Regular polytope|regular polytope]] into two halves, and seen as a regular polygon in orthogonal projection.

In the infinite limit ''regular skew polygons'' become skew [[Apeirogon|apeirogon]]s.
{{Clear}}

==Regular star polygons==
{| class="wikitable floatright"
|+ Regular star polygons
|-
! colspan=2 | 2 < 2q < p, gcd(p, q) = 1
{| class="wikitable" style="text-align:center;"
|-
| [[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}]]...
|}
|-
! [[Schläfli symbol]]
| {p/q}
|-
! [[Vertex (geometry)|Vertices]] and [[Edge (geometry)|Edges]]
| p
|-
! Density
| q
|-
! Coxeter diagram
| {{CDD|node_1|p|rat|dq|node}}
|-
! [[Symmetry group]]
| Dihedral (D<sub>p</sub>)
|-
! [[Dual polygon]]
| Self-dual
|-
! Internal angle<br>([[Degree (angle)|degree]]s)
| <math>\frac{180(p - 2q)}{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&pg=PA256&dq=star+polygon}}</ref>
|}
A non-convex regular polygon is a regular [[Star polygon|star polygon]]. The most common example is the [[Pentagram|pentagram]], which has the same vertices as a [[Pentagon|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"
|+ 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?] [[Biography:Branko Grünbaum|Branko Grünbaum]] (2003), Fig. 3</ref>
! Coxeter<br>''2''{3} or {6}[2{3}]{6}
|-
| [[File:Doubly wound hexagon.png|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 of two triangles, or [[Hexagram|hexagram]]. {{paragraph}}Coxeter clarifies this regular compound with a notation {kp}[k{p}]{kp} for the compound {p/k}, so the [[Hexagram|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|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.

==Duality of regular polygons ==
All regular polygons are self-dual to congruency, and for odd ''n'' they are self-dual to identity.

In addition, the regular star figures (compounds), being composed of regular polygons, are also self-dual.

==Regular polygons as faces of polyhedra==
A [[Uniform polyhedron|uniform polyhedron]] has regular polygons as faces, such that for every two vertices there is an [[Isometry|isometry]] mapping one into the other (just as there is for a regular polygon).

A [[Quasiregular polyhedron|quasiregular polyhedron]] is a uniform polyhedron which has just two kinds of face alternating around each vertex.

A [[Regular polyhedron|regular polyhedron]] is a uniform polyhedron which has just one kind of face.

The remaining (non-uniform) convex polyhedra with regular faces are known as the Johnson solids.

A polyhedron having regular triangles as faces is called a [[Deltahedron|deltahedron]].

==See also==
*[[Euclidean tilings by convex regular polygons]]
*[[Platonic solid]]
*[[Apeirogon]] – An infinite-sided polygon can also be regular, {∞}.
*[[List of regular polytopes and compounds]]
*[[Equilateral polygon]]
*[[Carlyle circle]]

==Notes==
{{Reflist}}

==References==
*{{Cite journal |first=H.S.M. |last=Coxeter |title=Regular Polytopes |publisher=Methuen and Co. |year=1948 |ref=harv |postscript=<!--None-->}}
*Grünbaum, B.; Are your polyhedra the same as my polyhedra?, ''Discrete and comput. geom: the Goodman-Pollack festschrift'', Ed. Aronov et al., Springer (2003), pp.&nbsp;461–488.
*Poinsot, L.; Memoire sur les polygones et polyèdres. ''J. de l'École Polytechnique'' '''9''' (1810), pp.&nbsp;16–48.

==External links==
*{{mathworld |urlname=RegularPolygon |title=Regular polygon}}
*[http://www.mathopenref.com/polygonregular.html Regular Polygon description] With interactive animation
*[http://www.mathopenref.com/polygonincircle.html Incircle of a Regular Polygon] With interactive animation
*[http://www.mathopenref.com/polygonregulararea.html Area of a Regular Polygon] Three different formulae, with interactive animation
*[http://mathdl.maa.org/convergence/1/?pa=content&sa=viewDocument&nodeId=1056&bodyId=1245 Renaissance artists' constructions of regular polygons] at [http://mathdl.maa.org/convergence/1/ Convergence]

{{Polygons}}
{{Polytopes}}
{{Ancient Greek mathematics}}

{{DEFAULTSORT:Regular Polygon}}
[[Category:Polygons]]

{{Sourceattribution|Regular polygon|1}}