View source for Icosagon

{{short description|Polygon with 20 edges}}
{{Use dmy dates|date=December 2018}}
{{Regular polygon db|Regular polygon stat table|p20}}
In [[geometry]], an '''icosagon''' or 20-gon is a twenty-sided [[polygon]]. The sum of any icosagon's interior angles is 3240 degrees.

==Regular icosagon==

The [[Regular polygon|regular]] icosagon has [[Schläfli symbol]] {{math|{{mset|20}}}}, and can also be constructed as a [[Truncation (geometry)|truncated]] [[decagon]], {{math|t{{mset|10}}}}, or a twice-truncated [[pentagon]], {{math|tt{{mset|5}}}}.

One interior angle in a [[regular polygon|regular]] icosagon is 162°, meaning that one exterior angle would be 18°.

The [[area]] of a regular icosagon with edge length {{math|''t''}} is
:<math>A={5}t^2(1+\sqrt{5}+\sqrt{5+2\sqrt{5}}) \simeq 31.5687 t^2.</math>

In terms of the radius {{math|''R''}} of its [[circumcircle]], the area is

:<math>A=\frac{5R^2}{2}(\sqrt{5}-1);</math>

since the area of the circle is <math>\pi R^2,</math> the regular icosagon fills approximately 98.36% of its circumcircle.

==Uses==
The Big Wheel on the popular US game show ''[[The Price Is Right (U.S. game show)|The Price Is Right]]'' has an icosagonal cross-section.

The Globe, the outdoor theater used by William Shakespeare's acting company, was discovered to have been built on an icosagonal foundation when a partial excavation was done in 1989.<ref>Muriel Pritchett, University of Georgia [http://researchmagazine.uga.edu/92f/globe.html "To Span the Globe"] {{Webarchive|url=https://web.archive.org/web/20100610105634/http://researchmagazine.uga.edu/92f/globe.html |date=10 June 2010 }}, see also Editor's Note, retrieved on 10 January 2016</ref>

As a [[golygon]]al path, the [[swastika]] is considered to be an irregular icosagon.<ref>{{MathWorld |title=Icosagon |urlname=Icosagon}}</ref>

[[File:4.5.20 vertex.png|160px]]
A regular square, pentagon, and icosagon can completely [[Euclidean tilings of convex regular polygons|fill a plane vertex]].

===Construction===
As {{math|1=20 = 2<sup>2</sup> × 5}}, regular icosagon is [[constructible polygon|constructible]] using a [[compass and straightedge]], or by an edge-[[bisection]] of a regular [[decagon]], or a twice-bisected regular [[pentagon]]:

{| class=wikitable
|[[File:Regular Icosagon Inscribed in a Circle.gif|350px]]<BR>Construction of a regular icosagon
|[[File:Regular Decagon Inscribed in a Circle.gif|350px]]<BR>Construction of a regular decagon
|}

== The golden ratio in an icosagon ==
* In the construction with given side length the circular arc around {{math|C}} with radius {{math|{{Overline|CD}}}}, shares the segment {{math|{{Overline|E<sub>20</sub>F}}}} in ratio of the golden ratio.
:<math>\frac{\overline{ E_{20}E_1}}{\overline{E_1 F}} = \frac{\overline{E_{20} F}}{\overline{ E_{20}E_1}} = \frac{1+ \sqrt{5}}{2} =\varphi \approx 1.618</math>
[[File:01-Zwanzigeck-Seite-gegeben Animation.gif |thumb|left|500px|Icosagon with given side length, animation (The construction is very similar to that of [[Decagon#The golden ratio in decagon|decagon with given side length)]]]]
{{clear}}

== Symmetry==
[[File:Symmetries_of_icosagon.png|thumb|360px|Symmetries of a regular icosagon. Vertices are colored by their symmetry positions. Blue mirrors are drawn through vertices, and purple mirrors are drawn through edge. Gyration orders are given in the center.]]

The ''regular icosagon'' has [[dihedral symmetry|{{math|Dih<sub>20</sub>}} symmetry]], order 40. There are 5 subgroup dihedral symmetries: {{math|(Dih<sub>10</sub>, Dih<sub>5</sub>)}}, and {{math|(Dih<sub>4</sub>, Dih<sub>2</sub>, and Dih<sub>1</sub>)}}, and 6 [[cyclic group]] symmetries: {{math|(Z<sub>20</sub>, Z<sub>10</sub>, Z<sub>5</sub>)}}, and ({{math|Z<sub>4</sub>, Z<sub>2</sub>, Z<sub>1</sub>)}}.

These 10 symmetries can be seen in 16 distinct symmetries on the icosagon, a larger number because the lines of reflections can either pass through vertices or edges. [[John Horton Conway|John Conway]] labels these by a letter and group order.<ref>John H. Conway, Heidi Burgiel, [[Chaim Goodman-Strauss]], (2008) The Symmetries of Things, {{ISBN|978-1-56881-220-5}} (Chapter 20, Generalized Schaefli symbols, Types of symmetry of a polygon pp. 275-278)</ref> Full symmetry of the regular form is {{math|'''r40'''}} and no symmetry is labeled {{math|'''a1'''}}. The dihedral symmetries are divided depending on whether they pass through vertices ({{math|'''d'''}} for diagonal) or edges ({{math|'''p'''}} for perpendiculars), and {{math|'''i'''}} when reflection lines path through both edges and vertices. Cyclic symmetries in the middle column are labeled as {{math|'''g'''}} for their central gyration orders.

Each subgroup symmetry allows one or more degrees of freedom for irregular forms. Only the {{math|'''g20'''}} subgroup has no degrees of freedom but can be seen as [[directed edge]]s.

The highest symmetry irregular icosagons are {{math|'''d20'''}}, an [[isogonal figure|isogonal]] icosagon constructed by ten mirrors which can alternate long and short edges, and {{math|'''p20'''}}, an [[isotoxal figure|isotoxal]] icosagon, constructed with equal edge lengths, but vertices alternating two different internal angles. These two forms are [[dual polygon|duals]] of each other and have half the symmetry order of the regular icosagon.

{{-}}

== Dissection==
{| class=wikitable align=right
|+ 20-gon with 180 rhombs
|- align=center
|[[File:20-gon rhombic dissection-size2.svg|160px]]<BR>regular
|[[File:Isotoxal 20-gon rhombic dissection-size2.svg|160px]]<BR>Isotoxal
|}

[[Coxeter]] states that every [[zonogon]] (a {{math|2''m''}}-gon whose opposite sides are parallel and of equal length) can be dissected into {{math|''m''(''m''-1)/2}} parallelograms.<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.  For the icosagon, {{math|1=''m''=10}}, and it can be divided into 45: 5 squares and 4 sets of 10 rhombs. This decomposition is based on a [[Petrie polygon]] projection of a [[10-cube]], with 45 of 11520 faces. The list {{OEIS2C|1=A006245}} enumerates the number of solutions as 18,410,581,880, including up to 20-fold rotations and chiral forms in reflection.
{| class=wikitable
|+ Dissection into 45 rhombs
|- align=center valign=top
|[[File:10-cube.svg|140px]]<BR>[[10-cube]]
|[[File:20-gon-dissection.svg|160px]]
|[[File:20-gon rhombic dissection3.svg|160px]]
|[[File:20-gon rhombic dissection4.svg|160px]]
|[[File:20-gon-dissection-random.svg|160px]]
|}

==Related polygons==
An '''icosagram''' is a 20-sided [[star polygon]], represented by symbol {{math|{{mset|20/n}}}}. There are three regular forms given by [[Schläfli symbol]]s: {{math|{{mset|20/3}}}}, {{math|{{mset|20/7}}}}, and {{math|{{mset|20/9}}}}. There are also five regular star figures (compounds) using the same [[vertex arrangement]]: {{math|2{{mset|10}}}}, {{math|4{{mset|5}}}}, {{math|5{{mset|4}}}}, {{math|2{{mset|10/3}}}}, {{math|4{{mset|5/2}}}}, and {{math|10{{mset|2}}}}.
{| class=wikitable
!n
!1
!2
!3
!4
!5
|-
!Form
!Convex polygon
!Compound
!Star polygon
!colspan=2|Compound
|- align=center
!Image
|BGCOLOR="#ffe0e0"|[[File:Regular_polygon_20.svg|120px]]<br>{20/1} = {20}
|[[File:Regular_star_figure_2(10,1).svg|120px]]<BR>{20/2} = 2{10}
|BGCOLOR="#ffe0e0"|[[File:Regular_star_polygon_20-3.svg|120px]]<BR>{20/3}
|[[File:Regular_star_figure_4(5,1).svg|120px]]<BR>{20/4} = 4{5}
|[[File:Regular_star_figure_5(4,1).svg|120px]]<BR>{20/5} = 5{4}
|- align=center
! [[Interior angle]]
| 162°
| 144°
| 126°
| 108°
| 90°
|-
!n
!6
!7
!8
!9
!10
|-
!Form
!Compound
!Star polygon
!Compound
!Star polygon
!Compound
|- align=center
!Image
|[[File:Regular_star_figure_2(10,3).svg|120px]]<BR>{20/6} = 2{10/3}
|BGCOLOR="#ffe0e0"|[[File:Regular_star_polygon_20-7.svg|120px]]<BR>{20/7} 
|[[File:Regular_star_figure_4(5,2).svg|120px]]<BR>{20/8} = 4{5/2}
|BGCOLOR="#ffe0e0"|[[File:Regular_star_polygon_20-9.svg|120px]]<BR>{20/9} 
|[[File:Regular_star_figure_10(2,1).svg|120px]]<BR>{20/10} = 10{2}
|- align=center
! Interior angle
| 72°
| 54°
| 36°
| 18°
| 0°
|}

Deeper truncations of the regular decagon and decagram can produce isogonal ([[vertex-transitive]]) intermediate icosagram forms with equally spaced vertices and two edge lengths.<ref>The Lighter Side of Mathematics: Proceedings of the Eugène Strens Memorial Conference on Recreational Mathematics and its History, (1994), ''Metamorphoses of polygons'', [[Branko Grünbaum]]</ref>

A regular icosagram, {{math|{{mset|20/9}}}}, can be seen as a quasitruncated decagon, {{math|1=t{{mset|10/9}}={{mset|20/9}}}}. Similarly a [[decagram (geometry)|decagram]], {{math|{{mset|10/3}}}} has a quasitruncation {{math|1=t{{mset|10/7}}={{mset|20/7}}}}, and finally a simple truncation of a decagram gives {{math|1=t{{mset|10/3}}={{mset|20/3}}}}.

{| class=wikitable width=360
|+ Icosagrams as truncations of a regular decagons and decagrams, {10}, {10/3}
!Quasiregular
|colspan=4|
!Quasiregular
|- align=center valign=top
|BGCOLOR="#ffe0e0"|[[File:regular_polygon_truncation_10_1.svg|100px]]<BR>t{10}={20}
|[[File:regular_polygon_truncation_10_2.svg|100px]]
|[[File:regular_polygon_truncation_10_3.svg|100px]]
|[[File:regular_polygon_truncation_10_4.svg|100px]]
|[[File:regular_polygon_truncation_10_5.svg|100px]]
|BGCOLOR="#ffe0e0"|[[File:regular_polygon_truncation_10_6.svg|100px]]<BR>t{10/9}={20/9}
|- align=center valign=top
|BGCOLOR="#ffe0e0"|[[File:regular_star_truncation_10-3_1.svg|100px]]<BR>t{10/3}={20/3}
|[[File:regular_star_truncation_10-3_2.svg|100px]]
|[[File:regular_star_truncation_10-3_3.svg|100px]]
|[[File:regular_star_truncation_10-3_4.svg|100px]]
|[[File:regular_star_truncation_10-3_5.svg|100px]]
|BGCOLOR="#ffe0e0"|[[File:regular_star_truncation_10-3_6.svg|100px]]<BR>t{10/7}={20/7}
|}

==Petrie polygons==
The regular icosagon is the [[Petrie polygon]] for a number of higher-dimensional polytopes, shown in [[orthogonal projection]]s in [[Coxeter plane]]s:

{| class=wikitable
|- align=center
!A<sub>19</sub>
!colspan=2|B<sub>10</sub>
!D<sub>11</sub>
!E<sub>8</sub>
!H<sub>4</sub>
!{{sfrac|1|2}}2H<sub>2</sub>
!colspan=2|2H<sub>2</sub>
|- align=center
|[[File:19-simplex_t0.svg|100px]]<br>[[19-simplex]]
|[[File:10-cube_t9.svg|100px]]<br>[[10-orthoplex]]
|[[File:10-cube_t0.svg|100px]]<br>[[10-cube]]
|[[File:11-demicube.svg|100px]]<br>[[11-demicube]]
|[[File:4_21_t0_p20.svg|100px]]<br>[[4 21 polytope|(4<sub>21</sub>)]]
|[[File:600-cell_t0_p20.svg|100px]]<br>[[600-cell]]
|[[File:Grand antiprism 20-gonal orthogonal projection.png|100px]]<br>[[Grand antiprism]]
|[[File:10-10 duopyramid ortho-3.png|100px]]<br>[[duopyramid|10-10 duopyramid]]
|[[File:10-10 duoprism ortho-3.png|100px]]<br>[[duoprism|10-10 duoprism]]
|}
It is also the Petrie polygon for the [[icosahedral 120-cell]], [[small stellated 120-cell]], [[great icosahedral 120-cell]], and [[great grand 120-cell]].

==References==
{{Reflist}}

==External links==
*[http://mathforum.org/dr.math/faq/faq.polygon.names.html Naming Polygons and Polyhedra] {{Webarchive|url=https://web.archive.org/web/20130525061110/http://mathforum.org/dr.math/faq/faq.polygon.names.html |date=25 May 2013 }}
*[https://books.google.com/books?id=ugBDAAAAIAAJ&dq=icosagon&pg=PA194 icosagon]

{{Clear}}
{{polygons}}

[[Category:Constructible polygons]]
[[Category:Polygons by the number of sides]]