Icosagon

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Regular icosagon
Regular polygon 20 annotated.svg
A regular icosagon
Type Regular polygon
Edges and vertices 20
Schläfli symbol {20}
t{10}
Coxeter diagram CDel node 1.pngCDel 20.pngCDel node.png
CDel node 1.pngCDel 10.pngCDel node 1.png
Symmetry group Dihedral (D20), order 2×20
Internal angle (degrees) 162°
Dual polygon self
Properties convex, cyclic, equilateral, isogonal, isotoxal

In geometry, an icosagon is a twenty-sided polygon. The sum of any icosagon's interior angles is 3240 degrees.

The regular icosagon is a constructible polygon, by an edge-bisection of a regular decagon, and can be seen as a truncated decagon, t{10}. One interior angle in a regular icosagon is 162°, meaning that one exterior angle would be 18°.

Regular Icosagon[edit]

The area of a regular icosagon is: (with t = edge length)

A={5}t^2(1+\sqrt{5}+\sqrt{5+2\sqrt{5}}) \simeq 31.56875757 t^2.

Uses[edit]

The Big Wheel on the popular US 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.[1]

As a golygonal path, the swastika is considered to be an irregular icosagon.[2]

4.5.20 vertex.png A regular square, pentagon, and icosagon can completely fill a plane vertex.

Construction[edit]

A regular icosagon is constructible using a compass and straightedge:

Regular Icosagon Inscribed in a Circle.gif
Construction of a regular icosagon

Related polygons[edit]

An icosagram is a 20-sided star polygon, represented by symbol {20/n}. There are 3 regular forms given by Schläfli symbols: {20/3}, {20/7}, and {20/9}. There are also 5 regular star figures (compounds) using the same vertex arrangement: 2{10}, 4{5}, 5{4}, 2{10/3}, 4{5/2}, and 10{2}.

n 1 2 3 4 5
Form Convex polygon Compound Star polygon Compound
Image Regular polygon 20.svg
{20/1} = {20}
Regular star figure 2(10,1).svg
{20/2} = 2{10}
Regular star polygon 20-3.svg
{20/3}
Regular star figure 4(5,1).svg
{20/4} = 4{5}
Regular star figure 5(4,1).svg
{20/5} = 5{4}
Interior angle 162° 144° 126° 108° 90°
n 6 7 8 9 10
Form Compound Star polygon Compound Star polygon Compound
Image Regular star figure 2(10,3).svg
{20/6} = 2{10/3}
Regular star polygon 20-7.svg
{20/7}
Regular star figure 4(5,2).svg
{20/8} = 4{5/2}
Regular star polygon 20-9.svg
{20/9}
Regular star figure 10(2,1).svg
{20/10} = 10{2}
Interior angle 72° 54° 36° 18°

Deeper truncations of the regular decagon and decagram can produce isogonal (vertex-transitive) intermediate icosagram forms with equally spaced vertices and two edge lengths.[3]

A regular icosagram, {20/9}, can be seen as a quasitruncated decagon, t{10/9}={20/9}. Similarly a decagram, {10/3} has a quasitruncation t{10/7}={20/7}, and finally a simple truncation of a decagram gives t{10/3}={20/3}.

Icosagrams as truncations of a regular decagons and decagrams, {10}, {10/3}
Quasiregular Quasiregular
Regular polygon truncation 10 1.svg
t{10}={20}
Regular polygon truncation 10 2.svg Regular polygon truncation 10 3.svg Regular polygon truncation 10 4.svg Regular polygon truncation 10 5.svg Regular polygon truncation 10 6.svg
t{10/9}={20/9}
Regular star truncation 10-3 1.svg
t{10/3}={20/3}
Regular star truncation 10-3 2.svg Regular star truncation 10-3 3.svg Regular star truncation 10-3 4.svg Regular star truncation 10-3 5.svg Regular star truncation 10-3 6.svg
t{10/7}={20/7}

Petrie polygons[edit]

The regular icosagon is the Petrie polygon for a number of higher-dimensional polytopes, shown in orthogonal projections in Coxeter planes:

A19 B10 D11 E8 H4 2H2
19-simplex t0.svg
19-simplex
10-cube t9.svg
10-orthoplex
10-cube t0.svg
10-cube
11-demicube.svg
11-demicube
4 21 t0 p20.svg
(421)
600-cell t0 p20.svg
600-cell
10-10 duopyramid ortho-3.png
10-10 duopyramid
10-10 duoprism ortho-3.png
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[edit]

  1. ^ [1]
  2. ^ Weisstein, Eric W., "Icosagon", MathWorld.
  3. ^ The Lighter Side of Mathematics: Proceedings of the Eugène Strens Memorial Conference on Recreational Mathematics and its History, (1994), Metamorphoses of polygons, Branko Grünbaum