Icosagon

Template:Short description Template:Use dmy dates Template:Regular polygon db 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 icosagon has Schläfli symbol Template:Mset, and can also be constructed as a truncated decagon, tTemplate:Mset, or a twice-truncated pentagon, ttTemplate:Mset.

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

The area of a regular icosagon with edge length t is

$A=5t^2(1+\sqrt{5}+\sqrt{5+2\sqrt{5}})\simeq 31.5687t^2.$

In terms of the radius R of its circumcircle, the area is

$A=\frac{5R^2}{2}(\sqrt{5}-1);$

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

Uses

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.

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

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

Construction

As 20 = 2² × 5, regular icosagon is constructible using a compass and straightedge, or by an edge-bisection of a regular decagon, or a twice-bisected regular pentagon:

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

File:Regular Decagon Inscribed in a Circle.gif Construction of a regular decagon

The golden ratio in an icosagon

• In the construction with given side length the circular arc around C with radius CD, shares the segment E₂₀F in ratio of the golden ratio.

$\frac{\overline{E_{20}{E}_1}}{\overline{E_{1}F}}=\frac{\overline{E_{20}F}}{\overline{E_{20}{E}_1}}=\frac{1+\sqrt{5}}{2}=\phi \approx 1.618$

File:01-Zwanzigeck-Seite-gegeben Animation.gif

Icosagon with given side length, animation (The construction is very similar to that of decagon with given side length)

Symmetry

File:Symmetries of icosagon.png

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 Dih₂₀ symmetry, order 40. There are 5 subgroup dihedral symmetries: (Dih₁₀, Dih₅), and (Dih₄, Dih₂, and Dih₁), and 6 cyclic group symmetries: (Z₂₀, Z₁₀, Z₅), and (Z₄, Z₂, Z₁).

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 Conway labels these by a letter and group order. Full symmetry of the regular form is r40 and no symmetry is labeled a1. The dihedral symmetries are divided depending on whether they pass through vertices (d for diagonal) or edges (p for perpendiculars), and i when reflection lines path through both edges and vertices. Cyclic symmetries in the middle column are labeled as g for their central gyration orders.

Each subgroup symmetry allows one or more degrees of freedom for irregular forms. Only the g20 subgroup has no degrees of freedom but can be seen as directed edges.

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

Dissection

20-gon with 180 rhombs

File:20-gon rhombic dissection-size2.svg regular

File:Isotoxal 20-gon rhombic dissection-size2.svg Isotoxal

Coxeter states that every zonogon (a 2m-gon whose opposite sides are parallel and of equal length) can be dissected into m(m-1)/2 parallelograms. In particular this is true for regular polygons with evenly many sides, in which case the parallelograms are all rhombi. For the icosagon, 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 OEIS: A006245 enumerates the number of solutions as 18,410,581,880, including up to 20-fold rotations and chiral forms in reflection.

Dissection into 45 rhombs

File:10-cube.svg 10-cube

File:20-gon-dissection.svg

File:20-gon rhombic dissection3.svg

File:20-gon rhombic dissection4.svg

File:20-gon-dissection-random.svg

Related polygons

An icosagram is a 20-sided star polygon, represented by symbol Template:Mset. There are three regular forms given by Schläfli symbols: Template:Mset, Template:Mset, and Template:Mset. There are also five regular star figures (compounds) using the same vertex arrangement: 2Template:Mset, 4Template:Mset, 5Template:Mset, 2Template:Mset, 4Template:Mset, and 10Template:Mset.

n	1	2	3	4	5
Form	Convex polygon	Compound	Star polygon	Compound
Image	File:Regular polygon 20.svg {20/1} = {20}	File:Regular star figure 2(10,1).svg {20/2} = 2{10}	File:Regular star polygon 20-3.svg {20/3}	File:Regular star figure 4(5,1).svg {20/4} = 4{5}	File: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	File:Regular star figure 2(10,3).svg {20/6} = 2{10/3}	File:Regular star polygon 20-7.svg {20/7}	File:Regular star figure 4(5,2).svg {20/8} = 4{5/2}	File:Regular star polygon 20-9.svg {20/9}	File:Regular star figure 10(2,1).svg {20/10} = 10{2}
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.

A regular icosagram, Template:Mset, can be seen as a quasitruncated decagon, tTemplate:Mset=Template:Mset. Similarly a decagram, Template:Mset has a quasitruncation tTemplate:Mset=Template:Mset, and finally a simple truncation of a decagram gives tTemplate:Mset=Template:Mset.

Icosagrams as truncations of a regular decagons and decagrams, {10}, {10/3}

Quasiregular					Quasiregular
File:Regular polygon truncation 10 1.svg t{10}={20}	File:Regular polygon truncation 10 2.svg	File:Regular polygon truncation 10 3.svg	File:Regular polygon truncation 10 4.svg	File:Regular polygon truncation 10 5.svg	File:Regular polygon truncation 10 6.svg t{10/9}={20/9}
File:Regular star truncation 10-3 1.svg t{10/3}={20/3}	File:Regular star truncation 10-3 2.svg	File:Regular star truncation 10-3 3.svg	File:Regular star truncation 10-3 4.svg	File:Regular star truncation 10-3 5.svg	File:Regular star truncation 10-3 6.svg t{10/7}={20/7}

Petrie polygons

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	Template:Sfrac2H2	2H2
File:19-simplex t0.svg 19-simplex	File:10-cube t9.svg 10-orthoplex	File:10-cube t0.svg 10-cube	File:11-demicube.svg 11-demicube	File:4 21 t0 p20.svg (421)	File:600-cell t0 p20.svg 600-cell	File:Grand antiprism 20-gonal orthogonal projection.png Grand antiprism	File:10-10 duopyramid ortho-3.png 10-10 duopyramid	File: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.

External links

• Naming Polygons and Polyhedra Template:Webarchive
• icosagon