# Icosagon

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Regular icosagon
A regular icosagon
Type Regular polygon
Edges and vertices 20
Schläfli symbol {20}, t{10}, tt{5}
Coxeter–Dynkin diagrams
Symmetry group Dihedral (D20), order 2×20
Internal angle (degrees)162°
Properties Convex, cyclic, equilateral, isogonal, isotoxal

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 {20}, and can also be constructed as a truncated decagon, t{10}, or a twice-truncated pentagon, tt{5}.

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

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

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

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

since the area of the circle is ${\displaystyle \pi R^{2},}$ 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. [1]

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

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:

 Construction of a regular icosagon 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.
${\displaystyle {\frac {\overline {E_{20}E_{1}}}{\overline {E_{1}F}}}={\frac {\overline {E_{20}F}}{\overline {E_{20}E_{1}}}}={\frac {1+{\sqrt {5}}}{2}}=\varphi \approx 1.618}$

## Symmetry

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. [3] 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 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

 regular 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. [4] 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 enumerates the number of solutions as 18,410,581,880, including up to 20-fold rotations and chiral forms in reflection.

 10-cube

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

n12345
FormConvex polygonCompoundStar polygonCompound
Image
{20/1} = {20}

{20/2} = 2{10}

{20/3}

{20/4} = 4{5}

{20/5} = 5{4}
Interior angle 162°144°126°108°90°
n678910
FormCompoundStar polygonCompoundStar polygonCompound
Image
{20/6} = 2{10/3}

{20/7}

{20/8} = 4{5/2}

{20/9}

{20/10} = 10{2}
Interior angle72°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. [5]

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}.

Quasiregular Quasiregular t{10}={20} t{10/9}={20/9} t{10/3}={20/3} 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:

A19B10D11E8H4½2H22H2

19-simplex

10-orthoplex

10-cube

11-demicube

(421)

600-cell

Grand antiprism

10-10 duopyramid

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.

## Related Research Articles

In geometry, an octahedron is a polyhedron with eight faces. The term is most commonly used to refer to the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex.

In geometry, a hexagon is a six-sided polygon or 6-gon. The total of the internal angles of any simple (non-self-intersecting) hexagon is 720°.

In geometry, the truncated icosidodecahedron is an Archimedean solid, one of thirteen convex, isogonal, non-prismatic solids constructed by two or more types of regular polygon faces.

In geometry, the snub dodecahedron, or snub icosidodecahedron, is an Archimedean solid, one of thirteen convex isogonal nonprismatic solids constructed by two or more types of regular polygon faces.

In geometry, an octagon is an eight-sided polygon or 8-gon.

In geometry, a decagon is a ten-sided polygon or 10-gon. The total sum of the interior angles of a simple decagon is 1440°.

In Euclidean geometry, a regular polygon is a polygon that is direct equiangular and equilateral. Regular polygons may be either convex, star or skew. In the limit, a sequence of regular polygons with an increasing number of sides approximates a circle, if the perimeter or area is fixed, or a regular apeirogon, if the edge length is fixed.

In geometry, the 600-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol {3,3,5}. It is also known as the C600, hexacosichoron and hexacosihedroid. It is also called a tetraplex (abbreviated from "tetrahedral complex") and a polytetrahedron, being bounded by tetrahedral cells.

In geometry, a dodecagon or 12-gon is any twelve-sided polygon.

In geometry, a tridecagon or triskaidecagon or 13-gon is a thirteen-sided polygon.

In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles. It can also be defined as a rectangle with two equal-length adjacent sides. It is the only regular polygon whose internal angle, central angle, and external angle are all equal (90°), and whose diagonals are all equal in length. A square with vertices ABCD would be denoted ABCD.

In geometry, a triacontagon or 30-gon is a thirty-sided polygon. The sum of any triacontagon's interior angles is 5040 degrees.

In geometry, a pentadecagon or pentakaidecagon or 15-gon is a fifteen-sided polygon.

In geometry, a tetradecagon or tetrakaidecagon or 14-gon is a fourteen-sided polygon.

In mathematics, a hexadecagon is a sixteen-sided polygon.

In geometry, an octadecagon or 18-gon is an eighteen-sided polygon.

In geometry, a decagram is a 10-point star polygon. There is one regular decagram, containing the vertices of a regular decagon, but connected by every third point. Its Schläfli symbol is {10/3}.

In geometry, a pentagon is any five-sided polygon or 5-gon. The sum of the internal angles in a simple pentagon is 540°.

In geometry, an icositetragon or 24-gon is a twenty-four-sided polygon. The sum of any icositetragon's interior angles is 3960 degrees.

In geometry, an infinite skew polygon or skew apeirogon is an infinite 2-polytope with vertices that are not all colinear. Infinite zig-zag skew polygons are 2-dimensional infinite skew polygons with vertices alternating between two parallel lines. Infinite helical polygons are 3-dimensional infinite skew polygons with vertices on the surface of a cylinder.

## References

1. Muriel Pritchett, University of Georgia "To Span the Globe" Archived 10 June 2010 at the Wayback Machine , see also Editor's Note, retrieved on 10 January 2016
2. Weisstein, Eric W. "Icosagon". MathWorld .
3. 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)
4. Coxeter, Mathematical recreations and Essays, Thirteenth edition, p.141
5. The Lighter Side of Mathematics: Proceedings of the Eugène Strens Memorial Conference on Recreational Mathematics and its History, (1994), Metamorphoses of polygons, Branko Grünbaum