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}, tt{5}
Coxeter–Dynkin diagrams 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°
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.

Contents

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

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

since the area of the circle is 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]

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:

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

The golden ratio in an icosagon

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

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. 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. [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

20-gon with 180 rhombs
20-gon rhombic dissection-size2.svg
regular
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. [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 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
10-cube.svg
10-cube
20-gon-dissection.svg 20-gon rhombic dissection3.svg 20-gon rhombic dissection4.svg 20-gon-dissection-random.svg

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 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°
n678910
FormCompoundStar polygonCompoundStar polygonCompound
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 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}.

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

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

A19B10D11E8H4½2H22H2
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
Grand antiprism 20-gonal orthogonal projection.png
Grand antiprism
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.

Related Research Articles

Octahedron Polyhedron with 8 triangular faces

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<span class="mw-page-title-main">Hexagon</span> Shape with six sides

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°.

Truncated icosidodecahedron Archimedean solid

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.

<span class="mw-page-title-main">Snub dodecahedron</span> Archimedean solid with 92 faces

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<span class="mw-page-title-main">Octagon</span> Polygon shape with eight sides

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

Decagon Shape with ten sides

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.

<span class="mw-page-title-main">600-cell</span> Four-dimensional analog of the icosahedron

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.

<span class="mw-page-title-main">Dodecagon</span> Polygon with 12 edges

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Tridecagon Polygon with 13 edges

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Square Regular quadrilateral

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Triacontagon Polygon with 30 edges

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

Pentadecagon Polygon with 15 edges

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

Tetradecagon Polygon with 14 edges .

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

Hexadecagon Polygon with 16 edges

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

Octadecagon Polygon with 18 edges

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

Decagram (geometry) 10-pointed star 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}.

Pentagon Shape with five sides

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

Icositetragon Polygon with 24 edges

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