Icosagon

Regular icosagon
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
Dual polygon	Self

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

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

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.^[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 $20 = 2 2 \times 5$ , 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 $E 20 F$ 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}}=\varphi \approx 1.618

Symmetry

The regular icosagon has $Dih 20$ symmetry, order 40. There are 5 subgroup dihedral symmetries: $(Dih 10, Dih 5)$ , and $(Dih 4, Dih 2, and Dih 1)$ , and 6 cyclic group symmetries: $(Z 20, Z 10, Z 5)$ , and ( $Z 4, Z 2, Z 1)$ .

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 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
regular	Isotoxal

Coxeter states that every zonogon (a $2 m$ -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

Related polygons

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 $10{2}$ .

n	1	2	3	4	5
Form	Convex polygon	Compound	Star polygon	Compound
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°
n	6	7	8	9	10
Form	Compound	Star polygon	Compound	Star polygon	Compound
Image	{20/6} = 2{10/3}	{20/7}	{20/8} = 4{5/2}	{20/9}	{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.^[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}
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:

A₁₉	B₁₀		D₁₁	E₈	H₄	⁠1/2⁠2H₂	2H₂
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.

References

^ 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
^ Weisstein, Eric W. "Icosagon". MathWorld.
^ 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)
^ Coxeter, Mathematical recreations and Essays, Thirteenth edition, p.141
^ The Lighter Side of Mathematics: Proceedings of the Eugène Strens Memorial Conference on Recreational Mathematics and its History, (1994), Metamorphoses of polygons, Branko Grünbaum

External links

[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

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