geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

, an icosagon or 20-gon is a twenty-sided

polygon In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed ''polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two toge ...

. The sum of any icosagon's interior angles is 3240 degrees.

Regular icosagon

The regular icosagon has

Schläfli symbol In geometry, the Schläfli symbol is a notation of the form \ that defines regular polytopes and tessellations. The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, who generalized Euclidean geometry to more ...

, and can also be constructed as a truncated

decagon In geometry, a decagon (from the Greek δέκα ''déka'' and γωνία ''gonía,'' "ten angles") is a ten-sided polygon or 10-gon.. The total sum of the interior angles of a simple decagon is 1440°. A self-intersecting ''regular decagon'' i ...

, , or a twice-truncated

pentagon In geometry, a pentagon (from the Greek πέντε ''pente'' meaning ''five'' and γωνία ''gonia'' meaning ''angle'') is any five-sided polygon or 5-gon. The sum of the internal angles in a simple pentagon is 540°. A pentagon may be simpl ...

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

area Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape A shape or figure is a graphics, graphical representation of an obje ...

of a regular icosagon with edge length is :

A=t^2(1+\sqrt+\sqrt) \simeq 31.5687 t^2.

In terms of the radius of its

circumcircle In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius. Not every polyg ...

, the area is :

A=\frac(\sqrt-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 ''The Price Is Right'' is a television game show franchise created by Bob Stewart, originally produced by Mark Goodson and Bill Todman; currently it is produced and owned by Fremantle. The franchise centers on television game shows, but also inc ...

'' 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

golygon A golygon, or more generally a serial isogon of 90°, is any polygon with all right angles (a rectilinear polygon) whose sides are consecutive integer lengths. Golygons were invented and named by Lee Sallows, and popularized by A.K. Dewdney in a 19 ...

al path, the

swastika The swastika (卐 or 卍) is an ancient religious and cultural symbol, predominantly in various Eurasian, as well as some African and American cultures, now also widely recognized for its appropriation by the Nazi Party and by neo-Nazis. It ...

is considered to be an irregular icosagon.

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

Construction

As , regular icosagon is constructible using a

compass and straightedge In geometry, straightedge-and-compass construction – also known as ruler-and-compass construction, Euclidean construction, or classical construction – is the construction of lengths, angles, and other geometric figures using only an ideali ...

, or by an edge-

bisection In geometry, bisection is the division of something into two equal or congruent parts, usually by a line, which is then called a ''bisector''. The most often considered types of bisectors are the ''segment bisector'' (a line that passes through ...

of a regular

, or a twice-bisected regular

The golden ratio in an icosagon

* In the construction with given side length the circular arc around with radius , shares the segment in ratio of the golden ratio. :

\frac = \frac = \frac =\varphi \approx 1.618

Symmetry

The ''regular icosagon'' has symmetry, order 40. There are 5 subgroup dihedral symmetries: , and , and 6

cyclic group In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C''n'', that is generated by a single element. That is, it is a set of invertible elements with a single associative bina ...

symmetries: , and (. 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 John Horton Conway (26 December 1937 – 11 April 2020) was an English mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. He also made contributions to many branches ...

labels these by a letter and group order. Full symmetry of the regular form is and no symmetry is labeled . The dihedral symmetries are divided depending on whether they pass through vertices ( for diagonal) or edges ( for perpendiculars), and when reflection lines path through both edges and vertices. Cyclic symmetries in the middle column are labeled as for their central gyration orders. Each subgroup symmetry allows one or more degrees of freedom for irregular forms. Only the subgroup has no degrees of freedom but can seen as

directed edge In mathematics, and more specifically in graph theory, a directed graph (or digraph) is a graph that is made up of a set of vertices connected by directed edges, often called arcs. Definition In formal terms, a directed graph is an ordered pa ...

s. The highest symmetry irregular icosagons are , an isogonal icosagon constructed by ten mirrors which can alternate long and short edges, and , an

isotoxal In geometry, a polytope (for example, a polygon or a polyhedron) or a tiling is isotoxal () or edge-transitive if its symmetries act transitively on its edges. Informally, this means that there is only one type of edge to the object: given two ...

icosagon, constructed with equal edge lengths, but vertices alternating two different internal angles. These two forms are

duals ''Duals'' is a compilation album by the Irish rock band U2. It was released in April 2011 to u2.com subscribers. Track listing :* "Where the Streets Have No Name" and "Amazing Grace" are studio mix of U2's performance at the Rose Bowl, Pas ...

of each other and have half the symmetry order of the regular icosagon.

Dissection

Coxeter Harold Scott MacDonald "Donald" Coxeter, (9 February 1907 – 31 March 2003) was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century. Biography Coxeter was born in Kensington to ...

states that every

zonogon In geometry, a zonogon is a centrally-symmetric, convex polygon. Equivalently, it is a convex polygon whose sides can be grouped into parallel pairs with equal lengths and opposite orientations. Examples A regular polygon is a zonogon if and ...

(a -gon whose opposite sides are parallel and of equal length) can be dissected into parallelograms. In particular this is true for regular polygons with evenly many sides, in which case the parallelograms are all rhombi. For the icosagon, , and it can be divided into 45: 5 squares and 4 sets of 10 rhombs. This decomposition is based on a

Petrie polygon In geometry, a Petrie polygon for a regular polytope of dimensions is a skew polygon in which every consecutive sides (but no ) belongs to one of the facets. The Petrie polygon of a regular polygon is the regular polygon itself; that of a reg ...

projection of a

10-cube In geometry, a 10-cube is a ten-dimensional hypercube. It has 1024 vertices, 5120 edges, 11520 square faces, 15360 cubic cells, 13440 tesseract 4-faces, 8064 5-cube 5-faces, 3360 6-cube 6-faces, 960 7-cube 7-faces, 180 8-cube 8-faces, and 20 9 ...

, 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.

Related polygons

An icosagram is a 20-sided

star polygon In geometry, a star polygon is a type of non-convex polygon. Regular star polygons have been studied in depth; while star polygons in general appear not to have been formally defined, certain notable ones can arise through truncation operations ...

, represented by symbol . There are three regular forms given by

s: , , and . There are also five regular star figures (compounds) using the same

vertex arrangement In geometry, a vertex arrangement is a set of points in space described by their relative positions. They can be described by their use in polytopes. For example, a ''square vertex arrangement'' is understood to mean four points in a plane, equ ...

: , , , , , and . Deeper truncations of the regular decagon and decagram can produce isogonal (

vertex-transitive In geometry, a polytope (e.g. a polygon or polyhedron) or a tiling is isogonal or vertex-transitive if all its vertices are equivalent under the symmetries of the figure. This implies that each vertex is surrounded by the same kinds of face in ...

) intermediate icosagram forms with equally spaced vertices and two edge lengths.The Lighter Side of Mathematics: Proceedings of the Eugène Strens Memorial Conference on Recreational Mathematics and its History, (1994), ''Metamorphoses of polygons'',

Branko Grünbaum Branko Grünbaum ( he, ברנקו גרונבאום; 2 October 1929 – 14 September 2018) was a Croatian-born mathematician of Jewish descentdecagram, has a quasitruncation , and finally a simple truncation of a decagram gives .

Petrie polygons

The regular icosagon is the

for a number of higher-dimensional polytopes, shown in

orthogonal projection In linear algebra and functional analysis, a projection is a linear transformation P from a vector space to itself (an endomorphism) such that P\circ P=P. That is, whenever P is applied twice to any vector, it gives the same result as if it wer ...

s in

Coxeter plane In mathematics, the Coxeter number ''h'' is the order of a Coxeter element of an irreducible Coxeter group. It is named after H.S.M. Coxeter. Definitions Note that this article assumes a finite Coxeter group. For infinite Coxeter groups, there a ...

s: It is also the Petrie polygon for the

icosahedral 120-cell In geometry, the icosahedral 120-cell, polyicosahedron, faceting, faceted 600-cell or icosaplex is a regular star 4-polytope with Schläfli symbol . It is one of 10 regular Schläfli-Hess polytopes. It is constructed by 5 icosahedron, icosahedra a ...

small stellated 120-cell In geometry, the small stellated 120-cell or stellated polydodecahedron is a regular star 4-polytope with Schläfli symbol . It is one of 10 regular Schläfli-Hess polytopes. Related polytopes It has the same edge arrangement as the great gr ...

great icosahedral 120-cell In geometry, the great icosahedral 120-cell, great polyicosahedron or great faceting, faceted 600-cell is a regular star 4-polytope with Schläfli symbol . It is one of 10 regular Schläfli-Hess polytopes. Related polytopes It has the same edge ...

, and great grand 120-cell.

References

External links

Naming Polygons and Polyhedraicosagon
{{polygons Constructible polygons Polygons by the number of sides