geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

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

polygon In geometry, a polygon () is a plane figure made up of line segments connected to form a closed polygonal chain. The segments of a closed polygonal chain are called its '' edges'' or ''sides''. The points where two edges meet are the polygon ...

. 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 List of regular polytopes and compounds, regular polytopes and tessellations. The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, wh ...

, and can also be constructed as a truncated decagon, , or a twice-truncated

pentagon In geometry, a pentagon () is any five-sided polygon or 5-gon. The sum of the internal angles in a simple polygon, simple pentagon is 540°. A pentagon may be simple or list of self-intersecting polygons, self-intersecting. A self-intersecting ...

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

area Area is the measure of a region's size on a surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while '' surface area'' refers to the area of an open surface or the boundary of a three-di ...

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 triangle is a circle that passes through all three vertex (geometry), vertices. The center of this circle is called the circumcenter of the triangle, and its radius is called the circumrad ...

, 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 an American television game show where contestants compete by guessing the prices of merchandise to win cash and prizes. A 1972 revival by Mark Goodson and Bill Todman of their The Price Is Right (1956 American game ...

'' 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 The swastika (卐 or 卍, ) is a symbol used in various Eurasian religions and cultures, as well as a few Indigenous peoples of Africa, African and Indigenous peoples of the Americas, American cultures. In the Western world, it is widely rec ...

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 Idealiz ...

, or by an edge-

bisection In geometry, bisection is the division of something into two equal or congruent parts (having the same shape and size). Usually it involves a bisecting line, also called a ''bisector''. The most often considered types of bisectors are the ''s ...

of a regular decagon, 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 abstract algebra, a cyclic group or monogenous group is a Group (mathematics), group, denoted C_n (also frequently \Z_n or Z_n, not to be confused with the commutative ring of P-adic number, -adic numbers), that is Generating set of a group, ge ...

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 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 be seen as directed edges. The highest symmetry irregular icosagons are , an isogonal icosagon constructed by ten mirrors which can alternate long and short edges, and , an isotoxal 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, ...

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

Dissection

Coxeter states that every zonogon (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, 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, Decagram (geometry)#Related figures, certain notable ones can ...

, 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: , , , , , and . Deeper truncations of the regular decagon and decagram can produce isogonal ( vertex-transitive) 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 (; 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 we ...

s in Coxeter planes: 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

External links

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