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A regular square, pentagon, and icosagon can completely fill a plane vertex.
The ''regular icosagon'' has symmetry, order 40. There are 5 subgroup dihedral symmetries: , and , and 6 cyclic group 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 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
Naming Polygons and Polyhedraicosagon
{{polygons Constructible polygons Polygons by the number of sides
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 ...
. The sum of any icosagon's interior angles is 3240 degrees.
Regular icosagon
The regular icosagon hasSchläfli 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 mo ...
, and can also be constructed as a truncated decagon, , 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 sim ...
, .
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 or planar lamina, while '' surface area'' refers to the area of an op ...
of a regular icosagon with edge length is
:
In terms of the radius 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. As agolygon
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 ...
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. I ...
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, 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 throug ...
of a regular decagon, or a twice-bisected regular 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 sim ...
:
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. :
Symmetry
The ''regular icosagon'' has symmetry, order 40. There are 5 subgroup dihedral symmetries: , and , and 6 cyclic group 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 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
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 ...
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
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 projection of a10-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 ...
, 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-sidedstar 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 operatio ...
, represented by symbol . There are three regular forms given by 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 mo ...
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
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 fa ...
) 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
A regular icosagram, , can be seen as a quasitruncated decagon, . Similarly a decagram, has a quasitruncation , and finally a simple truncation of a decagram gives .
Petrie polygons
The regular icosagon is the Petrie polygon for a number of higher-dimensional polytopes, shown in orthogonal projections in Coxeter planes: It is also the Petrie polygon for the icosahedral 120-cell, small stellated 120-cell,great icosahedral 120-cell
In geometry, the great icosahedral 120-cell, great polyicosahedron or great 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 arran ...
, and great grand 120-cell
In geometry, the great grand 120-cell or great grand 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 small ste ...
.
References
External links
Naming Polygons and Polyhedra
{{polygons Constructible polygons Polygons by the number of sides