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In geometry, an icosagon or 20-gon is a twenty-sided polygon. The sum of any icosagon's interior angles is 3240 degrees.
One interior angle in a regular polygon icosagon is 162°, meaning that one exterior angle would be 18°.
The area 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.
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.Muriel Pritchett, University of Georgia "To Span the Globe" , see also Editor's Note, retrieved on 10 January 2016
As a path, the swastika is considered to be an irregular icosagon.
A regular square, pentagon, and icosagon can completely fill a plane vertex.
Construction of a regular icosagon | Construction of a regular decagon |
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.John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, (2008) The Symmetries of Things, (Chapter 20, Generalized Schaefli symbols, Types of symmetry of a polygon pp. 275-278) 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 .
The highest symmetry irregular icosagons are , an isogonal figure icosagon constructed by ten mirrors which can alternate long and short edges, and , an isotoxal figure icosagon, constructed with equal edge lengths, but vertices alternating two different internal angles. These two forms are dual polygon of each other and have half the symmetry order of the regular icosagon.
| + 20-gon with 180 rhombs | |
regular | Isotoxal |
Coxeter states that every zonogon (a -gon whose opposite sides are parallel and of equal length) can be dissected into parallelograms.Coxeter, Mathematical recreations and Essays, Thirteenth edition, p.141 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 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.
| + Dissection into 45 rhombs | ||||
10-cube |
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
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 .
| + 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} |
19-simplex | 10-orthoplex | 10-cube | 11-demicube | (421) | 600-cell | Grand antiprism | duopyramid | duoprism |
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