Icosagon
An icosagon is a shape with 20 sides and 20 corners. It has interior angles of 162 and exterior angles of 18.
| Regular icosagon | |
|---|---|
 A regular icosagon | |
| Type | Regular polygon |
| Edges and vertices | 20 |
| Schläfli symbol | {20}, t{10}, tt{5} |
| Coxeter diagram |     |
| Symmetry group | Dihedral (D20), order 2×20 |
| Internal angle (degrees) | 162° |
| Dual polygon | Self |
| Properties | Convex, cyclic, equilateral, isogonal, isotoxal |
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}.
Area
The amount of space a regular icosagon takes up is
- [math]\displaystyle{ \text{Area} = {5}a^2(1+\sqrt{5}+\sqrt{5+2\sqrt{5}}). }[/math]
a is the length of one of its sides.
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.
Dissection
Coxeter states that every parallel-sided 2m-gon can be divided into m(m-1)/2 rhombs. 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. [3] The list A006245 enumerates the number of solutions as 18,410,581,880, including up to 20-fold rotations and chiral forms in reflection.
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Icosagon Media
Regular Icosagon Inscribed in a Circle
Regular Decagon Inscribed in a Circle
Icosagon with given side length, animation (The construction is very similar to that of decagon with given side length)
Symmetries of a regular icosagon. Vertices are colored by their symmetry positions. Blue mirrors are drawn through vertices, and purple mirrors are drawn through edge. Gyration orders are given in the center.
20-gon rhombic dissection-size2
Isotoxal 20-gon rhombic dissection-size2
20-gon rhombic dissection3
References
- ↑ Muriel Pritchett, University of Georgia "To Span the Globe", see also Editor's Note, retrieved on 10th January 2016
- ↑ Weisstein, Eric W., "Icosagon" from MathWorld.
- ↑ Coxeter, Mathematical recreations and Essays, Thirteenth edition, p.141