Английская Википедия:Icosagon
Шаблон:Short description Шаблон:Use dmy dates Шаблон:Regular polygon db In geometry, an icosagon or 20-gon is a twenty-sided polygon. The sum of any icosagon's interior angles is 3240 degrees.
Regular icosagon
The regular icosagon has Schläfli symbol Шаблон:Math, and can also be constructed as a truncated decagon, Шаблон:Math, or a twice-truncated pentagon, Шаблон:Math.
One interior angle in a regular icosagon is 162°, meaning that one exterior angle would be 18°.
The area of a regular icosagon with edge length Шаблон:Math is
- <math>A={5}t^2(1+\sqrt{5}+\sqrt{5+2\sqrt{5}}) \simeq 31.5687 t^2.</math>
In terms of the radius Шаблон:Math of its circumcircle, the area is
- <math>A=\frac{5R^2}{2}(\sqrt{5}-1);</math>
since the area of the circle is <math>\pi R^2,</math> 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.[1]
As a golygonal path, the swastika is considered to be an irregular icosagon.[2]
Файл:4.5.20 vertex.png A regular square, pentagon, and icosagon can completely fill a plane vertex.
Construction
As Шаблон:Math, regular icosagon is constructible using a compass and straightedge, or by an edge-bisection of a regular decagon, or a twice-bisected regular pentagon:
| Файл:Regular Icosagon Inscribed in a Circle.gif Construction of a regular icosagon |
Файл:Regular Decagon Inscribed in a Circle.gif Construction of a regular decagon |
The golden ratio in an icosagon
- In the construction with given side length the circular arc around Шаблон:Math with radius Шаблон:Math, shares the segment Шаблон:Math in ratio of the golden ratio.
- <math>\frac{\overline{ E_{20}E_1}}{\overline{E_1 F}} = \frac{\overline{E_{20} F}}{\overline{ E_{20}E_1}} = \frac{1+ \sqrt{5}}{2} =\varphi \approx 1.618</math>
Symmetry
The regular icosagon has [[dihedral symmetry|Шаблон:Math symmetry]], order 40. There are 5 subgroup dihedral symmetries: Шаблон:Math, and Шаблон:Math, and 6 cyclic group symmetries: Шаблон:Math, and (Шаблон:Math.
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.[3] Full symmetry of the regular form is Шаблон:Math and no symmetry is labeled Шаблон:Math. The dihedral symmetries are divided depending on whether they pass through vertices (Шаблон:Math for diagonal) or edges (Шаблон:Math for perpendiculars), and Шаблон:Math when reflection lines path through both edges and vertices. Cyclic symmetries in the middle column are labeled as Шаблон:Math for their central gyration orders.
Each subgroup symmetry allows one or more degrees of freedom for irregular forms. Only the Шаблон:Math subgroup has no degrees of freedom but can be seen as directed edges.
The highest symmetry irregular icosagons are Шаблон:Math, an isogonal icosagon constructed by ten mirrors which can alternate long and short edges, and Шаблон:Math, an isotoxal 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
| Файл:20-gon rhombic dissection-size2.svg regular |
Файл:Isotoxal 20-gon rhombic dissection-size2.svg Isotoxal |
Coxeter states that every zonogon (a Шаблон:Math-gon whose opposite sides are parallel and of equal length) can be dissected into Шаблон:Math parallelograms.[4] In particular this is true for regular polygons with evenly many sides, in which case the parallelograms are all rhombi. For the icosagon, Шаблон:Math, 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 Шаблон:OEIS2C enumerates the number of solutions as 18,410,581,880, including up to 20-fold rotations and chiral forms in reflection.
| Файл:10-cube.svg 10-cube |
Файл:20-gon-dissection.svg | Файл:20-gon rhombic dissection3.svg | Файл:20-gon rhombic dissection4.svg | Файл:20-gon-dissection-random.svg |
Related polygons
An icosagram is a 20-sided star polygon, represented by symbol Шаблон:Math. There are three regular forms given by Schläfli symbols: Шаблон:Math, Шаблон:Math, and Шаблон:Math. There are also five regular star figures (compounds) using the same vertex arrangement: Шаблон:Math, Шаблон:Math, Шаблон:Math, Шаблон:Math, Шаблон:Math, and Шаблон:Math.
| n | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| Form | Convex polygon | Compound | Star polygon | Compound | |
| Image | Файл:Regular polygon 20.svg {20/1} = {20} |
Файл:Regular star figure 2(10,1).svg {20/2} = 2{10} |
Файл:Regular star polygon 20-3.svg {20/3} |
Файл:Regular star figure 4(5,1).svg {20/4} = 4{5} |
Файл:Regular star figure 5(4,1).svg {20/5} = 5{4} |
| Interior angle | 162° | 144° | 126° | 108° | 90° |
| n | 6 | 7 | 8 | 9 | 10 |
| Form | Compound | Star polygon | Compound | Star polygon | Compound |
| Image | Файл:Regular star figure 2(10,3).svg {20/6} = 2{10/3} |
Файл:Regular star polygon 20-7.svg {20/7} |
Файл:Regular star figure 4(5,2).svg {20/8} = 4{5/2} |
Файл:Regular star polygon 20-9.svg {20/9} |
Файл:Regular star figure 10(2,1).svg {20/10} = 10{2} |
| Interior angle | 72° | 54° | 36° | 18° | 0° |
Deeper truncations of the regular decagon and decagram can produce isogonal (vertex-transitive) intermediate icosagram forms with equally spaced vertices and two edge lengths.[5]
A regular icosagram, Шаблон:Math, can be seen as a quasitruncated decagon, Шаблон:Math. Similarly a decagram, Шаблон:Math has a quasitruncation Шаблон:Math, and finally a simple truncation of a decagram gives Шаблон:Math.
| Quasiregular | Quasiregular | ||||
|---|---|---|---|---|---|
| Файл:Regular polygon truncation 10 1.svg t{10}={20} |
Файл:Regular polygon truncation 10 2.svg | Файл:Regular polygon truncation 10 3.svg | Файл:Regular polygon truncation 10 4.svg | Файл:Regular polygon truncation 10 5.svg | Файл:Regular polygon truncation 10 6.svg t{10/9}={20/9} |
| Файл:Regular star truncation 10-3 1.svg t{10/3}={20/3} |
Файл:Regular star truncation 10-3 2.svg | Файл:Regular star truncation 10-3 3.svg | Файл:Regular star truncation 10-3 4.svg | Файл:Regular star truncation 10-3 5.svg | Файл:Regular star truncation 10-3 6.svg t{10/7}={20/7} |
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, and great grand 120-cell.
References
External links
- ↑ Muriel Pritchett, University of Georgia "To Span the Globe" Шаблон:Webarchive, see also Editor's Note, retrieved on 10 January 2016
- ↑ Шаблон:MathWorld
- ↑ John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, (2008) The Symmetries of Things, Шаблон:ISBN (Chapter 20, Generalized Schaefli symbols, Types of symmetry of a polygon pp. 275-278)
- ↑ Coxeter, Mathematical recreations and Essays, Thirteenth edition, p.141
- ↑ The Lighter Side of Mathematics: Proceedings of the Eugène Strens Memorial Conference on Recreational Mathematics and its History, (1994), Metamorphoses of polygons, Branko Grünbaum
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