Twentieth

A regular twenty-corner

A twenty- corner or icosagon is a polygon with 20 sides and 20 corners. This often means a flat, regular twenty-corner, with all sides of the same length and all corner points on a common perimeter . In the following, only the regular twenty-four and the regular overturned twenty-four are considered.

angle

The central angle is

${\displaystyle \alpha ={\frac {360^{\circ }}{20}}=18^{\circ }}$

pages

The side length compared to the perimeter radius is: ${\displaystyle r_{u}}$

${\displaystyle a=2\cdot r_{u}\cdot \sin {\frac {\alpha }{2}}}$

${\displaystyle a=2\cdot r_{u}\cdot \sin 9^{\circ }}$

With

${\displaystyle \sin 9^{\circ }={\sqrt {\frac {1-{\frac {1}{4}}{\sqrt {10+2{\sqrt {5}}}}}{2}}}}$

Diagonals

The twenty-corner has 170 diagonals:

• 20 diagonals over 2 (or 18) sides
• 20 diagonals over 3 (or 17) sides
• 20 diagonals over 4 (or 16) sides
• 20 diagonals over 5 (or 15) sides
• 20 diagonals over 6 (or 14) sides
• 20 diagonals over 7 (or 13) sides
• 20 diagonals over 8 (or 12) sides
• 20 diagonals over 9 (or 11) sides
• 10 diagonals over 10 sides

The lengths in relation to the perimeter radius are:

• The diagonal across two sides corresponds to the side of a decagon with the same circumference:
  • ${\displaystyle s=2\cdot r_{u}\cdot \sin {18^{\circ }}=2\cdot r_{u}\cdot {\frac {1}{4}}\left({\sqrt {5}}-1\right)\approx r_{u}\cdot 0{,}618033989}$
• The diagonal over three sides:
  • ${\displaystyle s=2\cdot r_{u}\cdot \sin {27^{\circ }}\approx r_{u}\cdot 0{,}907980999}$
• The diagonal across four sides corresponds to the side of a pentagon with the same circumference:
  • ${\displaystyle s=2\cdot r_{u}\cdot \sin {36^{\circ }}=r_{u}\cdot {\sqrt {\frac {5-{\sqrt {5}}}{2}}}\approx r_{u}\cdot 1{,}1755705}$
• The diagonal across five sides corresponds to the side of a square with the same circumference:
  • ${\displaystyle s=2\cdot r_{u}\cdot \sin {45^{\circ }}=r_{u}\cdot {\sqrt {2}}}$
• The diagonal over six sides:
  • ${\displaystyle s=2\cdot r_{u}\cdot \sin {54^{\circ }}=r_{u}\cdot {\frac {1}{2}}\left(1+{\sqrt {5}}\right)}$
• The diagonal over seven sides:
  • ${\displaystyle s=2\cdot r_{u}\cdot \sin {63^{\circ }}\approx r_{u}\cdot 1{,}78201305}$
• The diagonal over eight sides:
  • ${\displaystyle s=2\cdot r_{u}\cdot \sin {72^{\circ }}=r_{u}\cdot {\frac {1}{2}}{\sqrt {10+2{\sqrt {5}}}}\approx r_{u}\cdot 1{,}90211303}$
• The diagonal over nine sides:
  • ${\displaystyle s=2\cdot r_{u}\cdot \sin {81^{\circ }}\approx r_{u}\cdot 1{,}97537668}$
• The diagonal over ten sides corresponds to the diameter of the circumference:
  • ${\displaystyle s=2\cdot r_{u}}$

surface

The area of ​​a regular twenty-corner with the side length and the perimeter radius is calculated by the following formulas. ${\displaystyle a}$ ${\displaystyle r_{u}}$

${\displaystyle A=5a^{2}\left(1+{\sqrt {5}}+{\sqrt {5+2{\sqrt {5}}}}\right)\approx 31{,}5687a^{2}.}$

${\displaystyle A={\frac {5}{2}}r_{u}^{2}\left({\sqrt {5}}-1\right).}$

construction

The regular twenty-corner can be represented as a construction with compass and ruler , the main construction features are already used in the pentagon or in the decagon .

Construction for a given perimeter

The construction in Figure 1 is almost the same as that of the decagon for a given circumference .

It begins with the given diameter and halving it in the center. After drawing the circumference around through , the center axis is drawn in perpendicular to the diameter ; The points of intersection are the first two corner points and the resulting twentagon. This is followed by halving the route in , resulting in the intersection points and on the perimeter. Now an arc is drawn around with the radius until it intersects the line in . The route is thus divided according to the golden ratio with external division . Another short arc around with the radius that cuts the perimeter in the corner , then the connection of the corner with the point , now also at the same time then the first side length of the twenty- corner is constructed. Finally, move the side length fifteen times counterclockwise onto the circumference and connect the neighboring corner points with each other, the regular twenty-corner is thus constructed. ${\displaystyle {\overline {AB}}}$ ${\displaystyle M.}$ ${\displaystyle M}$ ${\displaystyle A}$ ${\displaystyle {\overline {AB}}}$ ${\displaystyle E_{6}}$ ${\displaystyle E_{16}}$ ${\displaystyle {\overline {AM}}}$ ${\displaystyle F}$ ${\displaystyle C}$ ${\displaystyle D}$ ${\displaystyle F}$ ${\displaystyle |FE_{6}|}$ ${\displaystyle {\overline {AB}}}$ ${\displaystyle G}$ ${\displaystyle {\overline {AG}}}$ ${\displaystyle E_{6}}$ ${\displaystyle |E_{6}G|}$ ${\displaystyle E_{2}}$ ${\displaystyle E_{2}}$ ${\displaystyle B}$ ${\displaystyle E_{1},}$ ${\displaystyle a}$ ${\displaystyle a}$

Image 1: Regular twenty-four corner with a given circumference		Fig. 2: Alternative construction of the regular twenty-corner number of construction steps almost the same as in Fig. 1

Construction for a given side length

Image 3: Regular octagon with a given side length,
see animation

The construction in Figure 3 is almost the same as that of the decagon with a given side length . The dotted lines are not required for construction, they are only used to illustrate the following description.

First, the ends of the side are marked with the first corner points (right) and . This is followed by an arc with the radius around the points and ; the points of intersection and subsequently a is half-line from through pulled; it halves the length of the side in a vertical up down follows and creates the point of intersection.If you now extend the distance beyond by approximately the same amount and then turn an arc with the radius , the point of intersection is created on the extension. The route is thus divided according to the golden ratio with external division. Now an arc of a circle with the radius that intersects the vertical half-line is drawn . In the thus resulting isosceles triangle , the angle corresponding to the angle of the apex of the central angle of a regular decagon, ${\displaystyle a}$ ${\displaystyle E_{1}}$ ${\displaystyle E_{20}}$ ${\displaystyle a}$ ${\displaystyle E_{1}}$ ${\displaystyle E_{20}}$ ${\displaystyle A}$ ${\displaystyle B.}$ ${\displaystyle B}$ ${\displaystyle A}$ ${\displaystyle a}$ ${\displaystyle C.}$ ${\displaystyle {\overline {E_{1}E_{20}}}}$ ${\displaystyle E_{1}}$ ${\displaystyle D.}$ ${\displaystyle {\overline {E_{20}E_{1}}}}$ ${\displaystyle E_{1}}$ ${\displaystyle C}$ ${\displaystyle |E_{20}D|}$ ${\displaystyle F}$ ${\displaystyle {\overline {E_{20}F}}}$ ${\displaystyle E_{20}}$ ${\displaystyle {\overline {E_{20}F}}}$ ${\displaystyle G}$ ${\displaystyle E_{20}E_{1}G}$ ${\displaystyle G}$ ${\displaystyle \mu '}$

because for a side length applies in a right triangle ${\displaystyle a=1}$ ${\displaystyle E_{20}CG}$

${\displaystyle {\begin{aligned}\sin \left({\frac {1}{2}}\mu '\right)&={\frac {\frac {a}{2}}{{\frac {1}{2}}+{\frac {\sqrt {5}}{2}}}}={\frac {a}{2\cdot \left({\frac {1}{2}}+{\frac {\sqrt {5}}{2}}\right)}}\end{aligned}}}$

with inserted values

${\displaystyle {\begin{aligned}\sin \left({\frac {\mu '}{2}}\right)&={\frac {0{,}5}{{\frac {1}{2}}+{\frac {\sqrt {5}}{2}}}}\\\end{aligned}}}$

from this follows for angles ${\displaystyle {\frac {\mu '}{2}}}$

${\displaystyle {\begin{aligned}{\frac {\mu '}{2}}&=\arcsin \left({\frac {0{,}5}{{\frac {1}{2}}+{\frac {\sqrt {5}}{2}}}}\right)=18^{\circ }\end{aligned}}}$

thus the angle is equal to the central angle of the decagon. ${\displaystyle \mu '=36^{\circ }}$

It continues with the arc around the point with the radius that intersects the half-line that runs through in and into . Because of the principle of central angle, the angle at the apex is half as large as the central angle of the decagon. Because of this, the center of the wanted twentagon with its central angle is now just draw the circumference around the center , mark the side length sixteen times counter-clockwise on the circumference and connect the neighboring corner points with each other, then the regular twenty-one is constructed. ${\displaystyle G}$ ${\displaystyle {\overline {E_{20}G}}}$ ${\displaystyle B}$ ${\displaystyle A}$ ${\displaystyle O}$ ${\displaystyle H}$ ${\displaystyle {\overline {E_{20}G}}={\overline {GO}}={\overline {GH}}}$ ${\displaystyle \mu }$ ${\displaystyle O}$ ${\displaystyle \mu '}$ ${\displaystyle O}$ ${\displaystyle 18^{\circ }.}$ ${\displaystyle O}$ ${\displaystyle a}$

Regular overturned twenty corners

A regular, crossed-over twentieth corner results when at least one is skipped over each time when connecting the twenty corner points and the chords thus created are of the same length. Such regular stars are noted with Schläfli symbols , indicating the number of corner points and connecting every -th point. ${\displaystyle \left\{n/k\right\}}$ ${\displaystyle n}$ ${\displaystyle k}$

There are only three regular twenty-ray stars, also called icosagrams.

The "stars" with the Schläfli symbols {20/2} and {20/18} are regular decagons , those with the Schläfli symbols {20/4} and {20/16} are pentagons , those with the Schläfli symbols {20/5} and {20/15} are squares . The star with the Schläfli symbols {20/6} and {20/14} is a ten-ray star , also called a decagram.

• Regular twenty-ray stars
• 

  ${\displaystyle \left\{20/3\right\}{,}\ \left\{20/17\right\}}$

• 

  ${\displaystyle \left\{20/7\right\}{,}\ \left\{20/13\right\}}$

• 

  ${\displaystyle \left\{20/9\right\}{,}\ \left\{20/11\right\}}$

Occurrence

• Wuppertal-Heckinghausen gasometer

Individual evidence

1. ^ Henry Green: Euclid's Plane Geometry, Books III – VI, Practically Applied, or Gradations in Euclid, Part II. In: books.google.de. London: Simpkin, Marshall, & CO., 1861, p. 116 , accessed February 10, 2018 .