Dodecagon

A regular dodecagon

The dodecagon , or dodecagon is a geometric figure , and a polygon (polygon) with twelve corners and twelve sides.

Variations

The dodecagon can be represented as:

• concave dodecagon in which at least one interior angle is greater than 180 °. A dodecagon can have at most six such angles . A concave dodecagon can be regular or irregular .
• convex dodecagon in which all interior angles are less than 180 °. A convex dodecagon can be regular or irregular.
• Tendon-dodecagon in which all corners lie on a common perimeter , but the side lengths ( chords ) may be unequal.
• regular dodecagon, it is determined by twelve points on a circumference . The neighboring points are always the same distance from one another and are connected by means of lined up sides or edges .
• Regular, overturned dodecagon, it results when at least one is skipped over each time when connecting the twelve 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 is only one regular twelve-ray star, also called a dodecagram .

The "stars" with the symbols {12/2} and {12/10} are regular hexagons , {12/3} and {12/9} squares, and {12/4} and {12/8} equilateral triangles .

• Examples of equilateral and irregular concave dodecagons
• 

           equilateral

• 

           equilateral

• 

         irregular

• 

         irregular

• Regular twelve-ray star
• 

  ${\displaystyle \left\{12/5\right\}{,}\ \left\{12/7\right\}}$

Regular dodecagon

With a regular dodecagon, all sides are of the same length and all corner points lie on a common perimeter .

Formulas

Mathematical formulas for the regular dodecagon
Area	${\displaystyle A=(6+3\cdot {\sqrt {3}})\cdot a^{2}\approx 11{,}196\cdot a^{2}}$
	${\displaystyle A=(24-12\cdot {\sqrt {3}})\cdot r_{i}^{2}\approx 3{,}215\cdot r_{i}^{2}}$
	${\displaystyle A=3\cdot r_{u}^{2}}$
Length of the diagonal	${\displaystyle d_{2}={\frac {{\sqrt {6}}+{\sqrt {2}}}{2}}\cdot a\approx 1{,}932\cdot a}$
	${\displaystyle d_{3}=(1+{\sqrt {3}})\cdot a\approx 2{,}732\cdot a}$
	${\displaystyle d_{4}={\frac {{\sqrt {6}}+3\cdot {\sqrt {2}}}{2}}\cdot a\approx 3{,}346\cdot a}$
	${\displaystyle d_{5}=2\cdot r_{i}=(2+{\sqrt {3}})\cdot a\approx 3{,}732\cdot a}$
	${\displaystyle d=2\cdot r_{u}=({\sqrt {6}}+{\sqrt {2}})\cdot a\approx 3{,}864\cdot a}$
Inscribed radius	${\displaystyle r_{i}={\frac {2+{\sqrt {3}}}{2}}\cdot a\approx 1{,}866\cdot a}$
Perimeter radius	${\displaystyle r_{u}={\frac {{\sqrt {6}}+{\sqrt {2}}}{2}}\cdot a\approx 1{,}932\cdot a}$
Central angle	${\displaystyle \alpha ={\frac {360^{\circ }}{12}}=30^{\circ }}$
Interior angle	${\displaystyle \delta =180^{\circ }-\alpha =150^{\circ }}$

construction

A regular dodecagon is ruler and compass constructible :

Construction of a regular dodecagon with a given circumference		Construction of a regular dodecagon with a given side length , animation (The construction is very similar to that of an octagon with a given side length .)

Decomposition into regular polygons

Besides the regular hexagon, the regular dodecagon is the only regular polygon that can be completely broken down into smaller regular polygons:

• 

  Division into 6 equilateral triangles , 6 squares and 1 regular hexagon

• 

  Division into 12 equilateral triangles and 6 squares

Parquet with regular dodecagons

A large number of tiling options are possible with regular dodecagons . The first two are Archimedean tiling , the third a demiregular tiling :

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  3-12-12

• 

  4-6-12

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  3-3-4-12 and 3-3-3-3-3-3

The numbers below the figures indicate how many corners the regular polygons have, each of which meets at a point . The interior angles add up to 360 °. These tiling are periodic, rotationally symmetric and translationally symmetric and contain only regular polygons .

An example of a commercially used tiling is the eternity puzzle , a placement game in which 209 irregular polygon pieces are to be placed in a dodecagon.

Dodecagon in numismatics

Threepence from 1942, back

There are a variety of twelve-sided coins, e.g. B. the British threepence from 1942, the former 3 pence coin from Nigeria and the Australian 50 cent coin, the 50 ¢ (= Seniti ) coin from Tonga , as well as special collector coins such as. B. the Spanish 300-euro coin .

Dodecagon in architecture

Vera Cruz Church in Segovia

Examples of twelve-sided buildings are:

Germany:

• the Zwölfeckhaus in Dresden ,
• Apostle Church (Greding)
• many of the Hamburg bicycle sheds ,
• the Dianatempel (a twelve-sided pavilion ) in Munich ,
• Twelve Apostles Church (Mannheim) ,
• Reformation Memorial Church (Nuremberg)
• the castle Großsachsenheim ,
• the water tower Schifferstadt ,
• the Gertrudenkapelle in Wolgast ( Mecklenburg-Western Pomerania ),

Further:

• the Vera Cruz Church in Segovia ,
• the pavilion of the Vinohrady carousel ,
• the glass kiosk of the Auto Palace petrol station ,
• the Pagoda of Songyue temple of the city of Dengfeng .
• the choir head of St-Pierre-et-St-Paul de Maguelone , plan of bisected dodecagon

Dodecagon in chemistry

Molecular model of cyclododecane

Molecular model of phenals

The molecular model of cyclododecane is only twelve-sided when viewed from above. The three-dimensional shape of this molecule means that the carbon atoms are not all in one plane . In addition, at higher temperatures the molecule is in constant motion, namely in pseudorotation , i.e. that is, a variety of conformations exist .

An equilateral concave dodecagon is formed by phenal , a polycyclic aromatic hydrocarbon .

Web links

Commons : Dodecagons  - collection of images, videos, and audio files

• Eric W. Weisstein : Dodecagon . In: MathWorld (English).

Individual evidence