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 .
- 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 .
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 | ||
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Area | ||
Length of the diagonal | ||
Inscribed radius | ||
Perimeter radius | ||
Central angle | ||
Interior angle |
construction
A regular dodecagon is ruler and compass constructible :
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 :
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
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
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
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
- Eric W. Weisstein : Dodecagon . In: MathWorld (English).