# Dodecagram

A dodecagram is a star polygon that has 12 vertices. There is one regular form: {12/5}. A regular dodecagram has the same vertex arrangement as a regular dodecagon, which may be regarded as {12/1}.

Regular dodecagram A regular dodecagram
TypeRegular star polygon
Edges and vertices12
Schläfli symbol{12/5}
t{6/5}
Coxeter diagram          Symmetry groupDihedral (D12)
Internal angle (degrees)30°
Dual polygonself
Propertiesstar, cyclic, equilateral, isogonal, isotoxal

The name "dodecagram" combines the numeral prefix dodeca- with the Greek suffix -gram. The -gram suffix derives from γραμμῆς (grammēs), which denotes a line.

## Isogonal variations

A regular dodecagram can be seen as a quasitruncated hexagon, t{6/5}={12/5}. Other isogonal (vertex-transitive) variations with equally spaced vertices can be constructed with two edge lengths. t{6}   t{6/5}={12/5}

## Dodecagrams as compounds

There are four regular dodecagram star figures: {12/2}=2{6}, {12/3}=3{4}, {12/4}=4{3}, and {12/6}=6{2}. The first is a compound of two hexagons, the second is a compound of three squares, the third is a compound of four triangles, and the fourth is a compound of six straight-sided digons. The last two can be considered compounds of two hexagrams and the last as three tetragrams. 2{6} 3{4} 4{3} 6{2}

## Complete graph

Superimposing all the dodecagons and dodecagrams on each other – including the degenerate compound of six digons (line segments), {12/6} – produces the complete graph K12. black: the twelve corner points (nodes)red: {12} regular dodecagon green: {12/2}=2{6} two hexagons blue: {12/3}=3{4} three squares cyan: {12/4}=4{3} four triangles magenta: {12/5} regular dodecagram yellow: {12/6}=6{2} six digons

## Regular dodecagrams in polyhedra

Dodecagrams can also be incorporated into uniform polyhedra. Below are the three prismatic uniform polyhedra containing regular dodecagrams (there are no other dodecagram-containing uniform polyhedra).

Dodecagrams can also be incorporated into star tessellations of the Euclidean plane.