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

In geometry, a dodecagram (from Greek δώδεκα (dṓdeka) 'twelve', and γραμμῆς (grammēs) 'line'[1]) is a star polygon or compound with 12 vertices. There is one regular dodecagram polygon (with Schläfli symbol {12/5} and a turning number of 5). There are also 4 regular compounds {12/2}, {12/3}, {12/4}, and {12/6}.

## Regular dodecagram

There is one regular form: {12/5}, containing 12 vertices, with a turning number of 5. A regular dodecagram has the same vertex arrangement as a regular dodecagon, which may be regarded as {12/1}.

### Dodecagrams as regular 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 compound hexagrams and the last as three compound tetragrams.

## Dodecagrams as isotoxal figures

An isotoxal polygon has two vertices and one edge type within its symmetry class. There are 5 isotoxal dodecagram star with a degree of freedom of angles, which alternates vertices at two radii, one simple, 3 compounds, and 1 unicursal star.

Isotoxal dodecagrams
Type Simple Compounds Star
Density 1 2 3 4 5
Image
{(6)α}

2{3α}

3{2α}

2{(3/2)α}

{(6/5)α}

## Dodecagrams as isogonal figures

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}

## 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.

## Dodecagram Symbolism

Dodecagrams or twelve-pointed stars have been used as symbols for the following: