Dodecagram

Regular dodecagram
Regular dodecagram
	A regular dodecagram
Type	Regular star polygon
Edges and vertices	12
Schläfli symbol	{12/5}; t{6/5}
Coxeter–Dynkin diagrams	;
Symmetry group	Dihedral (D12)
Internal angle (degrees)	30°
Properties	star, cyclic, equilateral, isogonal, isotoxal
Dual polygon	self

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.

2{6}
3{4}
4{3}
6{2}

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{3_α}	3{2_α}	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.

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).

Dodecagrammic prism
Dodecagrammic antiprism
Dodecagrammic crossed-antiprism

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

Dodecagram Symbolism

The twelve-pointed star is a prominent feature on the ancient Vietnamese Dong Son drums

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

the twelve tribes of Israel, in Judaism
the twelve disciples, in Christianity
the twelve olympians, in Hellenic Polytheism
the twelve signs of the zodiac
the International Order of Twelve Knights and Daughters of Tabor, an African-American fraternal group
the fictional secret society Manus Sancti, in the Knights of Manus Sancti series by Bryn Donovan
The twelve tribes of Nauru on the national flag.

References

^ γραμμή, Henry George Liddell, Robert Scott, A Greek-English Lexicon, on Perseus

Weisstein, Eric W. "Dodecagram". MathWorld.
Grünbaum, B. and G.C. Shephard; Tilings and patterns, New York: W. H. Freeman & Co., (1987), ISBN 0-7167-1193-1.
Grünbaum, B.; Polyhedra with Hollow Faces, Proc of NATO-ASI Conference on Polytopes ... etc. (Toronto 1993), ed T. Bisztriczky et al., Kluwer Academic (1994) pp. 43–70.
John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 26. pp. 404: Regular star-polytopes Dimension 2)

[1] γραμμή, Henry George Liddell, Robert Scott, A Greek-English Lexicon, on Perseus