Dodecagram

Regular dodecagram
File:Regular star polygon 12-5.svg A regular dodecagram
Type	Regular star polygon
Edges and vertices	12
Schläfli symbol	{12/5} t{6/5}
Coxeter–Dynkin diagrams	File:CDel node 1.pngFile:CDel 12.pngFile:CDel rat.pngFile:CDel d5.pngFile:CDel node.png File:CDel node 1.pngFile:CDel 6.pngFile:CDel rat.pngFile:CDel d5.pngFile:CDel node 1.png
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') 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.

• Error creating thumbnail:
  2{6}
• Error creating thumbnail:
  3{4}
• Error creating thumbnail:
  4{3}
• 6{2}
  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	File:Isotoxal hexagram.svg {(6)α}	File:Concave isotoxal hexagon compound2.svg 2{3α}	File:Isotoxal rhombus compound3.svg 3{2α}	File:Intersecting isotoxal hexagon compound2.svg 2{(3/2)α}	File:Intersecting isotoxal dodecagon.svg {(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.

File:Regular polygon truncation 6 1.svg t{6}

File:Regular polygon truncation 6 2.svg

File:Regular polygon truncation 6 3.svg

File:Regular polygon truncation 6 4.svg 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

File:K12 coloured.svg

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 prism
• Dodecagrammic antiprism
  Dodecagrammic antiprism
• Dodecagrammic crossed-antiprism
  Dodecagrammic crossed-antiprism

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

Dodecagram Symbolism

File:DongSonBronzeDrum.JPG

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.

See also

• Stellation
• Star polygon
• List of regular polytopes and compounds

References

• Grünbaum, B. and G.C. Shephard; Tilings and patterns, New York: W. H. Freeman & Co., (1987), .
• 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, (Chapter 26. pp. 404: Regular star-polytopes Dimension 2)