From Wikipedia, the free encyclopedia
Jump to navigation Jump to search
Regular dodecagram
Regular star polygon 12-5.svg
A regular dodecagram
TypeRegular star polygon
Edges and vertices12
Schläfli symbol{12/5}
Coxeter–Dynkin diagramsCDel node 1.pngCDel 12.pngCDel rat.pngCDel d5.pngCDel node.png
CDel node 1.pngCDel 6.pngCDel rat.pngCDel d5.pngCDel node 1.png
Symmetry groupDihedral (D12)
Internal angle (degrees)30°
Propertiesstar, cyclic, equilateral, isogonal, isotoxal

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[edit]

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[edit]

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[edit]

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 Isotoxal hexagram.svg
Concave isotoxal hexagon compound2.svg
Isotoxal rhombus compound3.svg
Intersecting isotoxal hexagon compound2.svg
Intersecting isotoxal dodecagon.svg

Dodecagrams as isogonal figures[edit]

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.

Regular polygon truncation 6 1.svg
Regular polygon truncation 6 2.svg Regular polygon truncation 6 3.svg Regular polygon truncation 6 4.svg

Complete graph[edit]

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 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[edit]

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[edit]

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:

See also[edit]


  1. ^ 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-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 26. pp. 404: Regular star-polytopes Dimension 2)