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Regular dodecagram
Regular star polygon 12-5.svg
A regular dodecagram
Type Regular star polygon
Edges and vertices 12
Schläfli symbol {12/5}
Coxeter diagram CDel 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 group Dihedral (D12)
Internal angle (degrees) 30°
Dual polygon self
Properties star, cyclic, equilateral, isogonal, isotoxal

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}.

The name dodecagram combine a numeral prefix, dodeca-, with the Greek suffix -gram. The -gram suffix derives from γραμμῆς (grammēs) meaning a line.[1]

Isogonal variations[edit]

A regular dodecagram can be seen as a quasitruncated hexagon, t{6/5}={12/5}. Other isogonal (vertex-transitive) variations with equal 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

Dodecagrams as 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.

Regular star figure 2(6,1).svg
Regular star figure 3(4,1).svg
Regular star figure 4(3,1).svg
Regular star figure 6(2,1).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.

11-simplex graph.svg

Regular dodecagrams in polyhedra[edit]

Dodecagrams can also be incorporated into uniform polyhedra. Below are the three prismatic uniform polyhedra containing regular dodecagrams.

See also[edit]


  1. ^ γραμμή, Henry George Liddell, Robert Scott, A Greek-English Lexicon, on Perseus
  • 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)