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

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" combines the numeral prefix dodeca- with the Greek suffix -gram. The -gram suffix derives from γραμμῆς (grammēs), which denotes a line.

Isogonal variations

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}

Dodecagrams as 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 hexagrams and the last as three tetragrams.

2{6}

3{4}

4{3}

6{2}

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.

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.

See also

• Stellation
• Star polygon
• List of regular polytopes

References

• 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)