Dodecagram

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Regular dodecagram
Regular star polygon 12-5.svg
A regular dodecagram
TypeRegular star polygon
Edges and vertices12
Schläfli symbol{12/5}
t{6/5}
Coxeter diagramCDel 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°
Dual polygonself
Propertiesstar, 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" combines the numeral prefix dodeca- with the Greek suffix -gram. The -gram suffix derives from γραμμῆς (grammēs), which denotes 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 equally spaced vertices can be constructed with two edge lengths.

Regular polygon truncation 6 1.svg
t{6}
Regular polygon truncation 6 2.svg Regular polygon truncation 6 3.svg Regular polygon truncation 6 4.svg
t{6/5}={12/5}

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. The last two can be considered compounds of two hexagrams and the last as three tetragrams.

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

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 (there are no other dodecagram-containing uniform polyhedra).

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

See also[edit]

References[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)