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}.

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.

File:Regular star figure 2(6,1).svg|2{6}

File:Regular star figure 3(4,1).svg|3{4}

File:Regular star figure 4(3,1).svg|4{3}

File:Regular star figure 6(2,1).svg|6{2}

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.

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.

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.

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).

Image:Prism 12-5.png|Dodecagrammic prism

Image:Antiprism 12-5.png|Dodecagrammic antiprism

Image:Antiprism 12-7.png|Dodecagrammic crossed-antiprism

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

File:DongSonBronzeDrum.JPGs]]

Dodecagrams or twelve-pointed stars have been used as symbols for the following:

• [https://www.jewishvirtuallibrary.org/the-twelve-tribes-of-israel the twelve tribes of Israel, in Judaism]
• [http://www.religionfacts.com/twelve-point-star the twelve disciples, in Christianity]
• [https://www.britannica.com/list/12-greek-gods-and-goddesses the twelve olympians, in Hellenic Polytheism]
• [https://gottssymbols.tumblr.com/post/173871082894/the-zodiac-part-15-when-a-12-pointed-star-made the twelve signs of the zodiac]
• the International Order of Twelve Knights and Daughters of Tabor, an African-American fraternal group
• [https://www.bryndonovan.com/2017/03/27/star-symbolism-and-meaning-for-tattoos-or-whatever-you-like/ 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.

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

{{reflist}}

• {{MathWorld |title=Dodecagram |urlname=Dodecagram}}
• 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-Strauss, The Symmetries of Things 2008, {{ISBN|978-1-56881-220-5}} (Chapter 26. pp. 404: Regular star-polytopes Dimension 2)

{{Polygons}}

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