{{Short description|Star polygon with 12 vertices}} {{refimprove|date=August 2012}} {{Regular polygon db|Regular star polygon stat table|p12/5}} {{Star polygons}}

In geometry, a '''dodecagram''' ({{ety|el|''δώδεκα'' (dṓdeka)|twelve||''γραμμῆς'' (grammēs)|line}}<ref>[https://www.perseus.tufts.edu/hopper/text?doc=Perseus%3Atext%3A1999.04.0057%3Aentry%3Dgrammh%2F γραμμή], Henry George Liddell, Robert Scott, ''A Greek-English Lexicon'', on Perseus</ref>) is a star polygon or compound with 12 vertices. There is one regular dodecagram polygon (with Schläfli symbol {{math|{12/5} }} and a turning number of 5). There are also 4 regular compounds {{math|{12/2},}} {{math|{12/3},}} {{math|{12/4},}} and {{math|{12/6}.}}

== Regular 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}.

===Dodecagrams as regular compounds=== There are four regular compound 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.

<gallery mode=packed> 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} </gallery>

== Dodecagrams as isotoxal figures== 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. {| class=wikitable |+ Isotoxal dodecagrams !Type||Simple||colspan=3|Compounds||Star |- !Density||1||2||3||4||5 |- align=center valign=bottom !Image |85px<BR>{(6)<sub>α</sub>} |100px<BR>2{3<sub>α</sub>} |100px<BR>3{2<sub>α</sub>} |95px<BR>2{(3/2)<sub>α</sub>} |90px<BR>{(6/5)<sub>α</sub>} |}

== Dodecagrams as isogonal figures== 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. {| class=wikitable |- align=center valign=top |120px<BR>t{6} |120px |120px |120px<BR>t{6/5}={12/5} |}

==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 ''K''<sub>12</sub>.

{| class=wikitable align=center |+ ''K''<sub>12</sub> |- |256px |black: the twelve corner points (nodes)<br> red: {12} regular dodecagon<br> green: {12/2}=2{6} two hexagons<br> blue: {12/3}=3{4} three squares<br> cyan: {12/4}=4{3} four triangles<br> magenta: {12/5} regular dodecagram<br> yellow: {12/6}=6{2} six digons |}

==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). <gallery mode=packed> Image:Prism 12-5.png|Dodecagrammic prism Image:Antiprism 12-5.png|Dodecagrammic antiprism Image:Antiprism 12-7.png|Dodecagrammic crossed-antiprism </gallery> Dodecagrams can also be incorporated into star tessellations of the Euclidean plane.

==Dodecagram Symbolism== [[File:DongSonBronzeDrum.JPG|thumb|right|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: *[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.

==See also== *Stellation *Star polygon *List of regular polytopes and compounds

==References== {{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.&nbsp;43–70. *John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, ''The Symmetries of Things'' 2008, {{ISBN|978-1-56881-220-5}} (Chapter 26. pp.&nbsp;404: Regular star-polytopes Dimension 2)

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