geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...

, a dodecagram (γραμμή
Henry George Liddell, Robert Scott, ''A Greek-English Lexicon'', on Perseus) is a star polygon or compound with 12 vertices. There is one regular dodecagram polygon (with Schläfli symbol and a

turning number In mathematics, the winding number or winding index of a closed curve in the plane around a given point is an integer representing the total number of times that curve travels counterclockwise around the point, i.e., the curve's number of tu ...

of 5). There are also 4 regular compounds and

Regular dodecagram

There is one regular form: , containing 12 vertices, with a

of 5. A regular dodecagram has the same

vertex arrangement In geometry, a vertex arrangement is a set of points in space described by their relative positions. They can be described by their use in polytopes. For example, a ''square vertex arrangement'' is understood to mean four points in a plane, equa ...

as a regular

dodecagon In geometry, a dodecagon or 12-gon is any twelve-sided polygon. Regular dodecagon A regular dodecagon is a figure with sides of the same length and internal angles of the same size. It has twelve lines of reflective symmetry and rotational sym ...

, which may be regarded as .

Dodecagrams as regular compounds

There are four regular dodecagram star figures: =2, =3, =4, and =6. The first is a compound of two

hexagons In geometry, a hexagon (from Greek , , meaning "six", and , , meaning "corner, angle") is a six-sided polygon. The total of the internal angles of any simple (non-self-intersecting) hexagon is 720°. Regular hexagon A '' regular hexagon'' has ...

, the second is a compound of three

squares In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90- degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length a ...

, the third is a compound of four

triangles A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, any three points, when non-collinear ...

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

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.

Dodecagrams as isogonal figures

A regular dodecagram can be seen as a quasitruncated hexagon, t=. Other isogonal (

vertex-transitive In geometry, a polytope (e.g. a polygon or polyhedron) or a tiling is isogonal or vertex-transitive if all its vertices are equivalent under the symmetries of the figure. This implies that each vertex is surrounded by the same kinds of face in ...

) variations with equally spaced vertices can be constructed with two edge lengths.

Complete graph

Superimposing all the dodecagons and dodecagrams on each other – including the

degenerate Degeneracy, degenerate, or degeneration may refer to: Arts and entertainment * Degenerate (album), ''Degenerate'' (album), a 2010 album by the British band Trigger the Bloodshed * Degenerate art, a term adopted in the 1920s by the Nazi Party i ...

''compound of six

digon In geometry, a digon is a polygon with two sides (edges) and two vertices. Its construction is degenerate in a Euclidean plane because either the two sides would coincide or one or both would have to be curved; however, it can be easily visu ...

s'' (line segments), – produces the

complete graph In the mathematical field of graph theory, a complete graph is a simple undirected graph in which every pair of distinct vertices is connected by a unique edge. A complete digraph is a directed graph in which every pair of distinct vertices is ...

''K''₁₂.

Regular dodecagrams in polyhedra

Dodecagrams can also be incorporated into

uniform polyhedra In geometry, a uniform polyhedron has regular polygons as faces and is vertex-transitive (i.e., there is an isometry mapping any vertex onto any other). It follows that all vertices are congruent. Uniform polyhedra may be regular (if also ...

. 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.

Dodecagram Symbolism

Dodecagrams or twelve-pointed stars have been used as symbols for the following:
the twelve tribes of Israel, in Judaismthe twelve disciples, in Christianitythe twelve olympians, in Hellenic Polytheismthe twelve signs of the zodiacthe International Order of Twelve Knights and Daughters of Tabor, an African-American fraternal groupthe fictional secret society Manus Sancti, in the ''Knights of Manus Sancti'' series by Bryn Donovan
* The twelve tribes of Nauru on the national flag.

References

* * Grünbaum, B. and G.C. Shephard; ''Tilings and Patterns'', New York: W. H. Freeman & Co., (1987), . *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, (Chapter 26. pp. 404: Regular star-polytopes Dimension 2) {{Polygons 12