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Regular dodecagram | |
---|---|

A regular dodecagram | |

Type | Regular star polygon |

Edges and vertices | 12 |

Schläfli symbol | {12/5} t{6/5} |

Coxeter diagram | |

Symmetry group | Dihedral (D_{12}) |

Internal angle (degrees) | 30° |

Dual polygon | self |

Properties | star, cyclic, equilateral, isogonal, isotoxal |

Star polygons |
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A **dodecagram** is a star polygon or compound with 12 vertices. There is one regular dodecagram polygon, {12/5}, having a turning number of 5. There are also 4 regular compounds {12/2}, {12/3} {12/4}, and {12/6}

- Regular dodecagram
- Dodecagrams as regular compounds
- Dodecagrams as isotoxal figures
- Dodecagrams as isogonal figures
- Complete graph
- Regular dodecagrams in polyhedra
- Dodecagram Symbolism
- See also
- References

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] }

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.

- 2{6}
- 3{4}
- 4{3}
- 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.

Type | Simple | Compounds | Star | ||
---|---|---|---|---|---|

Density | 1 | 2 | 3 | 4 | 5 |

Image | {(6) _{α}} | 2{3 _{α}} | 3{2 _{α}} | 2{(3/2) _{α}} | {(6/5) _{α}} |

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} |

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*_{12}.

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.

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

- the twelve tribes of Israel, in Judaism
- the twelve disciples, in Christianity
- the twelve olympians, in Hellenic Polytheism
- the twelve signs of the zodiac
- the International Order of Twelve Knights and Daughters of Tabor, an African-American fraternal group
- 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.

In geometry, an **Archimedean solid** is one of the 13 solids first enumerated by Archimedes. They are the convex uniform polyhedra composed of regular polygons meeting in identical vertices, excluding the five Platonic solids and excluding the prisms and antiprisms. They differ from the Johnson solids, whose regular polygonal faces do not meet in identical vertices.

In geometry, any polyhedron is associated with a second **dual** figure, where the vertices of one correspond to the faces of the other and the edges between pairs of vertices of one correspond to the edges between pairs of faces of the other. Such dual figures remain combinatorial or abstract polyhedra, but not all are also geometric polyhedra. Starting with any given polyhedron, the dual of its dual is the original polyhedron.

In geometry, a **star polygon** is a type of non-convex polygon. Only the **regular star polygons** have been studied in any depth; star polygons in general appear not to have been formally defined, however certain notable ones can arise through truncation operations on regular simple and star polygons.

In geometry, the **Schläfli symbol** is a notation of the form {*p*,*q*,*r*,...} that defines regular polytopes and tessellations.

In geometry, a **dodecagon** or 12-gon is any twelve-sided polygon.

In geometry, a polytope 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 the same or reverse order, and with the same angles between corresponding faces.

Euclidean plane **tilings by convex regular polygons** have been widely used since antiquity. The first systematic mathematical treatment was that of Kepler in his *Harmonices Mundi*.

A **uniform polyhedron** has regular polygons as faces and is vertex-transitive. It follows that all vertices are congruent.

In geometry, an **octagram** is an eight-angled star polygon.

In geometry, an **alternation** or *partial truncation*, is an operation on a polygon, polyhedron, tiling, or higher dimensional polytope that removes alternate vertices.

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 visualised in elliptic space.

In geometry, a polytope of dimension 3 or higher is **isohedral** or **face-transitive** when all its faces are the same. More specifically, all faces must be not merely congruent but must be *transitive*, i.e. must lie within the same *symmetry orbit*. In other words, for any faces *A* and *B*, there must be a symmetry of the *entire* solid by rotations and reflections that maps *A* onto *B*. For this reason, convex isohedral polyhedra are the shapes that will make fair dice.

In geometry, a polytope, or a tiling, is **isotoxal** or **edge-transitive** if its symmetries act transitively on its edges. Informally, this means that there is only one type of edge to the object: given two edges, there is a translation, rotation and/or reflection that will move one edge to the other, while leaving the region occupied by the object unchanged.

A **uniform polytope** of dimension three or higher is a vertex-transitive polytope bounded by uniform facets. The uniform polytopes in two dimensions are the regular polygons.

In geometry, a **uniform tiling** is a tessellation of the plane by regular polygon faces with the restriction of being vertex-transitive.

In geometry, a **decagram** is a 10-point star polygon. There is one regular decagram, containing the vertices of a regular decagon, but connected by every third point. Its Schläfli symbol is {10/3}.

In geometry, an **icositetragon** or 24-gon is a twenty-four-sided polygon. The sum of any icositetragon's interior angles is 3960 degrees.

In geometry, the **density** of a star polyhedron is a generalization of the concept of winding number from two dimensions to higher dimensions, representing the number of windings of the polyhedron around the center of symmetry of the polyhedron. It can be determined by passing a ray from the center to infinity, passing only through the facets of the polytope and not through any lower dimensional features, and counting how many facets it passes through. For polyhedra for which this count does not depend on the choice of the ray, and for which the central point is not itself on any facet, the density is given by this count of crossed facets.

In geometry of 4 dimensions, a **6-6 duoprism** or **hexagonal duoprism** is a polygonal duoprism, a 4-polytope resulting from the Cartesian product of two hexagons.

- 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)

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