Dodecagram

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Regular dodecagram
Regular star polygon 12-5.svg
A regular dodecagram
Type Regular star polygon
Edges and vertices 12
Schläfli symbol {12/5}
t{6/5}
Coxeter diagram CDel 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 group Dihedral (D12)
Internal angle (degrees)30°
Dual polygon self
Properties star, cyclic, equilateral, isogonal, isotoxal

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}

Contents

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]

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 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.

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.

Isotoxal dodecagrams
TypeSimpleCompoundsStar
Density 12345
Image Isotoxal hexagram.svg
{(6)α}
Concave isotoxal hexagon compound2.svg
2{3α}
Isotoxal rhombus compound3.svg
3{2α}
Intersecting isotoxal hexagon compound2.svg
2{(3/2)α}
Intersecting isotoxal dodecagon.svg
{(6/5)α}

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.

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}

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 K12.

K12
K12 coloured.svg black: the twelve corner points (nodes)

red: {12} regular dodecagon
green: {12/2}=2{6} two hexagons
blue: {12/3}=3{4} three squares
cyan: {12/4}=4{3} four triangles
magenta: {12/5} regular dodecagram
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).

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

Dodecagram Symbolism

The twelve-pointed star is a prominent feature on the ancient Vietnamese Dong Son drums DongSonBronzeDrum.JPG
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:

See also

Related Research Articles

Archimedean solid Convex uniform polyhedra first enumerated by Archimedes

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.

Dual polyhedron

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.

Star polygon

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.

Schläfli symbol Notation that defines regular polytopes and tessellations

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

Dodecagon Polygon with 12 edges

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 tilings by convex regular polygons

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.

Uniform polyhedron Polyhedron which has regular polygons as faces and is vertex-transitive

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

Octagram star polygon

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

Alternation (geometry) Operation on a polyhedron or tiling that removes alternate vertices

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

Digon Polygon with 2 sides and 2 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.

Isohedral figure

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.

Uniform polytope

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.

Decagram (geometry) 10-pointed star polygon

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}.

Icositetragon Polygon with 24 edges

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

Density (polytope)

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.

6-6 duoprism

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.

References

  1. γραμμή, Henry George Liddell, Robert Scott, A Greek-English Lexicon, on Perseus