Hexagram

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Hexagram
Rank2
TypeRegular
Notation
Bowers style acronymShig
Coxeter diagramβ6o ()
Schläfli symbol{6/2}
Elements
Components2 triangles
Edges6
Vertices6
Vertex figureDyad, length 1
Measures (edge length 1)
Circumradius
Inradius
Area
Angle60°
Central density2
Number of external pieces12
Level of complexity2
Related polytopes
ArmyHig, edge length
DualHexagram
ConjugateHexagram
Convex coreHexagon
Abstract & topological properties
Flag count12
Euler characteristic0
OrientableYes
Properties
SymmetryG2, order 12
Flag orbits1
ConvexNo
NatureTame

A hexagram is any six-sided polygon, unicursal or compound, that has a density of two. The unqualified usage of "hexagram" usually refers only to the compound figure featured here, though the duals of the great ditrigonal icosidodecahedron and the small retrosnub icosicosidodecahedron also involve other hexagrammic faces.

The regular hexagram, or shig, also called the stellated hexagon, is the simplest possible polygon compound, being the compound of two triangles. As such it has 6 edges and 6 vertices. If the components are equilateral triangles, the compound will be regular.

It is the only stellation of the regular hexagon, which is the polygon with most sides to have no stellations aside from compounds.

Its quotient prismatic equivalent is the octahedron, which is three-dimensional.

Vertex coordinates[edit | edit source]

Coordinates for the vertices of a hexagram of edge length 1 centered at the origin are given by:

  • ,
  • .

Variations[edit | edit source]

The regular hexagram can be varied by changing the angle between the two component triangles from the usual 60°. These 2-triangle compounds generally have a ditrigon as their convex hull and remain uniform, but not regular, with triangular symmetry only.

External links[edit | edit source]