Hexagram

Hexagram
Rank2
TypeRegular
SpaceSpherical
Notation
Bowers style acronymShig
Coxeter diagramxo3ox
Schläfli symbol{6/2}
Elements
Components2 triangles
Edges6
Vertices6
Measures (edge length 1)
Circumradius${\displaystyle \frac{\sqrt3}{3} ≈ 0.57735}$
Inradius${\displaystyle \frac{\sqrt3}{6} ≈ 0.28868}$
Area${\displaystyle \frac{\sqrt3}{2} ≈ 0.86603}$
Angle60°
Central density2
Number of pieces12
Level of complexity2
Related polytopes
ArmyHig
DualHexagram
ConjugateHexagram
Convex coreHexagon
Abstract properties
Euler characteristic0
Topological properties
OrientableYes
Properties
SymmetryG2, order 12
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

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

• ${\displaystyle \left(0,\,±\frac{\sqrt3}{3}\right),}$
• ${\displaystyle \left(±\frac12,\,±\frac{\sqrt3}{6}\right).}$