Hexagram

Hexagram
(OFF file)
Rank	2
Type	Regular
Space	Spherical
Notation
Bowers style acronym	Shig
Coxeter diagram	xo3ox
Schläfli symbol	{6/2}
Elements
Components	2 triangles
Edges	6
Vertices	6
Vertex figure	Dyad, length 1
Measures (edge length 1)
Circumradius	$\frac{\sqrt3}{3} ≈ 0.57735$
Inradius	$\frac{\sqrt3}{6} ≈ 0.28868$
Area	$\frac{\sqrt3}{2} ≈ 0.86603$
Angle	60°
Central density	2
Number of external pieces	12
Level of complexity	2
Related polytopes
Army	Hig, edge length $\frac{\sqrt3}{3}$
Dual	Hexagram
Conjugate	Hexagram
Convex core	Hexagon
Abstract & topological properties
Flag count	12
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	G2, order 12
Convex	No
Nature	Tame

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:

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

Variations

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

• Bowers, Jonathan. "Regular Polygons and Other Two Dimensional Shapes".

• Nan Ma. "Hexagram {6/2}".

• Wikipedia Contributors. "Hexagram".