Octagram

Regular octagram
A regular octagram
Type	Regular polygon
Edges and vertices	8
Schläfli symbol	{8/3} t{4/3}
Coxeter diagram	{{{p8/3-CD2}}}
Symmetry group	Dihedral (D8)
Internal angle (degrees)	45°
Dual polygon	self
Properties	star, cyclic, equilateral, isogonal, isotoxal

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

The name octagram combine a Greek numeral prefix, octa-, with the Greek suffix -gram. The -gram suffix derives from γραμμή (grammḗ) meaning "line".

Detail

A regular octagram with each side length equal to 1

In general, an octagram is any self-intersecting octagon (8-sided polygon).

The regular octagram is labeled by the Schläfli symbol {8/3}, which means an 8-sided star, connected by every third point.

Variations

These variations have a lower dihedral, Dih₄, symmetry:

Narrow Wide (45 degree rotation)

Isotoxal

An old Flag of Chile contained this octagonal star geometry with edges removed.

The geometry can be adjusted so 3 edges cross at a single point, like the Auseklis symbol

An 8-point compass rose can be seen as an octagonal star, with 4 primary points, and 4 secondary points.

The symbol Rub el Hizb is a Unicode glyph ۞  at U+06DE.

As a quasitruncated square

Deeper truncations of the square can produce isogonal (vertex-transitive) intermediate star polygon forms with equal spaced vertices and two edge lengths. A truncated square is an octagon, t{4}={8}. A quasitruncated square, inverted as {4/3}, is an octagram, t{4/3}={8/3}.

The uniform star polyhedron stellated truncated hexahedron, t'{4,3}=t{4/3,3} has octagram faces constructed from the cube in this way.

Isogonal truncations of square and cube

Regular	Quasiregular	Isogonal	Quasiregular
{4}	t{4}={8}		t'{4}=t{4/3}={8/3}
Regular	Uniform	Isogonal	Uniform
{4,3}	t{4,3}		t'{4,3}=t{4/3,3}

Star polygon compounds

There are two regular octagrammic star figures (compounds) of the form {8/k}, the first constructed as two squares {8/2}=2{4}, and second as four degenerate digons, {8/4}=4{2}. There are other isogonal and isotoxal compounds including rectangular and rhombic forms.

Regular		Isogonal		Isotoxal
a{8}={8/2}=2{4} Star of Lakshmi	{8/4}=4{2}

Other presentations of an octagonal star

An octagonal star can be seen as a concave hexadecagon, with internal intersecting geometry erased. It can also be dissected by radial lines.

2{4}
{8/3}

See also

Usage

• Auseklis – usage of regular octagram by Latvians
• Guñelve – representation of Venus in Mapuche iconography.
• Selburose – usage of regular octagram in Norwegian design

Stars generally

• Star (polygon)
• Stellated polygons
• Two-dimensional regular polytopes

References

• 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)

External links

• Weisstein, Eric W., "Octagram", MathWorld.