Octagram

Regular octagram

A regular octagram
Type Regular polygon
Edges and vertices 8
Schläfli symbol {8/3}
t{4/3}
Coxeter diagram
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 numeral prefix, octa-, with the Greek suffix -gram. The -gram suffix derives from γραμμῆς (grammēs) meaning a line.[1]

Geometry

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 3rd 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, rotated 45 degrees, and 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}.[2]

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}

Regular star polygon compounds

There are two 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}.


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
Stars generally

References

  1. ↑ γραμμή, Henry George Liddell, Robert Scott, A Greek-English Lexicon, on Perseus
  2. ↑ The Lighter Side of Mathematics: Proceedings of the Eugène Strens Memorial Conference on Recreational Mathematics and its History, (1994), Metamorphoses of polygons, Branko Grünbaum

External links