# Octagram

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

Regular octagram A regular octagram
TypeRegular star polygon
Edges and vertices8
Schläfli symbol{8/3}
t{4/3}
Coxeter diagram          Symmetry groupDihedral (D8)
Internal angle (degrees)45°
Dual polygonself
Propertiesstar, cyclic, equilateral, isogonal, isotoxal

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

## Detail

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 Guñelve). 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. It may be considered for this reason as a three-dimensional analogue of the octagram.

Another three-dimensional version of the octagram is the nonconvex great rhombicuboctahedron (quasirhombicuboctahedron), which can be thought of as a quasicantellated (quasiexpanded) cube, t0,2{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}

{8/4}=4{2}

{8/2} or 2{4}, like Coxeter diagrams     +    , can be seen as the 2D equivalent of the 3D compound of cube and octahedron,       +      , 4D compound of tesseract and 16-cell,         +         and 5D compound of 5-cube and 5-orthoplex; that is, the compound of a n-cube and cross-polytope in their respective dual positions.

## 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}                ## Other uses

• In Unicode, the "Eight Spoked Asterisk" symbol ✳ is U+2733.