Octagram


In geometry, an octagram is an eight-angled star polygon.
The name octagram combine a Greek numeral prefix, ', with the Greek suffix '. The -gram suffix derives from γραμμή meaning "line".

Detail

In general, an octagram is any self-intersecting octagon.
The regular octagram is labeled by the Schläfli symbol, which means an 8-sided star, connected by every third point.

Variations

These variations have a lower dihedral, Dih4, symmetry:

Narrow

Wide


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 intermediate star polygon forms with equal spaced vertices and two edge lengths. A truncated square is an octagon, t=. A quasitruncated square, inverted as, is an octagram, t=.
The uniform star polyhedron stellated truncated hexahedron, t'=t 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.
RegularQuasiregularIsogonalQuasiregular


t=
t'=t=
RegularUniformIsogonalUniform


t
t'=t

Another three-dimensional version of the octagram is the nonconvex great rhombicuboctahedron, which can be thought of as a quasicantellated cube, t0,2.

Star polygon compounds

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

Other uses