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Regular octagram  

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


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In geometry, an octagram is an eightangled 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".^{[1]}
In general, an octagram is any selfintersecting octagon (8sided polygon).
The regular octagram is labeled by the Schläfli symbol {8/3}, which means an 8sided star, connected by every third point.
These variations have a lower dihedral, Dih_{4}, 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 8point 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.
Deeper truncations of the square can produce isogonal (vertextransitive) 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.
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} 
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 16cell, + and 5D compound of 5cube and 5orthoplex; that is, the compound of a ncube and crosspolytope in their respective dual positions.
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}  
Wikimedia Commons has media related to Octagrams. 