| Regular octagram | |
|---|---|
 A regular octagram | |
| Type | Regular star polygon |
| Edges andvertices | 8 |
| Schläfli symbol | {8/3} t{4/3} |
| Coxeter–Dynkin diagrams |       |
| Symmetry group | Dihedral (D8) |
| Internal angle (degrees) | 45° |
| Properties | star,cyclic,equilateral,isogonal,isotoxal |
| Dual polygon | self |
| Star polygons |
|---|
Ingeometry, anoctagram is an eight-angledstar polygon.
The nameoctagram combine a Greeknumeral prefix,octa-, with theGreek suffix-gram. The-gram suffix derives from γραμμή (grammḗ) meaning "line".[1]
In general, an octagram is any self-intersectingoctagon (8-sidedpolygon).
Theregular octagram is labeled by theSchläfli symbol {8/3}, which means an 8-sided star, connected by every third point.
These variations have a lower dihedral, Dih4, symmetry:
 Narrow  Wide (45 degree rotation) |   Isotoxal |  An oldFlag of Chile contained this octagonal star geometry with edges removed (theGuñelve). |  The regular octagonal star is very popular as a symbol of rowing clubs in theCologne Lowland, as seen on the club flag of theCologne Rowing Association. |  The geometry can be adjusted so 3 edges cross at a single point, like theAuseklis symbol |  An 8-pointcompass rose can be seen as an octagonal star, with 4 primary points, and 4 secondary points. |
The symbolRub el Hizb is aUnicode glyph ۞ at U+06DE.
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 uniformstar polyhedronstellated 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.
| 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} |
Another three-dimensional version of the octagram is thenonconvex great rhombicuboctahedron (quasirhombicuboctahedron), which can be thought of as a quasicantellated (quasiexpanded) cube, t0,2{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 degeneratedigons, {8/4}=4{2}. There are other isogonal and isotoxal compounds including rectangular and rhombic forms.
| Regular | Isogonal | Isotoxal | ||
|---|---|---|---|---|
.svg) a{8}={8/2}=2{4} | .svg) {8/4}=4{2} |  |  |  |
{8/2} or 2{4}, likeCoxeter diagrams +, can be seen as the 2D equivalent of the 3Dcompound of cube and octahedron,
+, 4D compound of tesseract and 16-cell, + and 5Dcompound of 5-cube and 5-orthoplex; that is, the compound of an-cube andcross-polytope in their respective dual positions.
Anoctagonal star can be seen as a concavehexadecagon, with internal intersecting geometry erased. It can also be dissected by radial lines.
| star polygon | Concave | Central dissections | ||
|---|---|---|---|---|
 Compound 2{4} |  |8/2| |  |  |  |
 Regular {8/3} |  |8/3| |  |  |  |
 Isogonal |  |  |  |  |
 Isotoxal |  |  |  |  |
