From Wikipedia, the free encyclopedia
| Regular octagram | |
|---|---|
A regular octagram | |
| Type | Regular star polygon |
| Edges and vertices | 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 |
In geometry, an octagram is an eight-angled 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 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.
These variations have a lower dihedral, Dih4, symmetry:
The symbol Rub el Hizb is a Unicode glyph ۞ at U+06DE.
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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. 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 the nonconvex great rhombicuboctahedron (quasirhombicuboctahedron), which can be thought of as a quasicantellated (quasiexpanded) cube, t0,2{4/3,3}.
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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} |
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{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.
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An octagonal star can be seen as a concave hexadecagon, with internal intersecting geometry erased. It can also be dissected by radial lines.
| star polygon | Concave | Central dissections | ||
|---|---|---|---|---|
Compound 2{4} |
|8/2| |
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Regular {8/3} |
|8/3| |
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Isogonal |
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Isotoxal |
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Wikimedia Commons has media related to Octagrams.
Look up octagram in Wiktionary, the free dictionary.
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