Octagram
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Regular octagram | |
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![]() 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 |
Star polygons |
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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]
Detail
![](http://upload.wikimedia.org/wikipedia/commons/thumb/9/90/Octagram_lengths.svg/384px-Octagram_lengths.svg.png)
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 regular octagonal star is very popular as a symbol of rowing clubs in the Cologne Lowland, as seen on the club flag of the Cologne Rowing Association. |
![]() 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}.[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 |
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![]() {4} |
![]() t{4}={8} |
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![]() t'{4}=t{4/3}={8/3} |
Regular | Uniform | Isogonal | Uniform |
![]() {4,3} |
![]() t{4,3} |
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![]() 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}.
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 | ||
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![]() 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.
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.
star polygon | Concave | Central dissections | ||
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![]() 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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Other uses
- In Unicode, the "Eight Spoked Asterisk" symbol ✳ is U+2733.
See also
![](http://upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png)
- Usage
- Rub el Hizb – Islamic character
- Star of Ishtar - symbol of the ancient Sumerian goddess Inanna and her East Semitic counterpart Ishtar and Roman Venus.
- Star of Lakshmi – Indian character
- Surya Majapahit – usage during Majapahit times in Indonesia to represent the Hindu gods of the directions
- Compass rose – usage in compasses to represent the cardinal directions for the eight principal winds
- Auseklis – usage of regular octagram by Latvians
- Guñelve – representation of Venus in Mapuche iconography.
- Selburose – usage of regular octagram in Norwegian design
- Stars generally
References
- ^ γραμμή, Henry George Liddell, Robert Scott, A Greek-English Lexicon, on Perseus
- ^ The Lighter Side of Mathematics: Proceedings of the Eugène Strens Memorial Conference on Recreational Mathematics and its History, (1994), Metamorphoses of polygons, Branko Grünbaum
- Grünbaum, B. and G.C. Shephard; Tilings and Patterns, New York: W. H. Freeman & Co., (1987), ISBN 0-7167-1193-1.
- Grünbaum, B.; Polyhedra with Hollow Faces, Proc of NATO-ASI Conference on Polytopes ... etc. (Toronto 1993), ed T. Bisztriczky et al., Kluwer Academic (1994) pp. 43–70.
- John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 26. pp. 404: Regular star-polytopes Dimension 2)