Octagram
Star polygon
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".
01Detail
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.
02Variations
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.
03As 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}.
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}.
04Star 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 | ||
|---|---|---|---|---|
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.
05Other 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 | ||
|---|---|---|---|---|
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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06Other uses
- One of the oldest depictions of several variations of the octagram nested within each other was found in a cave in Jordan. Dating back 6,000 years and associated with the Ghassulian culture, it is speculated to have represented some kind of calendar system.
- In Unicode, the "Eight Spoked Asterisk" symbol ✳ is U+2733.
- The unicode for Dingir, a Sumerian cuneiform eight-pointed ideogram meaning "sky" and later a logogram for a deity is U+1202D 𒀭.
- The 8-pointed diffraction spikes of the star images from the James Webb Space Telescope are due to the diffraction caused by the hexagonal shape of the mirror sections and the struts holding the secondary mirror.
- Used as a parol or star for the 2010 ABS-CBN Christmas Station ID Ngayong Pasko Magniningning Ang Pilipino (lit. 'This Christmas, the Filipinos will Shine') due to the usage of a sun from the Philippine flag, making it also a nationalism and patriotism-themed song aside from being a Christmas song.
Sources and credits
This article is adapted from the Wikipedia article “Octagram”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.
Images, from Wikimedia Commons:
- Regular star polygon 8-3.svg by Tomruen, CC BY-SA 4.0
- Octagram lengths.svg by Roshan220195, CC BY-SA 3.0
- The Ghassulian star.jpg by Original author unknown Copy from original: Alexis Mallon (1875-1934), Public domain
- Jupiter’s Rings And Moons (NIRCam) Commissioning Image (jupiter-hi-res-rings).tiff by NASA/ESA/CSA/STScI, CC BY 4.0
- JWST diffraction spikes.svg by Cmglee, CC BY-SA 4.0
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