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{{Short description|Star polygon}}
{{Regular polygon db|Regular star polygon stat table|p8/3}}
{{Star polygons}}

In [[Geometry|geometry]], an '''octagram''' is an eight-angled [[Star polygon|star polygon]].

The name ''octagram'' combine a Greek [[Numeral prefix|numeral prefix]], ''octa-'', with the  [[Social:Greek language|Greek]] suffix ''-gram''. The ''-gram'' suffix derives from γραμμή (''grammḗ'') meaning "line".<ref>[https://www.perseus.tufts.edu/hopper/text?doc=Perseus%3Atext%3A1999.04.0057%3Aentry%3Dgrammh%2F γραμμή], Henry George Liddell, Robert Scott, ''A Greek-English Lexicon'', on Perseus</ref>

==Detail==
[[File:Octagram lengths.svg|frame|left|A regular octagram with each side length equal to 1]]

In general, an octagram is any self-intersecting [[Octagon|octagon]] (8-sided [[Polygon|polygon]]).

The [[Regular polygon|regular]] octagram is labeled by the [[Schläfli symbol]] {8/3}, which means an 8-sided star, connected by every third point.
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== Variations ==
These variations have a lower dihedral, Dih<sub>4</sub>, symmetry:
{| class=wikitable width=600
|- valign=top align=center
|[[File:Regular truncation 4 1.5.svg|120px]]<br />Narrow<br />[[File:Regular truncation 4 2.svg|120px]]<br />Wide<br />(45 degree rotation)
|[[File:isotoxal octagram.png|120px]]<br />120px<br />[[Isotoxal figure|Isotoxal]]
|[[File:Ancient mapuche flag.svg|120px]]<br />An old Flag of Chile contained this octagonal star geometry with edges removed (the [[Astronomy:Guñelve|Guñelve]]).
|[[File:KRG1891.svg|120px]]<br />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.
|[[File:Star Guñelve.svg|120px]]<br />The geometry can be adjusted so 3 edges cross at a single point, like the Auseklis symbol
|[[File:Compass rose en 08p.svg|120px]]<br />An 8-point [[Earth:Compass rose|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 ۞ {{pad|7px}}at U+06DE.
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==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}.<ref>The Lighter Side of Mathematics: Proceedings of the Eugène Strens Memorial Conference on Recreational Mathematics and its History, (1994), ''Metamorphoses of polygons'', [[Biography:Branko Grünbaum|Branko Grünbaum]]</ref>

The uniform [[Star polyhedron|star polyhedron]] [[Stellated truncated hexahedron|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.

{| class=wikitable
|+ Isogonal truncations of square and cube
!Regular
!Quasiregular
!Isogonal
!Quasiregular
|- valign=top align=center
|[[File:Regular quadrilateral.svg|120px]]<br />{4}
|[[File:regular polygon truncation 4 1.svg|120px]]<br />t{4}={8}
|[[File:regular polygon truncation 4 2.svg|120px]]
|[[File:regular polygon truncation 4 3.svg|120px]]<br />t'{4}=t{4/3}={8/3}
|-
!Regular
!Uniform
!Isogonal
!Uniform
|- valign=top align=center
|[[File:Cube truncation 0.00.png|120px]]<br />{4,3}
|[[File:Cube truncation 0.50.png|120px]]<br />t{4,3}
|[[File:Cube truncation 3.50.png|120px]]
|[[File:Cube truncation 2.50.png|120px]]<br />t'{4,3}=t{4/3,3}
|}

Another three-dimensional version of the octagram is the [[Nonconvex great rhombicuboctahedron|nonconvex great rhombicuboctahedron]] (quasirhombicuboctahedron), which can be thought of as a quasicantellated (quasiexpanded) cube, t<sub>0,2</sub>{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 [[Digon|digon]]s, {8/4}=4{2}. There are other isogonal and isotoxal compounds including rectangular and rhombic forms.
{| class=wikitable
!colspan=2|Regular
!colspan=2|Isogonal
!Isotoxal
|- align=center valign=top
|[[File:Regular star figure 2(4,1).svg|120px]]<br />a{8}={8/2}=2{4}
|[[File:Regular star figure 4(2,1).svg|120px]]<br />{8/4}=4{2}
|[[File:Octagram rectangle compound.png|120px]]
|[[File:Octagram crossed-rectangle compound.png|120px]]
|[[File:Octagram rhombic star.png|120px]]
|}
{8/2} or 2{4}, like Coxeter diagrams {{CDD|node_1|4|node}} + {{CDD|node|4|node_1}}, can be seen as the 2D equivalent of the 3D [[Compound of cube and octahedron|compound of cube and octahedron]], {{CDD|node_1|4|node|3|node}} + {{CDD|node|4|node|3|node_1}}, 4D [[Compound of tesseract and 16-cell|compound of tesseract and 16-cell]], {{CDD|node_1|4|node|3|node|3|node}} + {{CDD|node|4|node|3|node|3|node_1}} and 5D [[Compound of 5-cube and 5-orthoplex|compound of 5-cube and 5-orthoplex]]; that is, the compound of a n-cube and [[Cross-polytope|cross-polytope]] in their respective dual positions.

==Other presentations of an octagonal star==
An '''octagonal star''' can be seen as a concave [[Hexadecagon|hexadecagon]], with internal intersecting geometry erased. It can also be dissected by radial lines.
{| class=wikitable
![[Star polygon|star polygon]]||Concave||colspan=3|Central dissections
|- align=center valign=top
|[[File:Squared octagonal-star3.svg|80px]]<br>Compound 2{4}
|[[File:Squared octagonal-star0.svg|80px]]<br>[[Star polygon#Simple isotoxal star polygons|<nowiki>|8/2|</nowiki>]]
|[[File:Squared octagonal-star1.svg|80px]]
|[[File:Squared octagonal-star4.svg|80px]]
|[[File:Squared octagonal-star2.svg|80px]]
|- align=center valign=top
|[[File:Regular octagram star3.svg|80px]]<br>[[Star polygon|Regular]] {8/3}
|[[File:Regular octagram star0.svg|80px]]<br>|8/3|
|[[File:Regular octagram star1.svg|80px]]
|[[File:Regular octagram star4.svg|80px]]
|[[File:Regular octagram star2.svg|80px]]
|- align=center valign=top
|[[File:Auseklis star3.svg|80px]]<br>[[Isogonal figure|Isogonal]]
|[[File:Auseklis star0.svg|80px]]
|[[File:Auseklis star1.svg|80px]]
|[[File:Auseklis star4.svg|80px]]
|[[File:Auseklis star2.svg|80px]]
|- align=center valign=top
|[[File:Square-compass-star3.svg|80px]]<br>[[Isotoxal figure|Isotoxal]]
|[[File:Square-compass-star0.svg|80px]]
|[[File:Square-compass-star1.svg|80px]]
|[[File:Square-compass-star4.svg|80px]]
|[[File:Square-compass-star2.svg|80px]]
|}

== Other uses ==
* In [[Unicode]], the "Eight Spoked Asterisk" symbol ✳ is U+2733.
[[File:Jupiter’s Rings And Moons (NIRCam) Commissioning Image (jupiter-hi-res-rings).tiff|thumb|The spikes are specially visible around Jupiter's moon Europa (on the left) in this NIRCam image.|alt=A big round white circle with faint rays around on a brown background. A black irregular shape stands on its left border. A black spot to its left issues six white spikes separated by 60 degrees and two fainter spikes in vertical.]]
* The 8-pointed [[Astronomy:Diffraction spike|diffraction spike]]s of the star images from the [[Astronomy:James Webb Space Telescope|James Webb Space Telescope]] are due to the [[Physics:Diffraction|diffraction]] caused by the hexagonal shape of the mirror sections and the struts holding the secondary mirror.<ref name="Lawrence">{{cite news |last1=Lawrence |first1=Pete |title=Why do all the stars have 8 points in the James Webb images? An astronomer explains |url=https://www.sciencefocus.com/space/diffraction-spikes-jwst/ |access-date=1 March 2023 |work=BBC Science Focus Magazine |date=13 September 2022 |language=en}}</ref>
* Used as a parol or star for the 2010 ABS-CBN Christmas Station ID ''Ngayong Pasko Magniningning Ang Pilipino'' ({{literal translation|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.

==See also==

;Usage
* [[Rub el Hizb]] – Islamic character
* [[Star of Ishtar]] – symbol of the ancient Sumerian goddess [[Social:Inanna|Inanna]] and her East Semitic counterpart Ishtar and Roman [[Astronomy:Venus|Venus]].
* Seshat – the [[Engineering:Hieroglyph|hieroglyph]] of this ancient Egyptian goddess depicts a seven-petaled flower, forming an octagram with its stem.  
* [[Astronomy:Star of Lakshmi|Star of Lakshmi]] – Indian character
* [[Surya Majapahit]] – usage during [[Social:Majapahit|Majapahit]] times in Indonesia to represent the Hindu gods of the directions
* [[Earth:Compass rose|Compass rose]] – usage in compasses to represent the cardinal directions for the eight principal winds
* Auseklis – usage of regular octagram by Latvians
* [[Astronomy:Guñelve|Guñelve]] – representation of [[Astronomy:Venus|Venus]] in Mapuche iconography.
* [[Selburose]] – usage of regular octagram in Norwegian design
* [[Religion:Utu|Utu]] – ancient Mesopotamian god symbol and symbol of the Sun god

;Stars generally
* Star (polygon)
* [[Stellation#Stellating polygons|Stellated polygons]]
* Two-dimensional regular polytopes

;Others
* [[Hexadecagram]]
* [[Dodecagram]]

==References==
{{Reflist}}
* [[Biography:Branko Grünbaum|Grünbaum, B.]] and G.C. Shephard; ''Tilings and Patterns'', New York: W. H. Freeman & Co., (1987), {{isbn|0-7167-1193-1}}.
* [[Biography:Branko Grünbaum|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.&nbsp;43–70.
* [[Biography:John Horton Conway|John H. Conway]], Heidi Burgiel, Chaim Goodman-Strauss, ''The Symmetries of Things'' 2008, {{isbn|978-1-56881-220-5}} (Chapter 26. pp.&nbsp;404: Regular star-polytopes Dimension 2)

==External links==
* {{Mathworld |urlname=Octagram |title=Octagram}}

{{Polygons}}

[[Category:Star polygons|08]]
[[Category:8 (number)]]
[[category: octogram is a (starlike) shape with 8 sides

{{Sourceattribution|Octagram}}