View source for Octagram

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

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

The name ''octagram'' combine a Greek [[numeral prefix]], ''[[wikt:octa-|octa-]]'', with the  [[Greek language|Greek]] suffix ''[[wikt:-gram|-gram]]''. The ''-gram'' suffix derives from γραμμή (''grammḗ'') meaning "line".<ref>{{Cite web |title=Henry George Liddell, Robert Scott, A Greek-English Lexicon, γραμμή |url=https://www.perseus.tufts.edu/hopper/text?doc=Perseus:text:1999.04.0057:entry=grammh/ |access-date=2024-10-31 |website=www.perseus.tufts.edu}}</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]] (8-sided [[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.
{{clear|left}}

== 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 />[[File:Octagram-in-square.svg|120px]]<br />[[Isotoxal figure|Isotoxal]]
|[[File:Ancient mapuche flag.svg|120px]]<br />An old [[Flag of Chile#Pre-Independence flags|Flag of Chile]] contained this octagonal star geometry with edges removed (the [[Guñelve]]).
|[[File:Flag of Kölner Rudergesellschaft 1891.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 [[Kölner Rudergesellschaft 1891|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 [[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'', [[Branko Grünbaum]]</ref>

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.

{| class=wikitable
|+ Isogonal truncations of square and cube
!Regular
!Quasiregular
!Isogonal
!Quasiregular
|- valign=top align=center
|[[File:Regular quadrilateral.svg|120px]]<br />[[Square|{4}]]
|[[File:regular polygon truncation 4 1.svg|120px]]<br />[[Octagon|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 />[[Cube|{4,3}]]
|[[File:Cube truncation 0.50.png|120px]]<br />[[Truncated cube|t{4,3}]]
|[[File:Cube truncation 3.50.png|120px]]
|[[File:Cube truncation 2.50.png|120px]]<br />[[Stellated truncated hexahedron|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, 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]]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 diagram]]s {{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]], {{CDD|node_1|4|node|3|node}} + {{CDD|node|4|node|3|node_1}}, 4D 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]]; 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.
{| class=wikitable
![[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 (planet)|Jupiter]]'s moon [[Europa (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.]]
[[File:JWST_diffraction_spikes.svg|thumb|Edges of the JWST primary mirror segments and spider colour-coded with their corresponding diffraction spikes]]
* The 8-pointed [[diffraction spike]]s 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.<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 [[Flag of the Philippines#Symbolism|sun]] from the [[Flag of the Philippines|Philippine flag]], making it also a nationalism and patriotism-themed song aside from being a Christmas song.

==See also==
{{Commons category|Octagrams}}

;Usage
* [[Rub el Hizb]] – Islamic character
* [[Seljuk star]]
* [[Shamsa]]
* [[Star of Ishtar]] – symbol of the ancient Sumerian goddess [[Inanna]] and her East Semitic counterpart [[Ishtar]] and Roman [[Venus]].
* [[Seshat]] – the [[hieroglyph]] of this ancient Egyptian goddess depicts a seven-petaled flower, forming an octagram with its stem.  
* [[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
* [[Utu]] – ancient [[Mesopotamian]] god symbol and symbol of the Sun god
* [[Russian criminal tattoos|Thieves' Star]] - [[Post-Soviet states|Post-Soviet]] [[thief in law]] symbol, ''«восьмиконечник»'' in [[Russian language|Russian]]

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

;Others
* [[Hexadecagram]]
* [[Dodecagram]]
* [[Symbol of Chaos]]

==References==
{{Reflist}}
* [[Branko Grünbaum|Grünbaum, B.]] and G.C. Shephard; ''[[Tilings and patterns]]'', New York: W. H. Freeman & Co., (1987), {{isbn|0-7167-1193-1}}.
* [[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.
* [[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==
{{wiktionary}}
* {{Mathworld |urlname=Octagram |title=Octagram}}

{{Polygons}}

[[Category:Star polygons|08]]
[[Category:8 (number)]]