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, octa-, with the Greek suffix -gram. The -gram suffix derives from γραμμή (grammḗ) meaning "line".{{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}}
Detail
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.
{{clear|left}}
Variations
These variations have a lower dihedral, Dih4, symmetry:
| class=wikitable width=600 |
| valign=top align=center
|120px |120px |120px |120px |120px |
The symbol Rub el Hizb is a Unicode glyph ۞ {{pad|7px}}at U+06DE.
{{clear|left}}
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}.The Lighter Side of Mathematics: Proceedings of the Eugène Strens Memorial Conference on Recreational Mathematics and its History, (1994), Metamorphoses of polygons, Branko Grünbaum
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
|120px |
| Regular
!Uniform !Isogonal !Uniform |
|---|
| valign=top align=center |
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.
| class=wikitable
!colspan=2|Regular !colspan=2|Isogonal !Isotoxal |
| align=center valign=top
|120px |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, {{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 | Concave | colspan=3|Central dissections |
| align=center valign=top
|80px |80px |80px |80px |80px | ||
| align=center valign=top
|80px |80px |80px |80px | ||
| align=center valign=top
|80px |80px |80px |80px | ||
| align=center valign=top
|80px |80px |80px |80px |
Other uses
File:Jupiter’s Rings And Moons (NIRCam) Commissioning Image (jupiter-hi-res-rings).tiff'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.]]
File:JWST_diffraction_spikes.svg
- 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.{{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}}
- 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
{{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
- Thieves' Star - Post-Soviet thief in law symbol, «восьмиконечник» in Russian
;Stars generally
;Others
References
{{Reflist}}
- 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-Strauss, The Symmetries of Things 2008, {{isbn|978-1-56881-220-5}} (Chapter 26. pp. 404: Regular star-polytopes Dimension 2)