| 释义 |
- Detail
- Variations
- As a quasitruncated square
- Star polygon compounds
- Other presentations of an octagonal star
- Other uses
- See also
- References
- External links
{{short description|star polygon}}{{Regular polygon db|Even 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".[1] DetailIn 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 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 ۞ {{pad|7px}}at U+06DE. As a quasitruncated squareDeeper 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. Isogonal truncations of square and cube| 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} | 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 |
a{8}={8/2}=2{4} |
{8/4}=4{2} |
{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 starAn octagonal star can be seen as a concave hexadecagon, with internal intersecting geometry erased. It can also be dissected by radial lines. Other uses - In Unicode, the "Eight Spoked Asterisk" symbol ✳ is U+2733.
See also{{Commonscat|Octagrams}}- Usage
- Rub el Hizb – Islamic character
- 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
- Star (polygon)
- Stellated polygons
- Two-dimensional regular polytopes
References1. ^γραμμή, Henry George Liddell, Robert Scott, A Greek-English Lexicon, on Perseus
2. ^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)
External links- {{Mathworld |urlname=Octagram |title=Octagram}}
{{Polygons}} 3 : Star symbols|Polygons|8 (number) |