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

Regular octagram
Regular star polygon 8-3.svg
A regular octagram
TypeRegular star polygon
Edges and vertices8
Schläfli symbol{8/3}
Coxeter diagramCDel node 1.pngCDel 8.pngCDel rat.pngCDel d3.pngCDel node.png
CDel node 1.pngCDel 4.pngCDel rat.pngCDel d3.pngCDel node 1.png
Symmetry groupDihedral (D8)
Internal angle (degrees)45°
Dual polygonself
Propertiesstar, cyclic, equilateral, isogonal, isotoxal

The name octagram combine a Greek numeral prefix, octa-, with the Greek suffix -gram. The -gram suffix derives from γραμμή (grammḗ) meaning "line".[1]


A regular octagram with each side length equal to 1

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.


These variations have a lower dihedral, Dih4, symmetry:

(45 degree rotation)
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 ۞  at U+06DE.

As a quasitruncated squareEdit

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}.[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
Regular Uniform Isogonal Uniform

Star polygon compoundsEdit

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

{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.

Other presentations of an octagonal starEdit

An octagonal star can be seen as a concave hexadecagon, with internal intersecting geometry erased. It can also be dissected by radial lines.


Other usesEdit

  • In Unicode, the "Eight Spoked Asterisk" symbol ✳ is U+2733.

See alsoEdit

Stars generally


  1. ^ 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 linksEdit