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Regular octagram
Regular star polygon 8-3.svg
A regular octagram
Type Regular polygon
Edges and vertices 8
Schläfli symbol {8/3}
Coxeter diagram CDel 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 group Dihedral (D8)
Internal angle (degrees) 45°
Dual polygon self
Properties star, cyclic, equilateral, isogonal, isotoxal

In geometry, an octagram is an eight-sided 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]


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 3rd point.

Octagram lengths.svg


These variations have a lower dihedral, Dih4, symmetry:

Regular truncation 4 1.5.svg
Regular truncation 4 2.svg
(45 degree rotation)
Isotoxal octagram.png
Ancient mapuche flag.svg
An old Flag of Chile contained this octagonal star geometry with edges removed.
Star Guñelve.svg
The geometry can be adjusted so 3 edges cross at a single point, like the Auseklis symbol
Compass rose en 08p.svg
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 square[edit]

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 quadrilateral.svg
Regular polygon truncation 4 1.svg
Regular polygon truncation 4 2.svg Regular polygon truncation 4 3.svg
Regular Uniform Isogonal Uniform
Cube truncation 0.00.png
Cube truncation 0.50.png
Cube truncation 3.50.png Cube truncation 2.50.png

Star polygon compounds[edit]

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 rectanglar and rhombic forms.

Regular Isogonal Isotoxal
Regular star figure 2(4,1).svg
Star of Lakshmi
Regular star figure 4(2,1).svg
Octagram rectangle compound.png Octagram crossed-rectangle compound.png Octagram rhombic star.png

Other presentations of an octagonal star[edit]

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

2{4} Ashthalakshmi - Star of Laxmi.svg Squared octagonal star.png Squared octagonal star1.png Squared octagonal star2.png
{8/3} Octagram graph.png Octagonal star.png Octagonal star2.png Octagonal star3.png
Auseklis star.svg Octagonal star-b.png Octagonal star-b2.png Octagonal star-b3.png
Isotoxal octagram.png Octagonal star-c.png Octagonal star-c2.png Octagonal star-c3.png

See also[edit]

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 links[edit]