Octagram

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

Regular octagram
A regular octagram
TypeRegular star polygon
Edges and vertices8
Schläfli symbol{8/3}
t{4/3}
Coxeter diagram
Symmetry groupDihedral (D8)
Internal angle (degrees)45°
Dual polygonself
Propertiesstar, cyclic, equilateral, isogonal, isotoxal
Star polygons

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

Detail
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.

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 ۞  at U+06DE.

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}.[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. It may be considered for this reason as a three-dimensional analogue of the octagram.

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}

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.

Regular Isogonal Isotoxal

a{8}={8/2}=2{4}
{8/4}=4{2}

{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 star

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}
{8/3}

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

See also
Usage
Stars generally

References
  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)
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