Octagram

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Regular octagram
Star polygon 8-3.svg
A regular octagram
Type Regular polygon
Edges and vertices 8
Schläfli symbol {8/3}
t{4/3}
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.

Geometry[edit]

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

Variations[edit]

These variations have a lower dihedral, Dih4, symmetry:

Auseklis star.svg Ancient mapuche flag.svg Star Guñelve.svg Isotoxal octagram.png
The geometry can be adjusted so 3 edges cross at a single point, like the Auseklis symbol An old Flag of Chile contained this octagonal star geometry, rotated 45 degrees, and edges removed. An isotoxal variation has a single edge length, but vertices alternate at two different distances from the center.

Other octagrams[edit]

Other symmetric octagrams can be constructed, with the same vertex arrangement, but different sequencing rules, like alternating two stepping rules: steps forward, and steps back. This can allow more than two edges per vertex, or it can be seen as a double-covering of 16 vertices in coinciding pairs.

Octagram flipped edges.png Octagram 3-2.png
Forward 3, back 1 Forward 3, back 2

Star figures[edit]

An 8-point compass rose can be seen as an octagonal star, with 4 primary points, and 4 secondary points

There is one compound star figure, 2{4} (or {8/2} in older writings), composed of two squares, called the Star of Lakshmi.

Ashthalakshmi - Star of Laxmi.svg

The symbol Rub el Hizb is a Unicode glyph ۞  at U+06DE.

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]

Usage
Stars generally

References[edit]

  • 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.

External links[edit]