Octagram

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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}
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:

Regular truncation 4 1.5.svg
Narrow
Regular truncation 4 2.svg
Wide
(45 degree rotation)
Isotoxal octagram.png
Isotoxal
Star Guñelve.svg
An old Flag of Chile contained this octagonal star geometry, rotated 45 degrees, and edges removed.
Ancient mapuche flag.svg
The geometry can be adjusted so 3 edges cross at a single point, like the Auseklis symbol

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.

An anti-truncated or hyper-truncated square makes an octagonal star. It has the same vertex arrangement as a regular octagon, but vertices are connected in a forward 3, back 1 sequencing.

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

Star figures[edit]

Regular octagram
Regular star figure 2(4,1).svg
A regular octagram
Type Regular polygonal figure
Edges and vertices 8
Schläfli symbol {8/2}, 2{4} or {{4}}
Coxeter diagram CDel node h3.pngCDel 8.pngCDel node.png or CDel node h3.pngCDel 4.pngCDel node h3.png
Symmetry group Dihedral (D8)
Internal angle (degrees) 90°
Dual polygon self
Properties star, compound, cyclic, equilateral, isogonal, isotoxal

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

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.

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.
  • 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]