Octagram

Octagram
(OFF file)
Rank	2
Type	Regular
Space	Spherical
Notation
Bowers style acronym	Og
Coxeter diagram	x8/3o ()
Schläfli symbol	{8/3}
Elements
Edges	8
Vertices	8
Vertex figure	Dyad, length √2–√2
Measures (edge length 1)
Circumradius	$\sqrt{\frac{2-\sqrt2}{2}} ≈ 0.54120$
Inradius	$\frac{\sqrt2-1}{2} ≈ 0.20711$
Area	$2(\sqrt2-1) ≈ 0.82843$
Angle	45°
Central density	3
Number of pieces	16
Level of complexity	2
Related polytopes
Army	Oc
Dual	Octagram
Conjugate	Octagon
Convex core	Octagon
Abstract properties
Flag count	16
Euler characteristic	0
Topological properties
Orientable	Yes
Properties
Symmetry	I2(8), order 16
Convex	No
Nature	Tame

The octagram is a non-convex polygon with 8 sides. It's created by stellating the octagon. A regular octagram has equal sides and equal angles.

This is the second stellation of the octagon, and the only one that is not a compound. The only other polygons with a single non-compound stellation are the pentagon, the decagon, and the dodecagon.

The octagram is the uniform quasitruncation of the square, and as such is the only regular star polygon to regularly appear in non-prismatic uniform polytopes in 5 dimensions and higher.

Vertex coordinates

Coordinates for an octagram of unit edge length, centered at the origin, are all permutations of

• $\left(±\frac{\sqrt2-1}{2},\,±\frac12\right).$

Representations

An octagram has the following Coxeter diagrams:

• x8/3o (full symmetry)
• x4/3x () (B₂ symmetry)

External links

• Bowers, Jonathan. "Regular Polygons and Other Two Dimensional Shapes".

• Klitzing, Richard. "Polygons"
• Wikipedia Contributors. "Octagram".