Octagon

The octagon is a polygon with 8 sides. A regular octagon has equal sides and equal angles.

Octagon
(OFF file)
Rank	2
Type	Regular
Space	Spherical
Notation
Bowers style acronym	Oc
Coxeter diagram	x8o ()
Schläfli symbol	{8}
Elements
Edges	8
Vertices	8
Vertex figure	Dyad, length √2+√2
Measures (edge length 1)
Circumradius	$\sqrt{\frac{2+\sqrt2}{2}} ≈ 1.30656$
Inradius	$\frac{1+\sqrt2}{2} ≈ 1.20711$
Area	$2(1+\sqrt2) ≈ 4.82843$
Angle	135°
Central density	1
Number of external pieces	8
Level of complexity	1
Related polytopes
Army	Oc
Dual	Octagon
Conjugate	Octagram
Abstract & topological properties
Flag count	16
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	I2(8), order 16
Convex	Yes
Nature	Tame

The combining prefix in BSAs is o-, as in odip.

The only non-compound stellation of the octagon is the octagram. The only other polygons with a single non-compound stellation are the pentagon, the decagon, and the dodecagon.

It can also be constructed as a uniform truncation of the square. It appears in higher uniform polytopes with hypercube symmetry in this form.

Naming

The name octagon is derived from the Ancient Greek ὀκτώ (8) and γωνία (angle), referring to the number of vertices.

Other names include:

• Oc, Bowers style acronym, short for "octagon".

Vertex coordinates

Coordinates for a regular octagon of unit edge length, centered at the origin, are all permutations of

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

Representations

A regular octagon can be represented by the following Coxeter diagrams:

• x8o (full symmetry)
• x4x (B2 symmetry, generally a ditetragon)
• ko4ok&#zx (B2, generally a tetrambus)
• xw wx&#zx (digonal symmetry)
• okK Kko#&zx (digonal symmetry, K=qk)
• xwwx&#xt (axial edge-first)
• okKko&#xt (axial vertex-first)

Variations

Two main variants of the octagon have square symmetry: the ditetragon, with two alternating side lengths and equal angles, and the dual tetrambus, with two alternating angles and equal edges. Other less regular variations with chiral square, rectangular, inversion, mirror, or no symmetry also exist.

Stellations

• 1st stellation: Stellated octagon (compound of two squares)
• 2nd stellation: Octagram

External links

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

• Klitzing, Richard. "Polygons"
• Wikipedia Contributors. "Octagon".
• Hi.gher.Space Wiki Contributors. "Octagon".

• Hartley, Michael. "{8}*16".