Octagon

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Octagon
Regular octagon.svg
Rank2
TypeRegular
SpaceSpherical
Notation
Bowers style acronymOc
Coxeter diagramx8o (CDel node 1.pngCDel 8.pngCDel node.png)
Schläfli symbol{8}
Elements
Edges8
Vertices8
Vertex figureDyad, length 2+2
Measures (edge length 1)
Circumradius
Inradius
Area
Angle135°
Central density1
Number of pieces8
Level of complexity1
Related polytopes
ArmyOc
DualOctagon
ConjugateOctagram
Abstract properties
Flag count16
Euler characteristic0
Topological properties
OrientableYes
Properties
SymmetryI2(8), order 16
ConvexYes
NatureTame

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

The combining prefix 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.

Naming[edit | edit source]

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[edit | edit source]

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

Representations[edit | edit source]

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[edit | edit source]

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[edit | edit source]

External links[edit | edit source]