# Octagon

Octagon | |
---|---|

Rank | 2 |

Type | Regular |

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 | |

Inradius | |

Area | |

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 | I_{2}(8), order 16 |

Flag orbits | 1 |

Convex | Yes |

Nature | Tame |

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

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[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".**8-gon**

The combining prefix in BSAs is **o-**, as in **o**dip.

## Vertex coordinates[edit | edit source]

Coordinates for a regular 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.

### Skew octagons[edit | edit source]

There are 12 regular skew octagons in Euclidean space.

Name | Extended Schläfli symbol | Dimensions |
---|---|---|

octagon | 2 | |

octagram | 2 | |

octagonal-square coil | 4 | |

octagonal-octagrammic coil | 4 | |

skew octagon | 3 | |

square-octagrammic coil | 4 | |

skew octagram | 3 | |

octagonal-square-octagrammic coil | 6 | |

skew octagonal-square coil | 5 | |

skew octagonal-octagrammic coil | 5 | |

skew square-octagrammic coil | 5 | |

skew octagonal-square-octagrammic coil | 7 |

## Stellations[edit | edit source]

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

## External links[edit | edit source]

- 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".