# Dodecagon

Dodecagon | |
---|---|

Rank | 2 |

Type | Regular |

Space | Spherical |

Bowers style acronym | Dog |

Info | |

Coxeter diagram | x12o |

Schläfli symbol | {12} |

Symmetry | I2(12), order 24 |

Army | Dog |

Elements | |

Vertex figure | Dyad, length (√2+√6)/2 |

Edges | 12 |

Vertices | 12 |

Measures (edge length 1) | |

Circumradius | |

Inradius | |

Area | |

Angle | 150° |

Central density | 1 |

Euler characteristic | 0 |

Number of pieces | 12 |

Level of complexity | 1 |

Related polytopes | |

Dual | Dodecagon |

Conjugate | Dodecagram |

Properties | |

Convex | Yes |

Orientable | Yes |

Nature | Tame |

The **dodecagon**, or **dog**, is a polygon with 12 sides. A regular dodecagon has equal sides and equal angles.

The combining prefix is **twa-**, as in twaddip.

The only non-compound stellation of the dodecagon is the dodecagram. This makes it the largest polygon with a single non-compound stellation. The only other polygons with only one are the pentagon, the octagon, and the decagon.

Regular dodecagons dgenerally do not occur in higher spherical polytopes aside from prisms, though some Euclidean tilings with hexagonal tiling symmetry do use dodecagonal faces.

## Naming[edit | edit source]

The name *decagon* is derived from the Ancient Greek *δώδεκα* (12) and *γωνία* (angle), referring to the number of vertices.

Other names include:

**dog**, Bowers style acronym, short for "dodecagon"

## Vertex coordinates[edit | edit source]

Coordinates for a dodecagon of unit edge length, centered at the origin, are all permutations of:

## Representations[edit | edit source]

A dodecagon has the following Coxeter diagrams:

## Variations[edit | edit source]

Two main variants of the dodecagon have hexagon symmetry: the dihexagon, with two alternating side lengths and equal angles, and the dual hexambus, with two alternating angles and equal edges. Other less regular variations with square, triangular, rectangular, mirror, or no symmetry also exist.

## Stellations[edit | edit source]

- 1st stellation: Stellated dodecagon
*(compound of two hexagons)* - 2nd stellation: Trisquare
*(compound of three squares)* - 3rd stellation: Tetratriangle
*(compound of four triangles)* - 4th stellation: Dodecagram

## External links[edit | edit source]

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

- Wikipedia Contributors. "Dodecagon".