# Dodecagon

Dodecagon | |
---|---|

Rank | 2 |

Type | Regular |

Notation | |

Bowers style acronym | Dog |

Coxeter diagram | x12o () |

Schläfli symbol | {12} |

Elements | |

Edges | 12 |

Vertices | 12 |

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

Measures (edge length 1) | |

Circumradius | |

Inradius | |

Area | |

Angle | 150° |

Central density | 1 |

Number of external pieces | 12 |

Level of complexity | 1 |

Related polytopes | |

Army | Dog |

Dual | Dodecagon |

Conjugate | Dodecagram |

Abstract & topological properties | |

Flag count | 24 |

Euler characteristic | 0 |

Orientable | Yes |

Properties | |

Symmetry | I_{2}(12), order 24 |

Flag orbits | 1 |

Convex | Yes |

Nature | Tame |

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

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.

The dodecagon is not found in the non-prismatic uniform polyhedra nor the Johnson solids. It does appear in regular-faced tilings such as the uniform truncated hexagonal tiling (since it's the uniform truncation of the hexagon) and is the largest polygon to do so.

The dodecagon appears as the face of several paracompact regular skew polyhedra, and is the largest planar polygon to appear in a spherical, euclidean, compact, or paracompact polyhedron in 3D space.

## 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".

The combining prefix in BSAs is **tw-**, as in twip or twaddip.

## Vertex coordinates[edit | edit source]

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

- ,
- .

## Representations[edit | edit source]

A dodecagon has the following Coxeter diagrams:

- x12o () (full symmetry)
- x6x () (G
_{2}symmetry, generally a dihexagon) - xy3yx&#zx (A2 symmetry, y = )

## 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 chiral hexagonal, square, triangular, rectangular, inversion, 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".
- Hartley, Michael. "{12}*24".