# Decagon

Decagon | |
---|---|

Rank | 2 |

Type | Regular |

Notation | |

Bowers style acronym | Dec |

Coxeter diagram | x10o () |

Schläfli symbol | {10} |

Elements | |

Edges | 10 |

Vertices | 10 |

Vertex figure | Dyad, length √(5+√5)/2 |

Measures (edge length 1) | |

Circumradius | |

Inradius | |

Area | |

Angle | 144° |

Central density | 1 |

Number of external pieces | 10 |

Level of complexity | 1 |

Related polytopes | |

Army | Dec |

Dual | Decagon |

Conjugate | Decagram |

Abstract & topological properties | |

Flag count | 20 |

Euler characteristic | 0 |

Orientable | Yes |

Properties | |

Symmetry | I_{2}(10), order 20 |

Flag orbits | 1 |

Convex | Yes |

Nature | Tame |

The **decagon** is a polygon with 10 sides. A regular decagon has equal sides and equal angles. It has the most sides of any polygon that occurs as a face of a non-prismatic uniform polytope, although dodecagons appear in some tilings and hexadecagons appear in scaliform polychora.

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

It can be constructed as the uniform truncation of the regular pentagon. This is how it appears in many uniform polyhedra and polychora.

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

## Naming[edit | edit source]

The name *decagon* is derived from the Ancient Greek *δέκα* (10) and *γωνία* (angle), referring to the number of vertices.

Other names include:

**Dec**, Bowers style acronym, short for "decagon".

The combining prefix in BSAs is **da-**, as in dadip.

## Vertex coordinates[edit | edit source]

Coordinates for a regular decagon of unit edge length, centered at the origin are all sign changes of:

- ,
- ,
- .

## Representations[edit | edit source]

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

- x10o () (regular),
- x5x () (H2 symmetry, generally a dipentagon),
- to5ot&#zx (t=, generally a pentambus),
- xFV Tto&#zx (rectangular symmetry, t as above, T=ft),
- xFVFx&#xt (axial edge-first),
- otTTto&#xt (axial vertex-first).

## Variations[edit | edit source]

Two main variants of the decagon have pentagon symmetry: the dipentagon, with two alternating side lengths and equal angles, and the dual pentambus, with two alternating angles and equal edges. Other less regular variations with chiral pentagonal, rectangular, central inversion, mirror, or no symmetry also exist.

## Stellations[edit | edit source]

- 1st stellation: Stellated decagon
*(compound of two pentagons)* - 2nd stellation: Decagram
- 3rd stellation: Stellated decagram
*(compound of two pentagrams)*

## External links[edit | edit source]

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

- Klitzing, Richard. "Polygons"
- Wikipedia contributors. "Decagon".
- Hi.gher.Space Wiki Contributors. "Decagon".

- Hartley, Michael. "{10}*20".