# Monogon

Monogon | |
---|---|

Rank | 2 |

Type | Regular |

Space | Spherical |

Notation | |

Schläfli symbol | {1} |

Elements | |

Edges | 1 |

Vertices | 1 |

Vertex figure | Dyad, length 0 |

Related polytopes | |

Dual | Monogon |

Conjugate | None |

Abstract properties | |

Flag count | 2 |

Net count | 1 |

Topological properties | |

Orientable | Yes |

Properties | |

Symmetry | A_{1}×I, order 2 |

The **monogon** is a 1-sided polygon. It is highly degenerate. It can be embedded in a spherical geometry, with a single edge that covers an entire great circle and joins a vertex to itself. It is unique compared to other polygons, by having a vertex figure where two different vertices represent the same element.

Most definitions of a polytope do not allow for monogons, as they violate the diamond condition. However, a monogon can be seen as a map on a surface. It may also be seen as a generalized complex (see below).

## Monogonal complex[edit | edit source]

A complex is an properly edge-colored graph. If we generalize this notion to multigraphs, one can build a complex with two vertices joined with edges of two colors. As it turns out, the omnitruncate of this generalized complex gives a digon. This suggests that this complex may be regarded as the true monogon under a more general notion of polytopes.

One surprising consequence of this would be that a monogon would have two flags instead of one. We might think of them as belonging to two halves of a region enclosed by the monogon.

By adding more parallel edges of different colors, one might also build monohedra and higher-dimensional monotopes.