# Hosohedron

n-gonal hosohedron | |
---|---|

Rank | 3 |

Type | Regular |

Space | Spherical |

Notation | |

Coxeter diagram | (ono2x) |

Schläfli symbol | {2,n} |

Elements | |

Faces | n digons |

Edges | n |

Vertices | 2 |

Vertex figure | n-gon |

Measures (edge length 1) | |

Dihedral angle | |

Central density | 1 |

Related polytopes | |

Army | n-gonal hosohedron |

Regiment | n-gonal hosohedron |

Dual | n-gonal dihedron |

Abstract & topological properties | |

Euler characteristic | 2 |

Surface | Sphere |

Orientable | Yes |

Genus | 0 |

Properties | |

Symmetry | I_{2}(n)×A_{1}, order 4n |

Convex | Yes |

A **hosohedron** is a polyhedron made of two or more digons or lunes, all sharing the same two vertices. Hosohedra are usually considered in the context of spherical polyhedra where the two vertices are antipodal and joined by two or more edges. Although hosohedra are valid abstract polytopes, they are degenerate in Euclidean space. One possible geometric interpretation in Euclidean space is to take the intersection of three or more half-spaces that are all parallel to one line such that an infinite convex polygonal prism is formed.

A spherical hosohedron is regular if all its lunes are congruent. Regular hosohedra exist for all regular n-gons with n > 2. The digonal hosohedron, with 2 digonal faces, is identical to the digonal dihedron.

It is the three-dimensional case of a hosotope.

## Related polyhedra[edit | edit source]

The truncation of an n-gonal hosohedron is isomorphic to an n-gonal prism. Similarly, the snub of an n-gonal hosohedron is isomorphic to an n-gonal antiprism.

## External links[edit | edit source]

- Wikipedia contributors. "Hosohedron".