# Hosohedron

n-gonal hosohedron
Rank3
TypeRegular
SpaceSpherical
Notation
Coxeter diagram (ono2x)
Schläfli symbol{2,n}
Elements
Facesn digons
Edgesn
Vertices2
Vertex figuren-gon
Measures (edge length 1)
Dihedral angle${\displaystyle \frac{(n-2)π}{n}}$
Central density1
Related polytopes
Armyn-gonal hosohedron
Regimentn-gonal hoshedron
Dualn-gonal dihedron
Abstract properties
Euler characteristic2
Topological properties
SurfaceSphere
OrientableYes
Genus0
Properties
SymmetryI2(n)×A1, order 4n
ConvexYes
NatureTame

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

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.