User:Tomruen/monogon

In geometry a henagon (or monogon) is a polygon with one edge and one vertex. It has Schläfli symbol {1}. Since a henagon has only one side and only one vertex, every henagon is regular by definition.

In Euclidean geometry
In Euclidean geometry a henagon is usually considered to be an impossible object, because its endpoints must coincide, unlike any Euclidean line segment. For this reason, most authorities do not consider the henagon as a proper polygon in Euclidean geometry.

In spherical geometry
In spherical geometry, a finite henagon can be drawn by placing a single vertex anywhere on a great circle. This forms a henagonal dihedron (Schläfli symbol {1,2}), with two hemispherical henagonal faces which share one 360° edge and one vertex. Its dual {2,1}, the henagonal hosohedron, has one digonal face (a full 360° lens), one 180° edge, and two vertices.

Simplest of all, the henagonal henahedron {1,1} is similar to the henagon but consists of a single vertex on the sphere, no edges and a single face as the sphere outside the vertex. The henagonal henahedron is self-dual, i.e. the point and face center can be swapped creating itself as a central inversion.