User:Tomruen/monogon

Henagon
On a circle, a henagon is a tessellation with a single vertex, and one 360-degree arc.
Type	Regular polygon
Edges and vertices	1
Schläfli symbol	{1}
Coxeter–Dynkin diagrams
Dual polygon	Self-dual

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.

Henagonal dihedron	Henagonal hosohedron Digonal henahedron	Henagonal henahedron
Dual tilings		Self-dual
{1,2}	{2,1}	{1,1}
(F:2, E:1, V=1)	(F:1, E:1, V=2)	(F:1, E:0, V=1)

See also

• Regular polygon
• Digon
• Degeneracy (mathematics)

References

• Olshevsky, George. "Monogon". Glossary for Hyperspace. Archived from the original on 4 February 2007.
• Herbert Busemann, The geometry of geodesics. New York, Academic Press, 1955