In
geometry, a monogon, also known as a henagon, is a
polygon with one
edge and one
vertex. It has
Schläfli symbol
In geometry, the Schläfli symbol is a notation of the form \ that defines regular polytopes and tessellations.
The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, who generalized Euclidean geometry to more ...
.
[Coxeter, ''Introduction to geometry'', 1969, Second edition, sec 21.3 ''Regular maps'', p. 386-388]
In Euclidean geometry
In
Euclidean geometry a ''monogon'' is a
degenerate
Degeneracy, degenerate, or degeneration may refer to:
Arts and entertainment
* ''Degenerate'' (album), a 2010 album by the British band Trigger the Bloodshed
* Degenerate art, a term adopted in the 1920s by the Nazi Party in Germany to descr ...
polygon because its endpoints must coincide, unlike any Euclidean line segment. Most definitions of a polygon in Euclidean geometry do not admit the monogon.
In spherical geometry
In
spherical geometry
300px, A sphere with a spherical triangle on it.
Spherical geometry is the geometry of the two-dimensional surface of a sphere. In this context the word "sphere" refers only to the 2-dimensional surface and other terms like "ball" or "solid sp ...
, a monogon can be constructed as a vertex on a
great circle
In mathematics, a great circle or orthodrome is the circular intersection of a sphere and a plane passing through the sphere's center point.
Any arc of a great circle is a geodesic of the sphere, so that great circles in spherical geomet ...
(
equator
The equator is a circle of latitude, about in circumference, that divides Earth into the Northern and Southern hemispheres. It is an imaginary line located at 0 degrees latitude, halfway between the North and South poles. The term can als ...
). This forms a
dihedron, , with two
hemispherical
A sphere () is a Geometry, geometrical object that is a solid geometry, three-dimensional analogue to a two-dimensional circle. A sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...
monogonal faces which share one 360° edge and one vertex. Its dual, a
hosohedron, has two
antipodal vertices at the poles, one 360°
lune face, and one edge (
meridian) between the two vertices.
See also
*
Digon
References
*
Herbert Busemann, The geometry of geodesics. New York, Academic Press, 1955
* Coxeter, H.S.M; ''Regular Polytopes'' (third edition). Dover Publications Inc.
{{polyhedra
Polygons by the number of sides
1 (number)