View source for Monogon

{{short description|Polygon with one edge and one vertex}}
{{Infobox polygon
| name       = Monogon
| image      = Monogon.svg
| caption    = On a circle, a '''monogon''' is a [[Tessellation|tessellation]] with a single vertex, and one 360-degree arc edge.
| type       = [[Regular polygon]]
| euler      =
| edges      = 1
| schläfli   = {1} or [[Alternation (geometry)|h]]{2}
| wythoff    = 
| coxeter    = {{CDD|node}} or {{CDD|node_h|2x|node}}
| symmetry   = [ ], C<sub>s</sub>
| area       = 
| angle      = 
| dual       = Self-dual
| properties = }}
In [[Geometry|geometry]], a '''monogon''', also known as a '''henagon''', is a [[Polygon|polygon]] with one [[Edge (geometry)|edge]] and one [[Vertex (geometry)|vertex]]. It has [[Schläfli symbol]] {1}.<ref name=cox>Coxeter, ''Introduction to geometry'', 1969, Second edition, sec 21.3 ''Regular maps'', p. 386-388</ref>

==In Euclidean geometry==
In [[Euclidean geometry]] a ''monogon'' is a [[Degeneracy (mathematics)|degenerate]] 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|spherical geometry]], a monogon can be constructed as a vertex on a [[Great circle|great circle]] ([[Astronomy:Equator|equator]]). This forms a [[Dihedron#As a tiling on a sphere|dihedron]], {1,2}, with two hemispherical monogonal faces which share one 360° edge and one vertex. Its dual, a [[Hosohedron|hosohedron]], {2,1} has two [[Antipodal point|antipodal]] vertices at the poles, one 360° lune face, and one edge (meridian) between the two vertices.<ref name=cox />

{| class=wikitable
|- align=center
|[[File:Hengonal dihedron.png|160px]]<BR>Monogonal [[Dihedron|dihedron]], {1,2}
|[[File:Henagonal hosohedron.png|160px]]<BR>Monogonal [[Hosohedron|hosohedron]], {2,1}
|}

==See also==
* [[Digon]]

==References==
{{reflist}}
* [[Biography:Herbert Busemann|Herbert Busemann]], The geometry of geodesics. New York, Academic Press, 1955
* Coxeter, H.S.M; ''Regular Polytopes'' (third edition). Dover Publications Inc. {{isbn|0-486-61480-8}}

{{clear}}
{{polygons}}
{{polyhedra}}

[[Category:1 (number)]]

{{Sourceattribution|Monogon}}