Monogon

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Monogon
Henagon.svg
On a circle, a monogon is a tessellation with a single vertex, and one 360-degree arc edge.
Type Regular polygon
Edges and vertices 1
Schläfli symbol {1} or h{2}
Coxeter diagram CDel node.png or CDel node h.pngCDel 2x.pngCDel node.png
Symmetry group [ ], Cs
Dual polygon Self-dual

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

In Euclidean geometry[edit]

In Euclidean geometry a monogon is a 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[edit]

In spherical geometry, a monogon can be constructed as a vertex on a great circle (equator). This forms a dihedron, {1,2}, with two hemispherical monogonal faces which share one 360° edge and one vertex. Its dual, a hosohedron, {2,1} has two antipodal vertices at the poles, one 360 degree lune face, and one edge (meridian) between the two vertices.[1]

Hengonal dihedron.png
Monogonal dihedron, {1,2}
Spherical henagonal hosohedron.png
Monohedral hosohedron, {2,1}

See also[edit]

References[edit]

  1. ^ a b Coxeter, Introduction to geometry, 1969, Second edition, sec 21.3 Regular maps, p. 386-388
  • 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