Monogon: Difference between revisions
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#redirect [[Digon#Monogon]] |
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{{Infobox polygon |
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| name = Monogon |
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| image = Henagon.svg |
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| caption = On a circle, a '''monogon''' is a [[tessellation]] with a single vertex, and one 360-degree arc edge. |
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| type = [[Regular polygon]] |
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| euler = |
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| edges = 1 |
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| schläfli = {1} or [[Alternation (geometry)|h]]{2} |
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| wythoff = |
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| coxeter = {{CDD|node}} or {{CDD|node_h|2x|node}} |
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| symmetry = [ ], C<sub>s</sub> |
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| area = |
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| angle = |
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| dual = Self-dual |
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| properties = }} |
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In [[geometry]] a '''monogon''' is a [[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> Since a ''monogon'' has only one side and only one vertex, every ''monogon'' is [[regular polygon|regular]] by definition. |
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==In Euclidean geometry== |
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In [[Euclidean geometry]] a ''monogon'' 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 ''monogon'' as a proper polygon in Euclidean geometry. |
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==In spherical geometry== |
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In [[spherical geometry]], a ''monogon'' can be constructed by a vertex a [[great circle]] ([[equator]]). This forms a [[Dihedron#As_a_tiling_on_a_sphere|dihedron]], {1,2}, with two [[hemisphere|hemispherical]] ''monogonal'' faces which share one 360° edge and one vertex. Its dual, a [[hosohedron]], {2,1} has two [[antipodal point|antipodal]] vertices at the poles, one 360 degree [[Lune_(mathematics)#Spherical_geometry|lune]] face, and one edge ([[meridian]]) between the two vertices.<ref name=cox></ref> |
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{| class=wikitable |
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|- align=center |
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|[[File:Hengonal dihedron.png|160px]]<BR>[[Dihedron]], {1,2} |
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|[[File:Spherical henagonal hosohedron.png|160px]]<BR>[[Hosohedron]], {2,1} |
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|} |
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==See also== |
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* [[Regular polygon]] |
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* [[Digon]] |
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* [[Degeneracy (mathematics)]] |
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==References== |
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{{reflist}} |
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* {{GlossaryForHyperspace|anchor=Monogon|title=Monogon}} |
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* [[Herbert Busemann]], The geometry of geodesics. New York, Academic Press, 1955 |
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* Coxeter, H.S.M; ''Regular Polytopes'' (third edition). Dover Publications Inc. ISBN 0-486-61480-8 |
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{{clear}} |
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{{polygons}} |
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{{polyhedra}} |
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[[Category:Polygons]] |
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Revision as of 10:34, 6 January 2015
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