Editing Non-Euclidean geometry (section)

==Models==
{{comparison_of_geometries.svg}}
[[File:Triangles (spherical geometry).jpg|thumb|350px|On a sphere, the sum of the angles of a triangle is not equal to 180°. The surface of a sphere is not a Euclidean space, but locally the laws of the Euclidean geometry are good approximations. In a small triangle on the face of the earth, the sum of the angles is very near 180°.]]

'''Models of non-Euclidean geometry''' are [[mathematical model]]s of geometries which are non-Euclidean in the sense that it is not the case that exactly one line can be drawn [[parallel lines|parallel]] to a given line ''l'' through a point that is not on ''l''. In hyperbolic geometric models, by contrast, there are [[infinity|infinitely]] many lines through ''A'' parallel to ''l'', and in elliptic geometric models, parallel lines do not exist. (See the entries on [[hyperbolic geometry]] and [[elliptic geometry]] for more information.)

Euclidean geometry is modelled by our notion of a "flat [[plane (mathematics)|plane]]." 
The simplest model for elliptic geometry is a sphere, where lines are "[[great circle]]s" (such as the [[equator]] or the [[meridian (geography)|meridian]]s on a [[globe]]), and points opposite each other are identified (considered to be the same). 
The [[pseudosphere]] has the appropriate [[curvature]] to model hyperbolic geometry.

===Elliptic geometry===
{{main|Elliptic geometry}}
The simplest model for [[elliptic geometry]] is a sphere, where lines are "[[great circle]]s" (such as the [[equator]] or the [[meridian (geography)|meridian]]s on a [[globe]]), and points opposite each other (called [[antipodal points]]) are identified (considered the same). This is also one of the standard models of the [[real projective plane]]. The difference is that as a model of elliptic geometry a metric is introduced permitting the measurement of lengths and angles, while as a model of the projective plane there is no such metric.

In the elliptic model, for any given line {{mvar|l}} and a point ''A'', which is not on {{mvar|l}}, all lines through ''A'' will intersect {{mvar|l}}.

===Hyperbolic geometry===
{{main|Hyperbolic geometry}}
Even after the work of Lobachevsky, Gauss, and Bolyai, the question remained: "Does such a model exist for [[hyperbolic geometry]]?". The model for [[hyperbolic geometry]] was answered by [[Eugenio Beltrami]], in 1868, who first showed that a surface called the [[pseudosphere]] has the appropriate [[curvature]] to model a portion of [[hyperbolic space]] and in a second paper in the same year, defined the [[Klein model]], which models the entirety of hyperbolic space, and used this to show that Euclidean geometry and hyperbolic geometry were [[equiconsistency|equiconsistent]] so that hyperbolic geometry was [[logically consistent]] if and only if Euclidean geometry was. (The reverse implication follows from the [[horosphere]] model of Euclidean geometry.)

In the hyperbolic model, within a two-dimensional plane, for any given line {{mvar|l}} and a point ''A'', which is not on {{mvar|l}}, there are [[Infinite set|infinitely]] many lines through ''A'' that do not intersect {{mvar|l}}.

In these models, the concepts of non-Euclidean geometries are represented by Euclidean objects in a Euclidean setting. This introduces a perceptual distortion wherein the straight lines of the non-Euclidean geometry are represented by Euclidean curves that visually bend. This "bending" is not a property of the non-Euclidean lines, only an artifice of the way they are represented.

===Three-dimensional non-Euclidean geometry===
{{main|Thurston geometry}}
In three dimensions, there are eight models of geometries.<ref>* [[William Thurston]]. ''Three-dimensional geometry and topology. Vol. 1''. Edited by Silvio Levy. Princeton Mathematical Series, 35. Princeton University Press, Princeton, NJ, 1997. x+311&nbsp;pp. {{isbn|0-691-08304-5}} (in depth explanation of the eight geometries and the proof that there are only eight)
</ref> There are Euclidean, elliptic, and hyperbolic geometries, as in the two-dimensional case; mixed geometries that are partially Euclidean and partially hyperbolic or spherical; twisted versions of the mixed geometries; and one unusual geometry that is completely [[anisotropy|anisotropic]] (i.e. every direction behaves differently).