Non-Euclidean geometry: Difference between revisions - Wikipedia

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===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]], the [[Poincaré disk model]], and the [[Poincaré half-plane model]] which model 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 Euclidean geometry was. (The reverse implication follows from the [[horosphere]] model of Euclidean geometry.) Another practical model of hyperbolic space was developed by Dr. Diana Taimina in 1997 using crochet (please see http://en.wikipedia.org/w/index.php?title=Non-Euclidean_geometry&action=edit&section=5 for more information).

In the hyperbolic model, for any given line ''l'' and a point ''A'', which is not on ''l'', there are [[infinity|infinitely]] many lines through ''A'' that do not intersect ''l''.