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Nasir al-Din attempted to derive a proof by contradiction of the parallel postulate. [18] He also considered the cases of what are now known as elliptical and hyperbolic geometry, though he ruled out both of them. [17] Euclidean, elliptical and hyperbolic geometry. The Parallel Postulate is satisfied only for models of Euclidean geometry.
Hyperbolic geometry is more closely related to Euclidean geometry than it seems: the only axiomatic difference is the parallel postulate. When the parallel postulate is removed from Euclidean geometry the resulting geometry is absolute geometry. There are two kinds of absolute geometry, Euclidean and hyperbolic.
Hyperbolic geometry is a non-Euclidean geometry where the first four axioms of Euclidean geometry are kept but the fifth axiom, the parallel postulate, is changed.The fifth axiom of hyperbolic geometry says that given a line L and a point P not on that line, there are at least two lines passing through P that are parallel to L. [1]
hyperbolic plane: plane, where the parallel postulate does not hold E 3: Euclidean 3 space: space defined by three perpendicular coordinate axes MLD: Mutually locally derivable: two tilings are said to be mutually locally derivable from each other, if one tiling can be obtained from the other by a simple local rule (such as deleting or ...
Poincaré disk with hyperbolic parallel lines Poincaré disk model of the truncated triheptagonal tiling.. In geometry, the Poincaré disk model, also called the conformal disk model, is a model of 2-dimensional hyperbolic geometry in which all points are inside the unit disk, and straight lines are either circular arcs contained within the disk that are orthogonal to the unit circle or ...
Hyperbolic space, developed independently by Nikolai Lobachevsky, János Bolyai and Carl Friedrich Gauss, is a geometric space analogous to Euclidean space, but such that Euclid's parallel postulate is no longer assumed to hold. Instead, the parallel postulate is replaced by the following alternative (in two dimensions):
For examples, elliptic geometry (no parallels) and hyperbolic geometry (many parallels). Both elliptic and hyperbolic geometry are consistent systems, showing that the parallel postulate is independent of the other axioms. [2] Proving independence is often very difficult. Forcing is one commonly used technique. [3]
In the 19th century mathematicians began to seriously question the parallel postulate and found that other forms of geometry are possible. For example in elliptical geometry: Given a line and a point not on that line, all lines drawn through that point will intersect the original line. (case 2) And in hyperbolic geometry: