Search results
Results from the WOW.Com Content Network
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. All theorems of absolute ...
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]
Euclid gave the definition of parallel lines in Book I, Definition 23 [2] just before the five postulates. [3] Euclidean geometry is the study of geometry that satisfies all of Euclid's axioms, including the parallel postulate. The postulate was long considered to be obvious or inevitable, but proofs were elusive.
The new axiom is Lobachevsky's parallel postulate (also known as the characteristic postulate of hyperbolic geometry): [75] Through a point not on a given line there exists (in the plane determined by this point and line) at least two lines which do not meet the given line. With this addition, the axiom system is now complete.
In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those that specify Euclidean geometry.As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean geometry arises by either replacing the parallel postulate with an alternative, or relaxing the metric requirement.
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 ...
The book begins with a historical overview of the long struggles with the parallel postulate in Euclidean geometry, [3] and of the foundational crisis of the late 19th and early 20th centuries, [6] Then, after reviewing background material in real analysis and computability theory, [1] the book concentrates on the reverse mathematics of theorems in real analysis, [3] including the Bolzano ...
In his historic work Elements, Euclid assumed the existence of parallel lines with his fifth postulate. His parallel postulate is equivalent to: His parallel postulate is equivalent to: Given a line and a point not on that line, exactly one line can be drawn through that point which does not intersect the original line.