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Elliptic geometry is an example of a geometry in which Euclid's parallel postulate does not hold. Instead, as in spherical geometry , there are no parallel lines since any two lines must intersect. However, unlike in spherical geometry, two lines are usually assumed to intersect at a single point (rather than two).
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 various attempted proofs of the parallel postulate produced a long list of theorems that are equivalent to the parallel postulate. Equivalence here means that in the presence of the other axioms of the geometry each of these theorems can be assumed to be true and the parallel postulate can be proved from this altered set of axioms.
Saccheri quadrilaterals. A Saccheri quadrilateral is a quadrilateral with two equal sides perpendicular to the base.It is named after Giovanni Gerolamo Saccheri, who used it extensively in his 1733 book Euclides ab omni naevo vindicatus (Euclid freed of every flaw), an attempt to prove the parallel postulate using the method reductio ad absurdum.
The archetypical example is the real projective plane, also known as the extended Euclidean plane. [4] This example, in slightly different guises, is important in algebraic geometry, topology and projective geometry where it may be denoted variously by PG(2, R), RP 2, or P 2 (R), among other notations.
Eventually in 1829, mathematician Nikolai Lobachevsky published a description of acute geometry (or hyperbolic geometry), a geometry which assumed a different form of the parallel postulate. It is in fact possible to create a valid geometry without the fifth postulate entirely, or with different versions of the fifth postulate (elliptic geometry).
Historically, the fourth angle of a Lambert quadrilateral was of considerable interest since if it could be shown to be a right angle, then the Euclidean parallel postulate could be proved as a theorem. It is now known that the type of the fourth angle depends upon the geometry in which the quadrilateral exists.
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]