Non-Euclidean geometry
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Category:Geometry
The term
non-Euclidean geometry (also spelled:
non-Euclidian geometry) describes both
hyperbolic and
elliptic geometry, which are contrasted with
Euclidean geometry. \nThe essential difference between Euclidean and non-Euclidean geometry is the nature of
parallel lines. \nIn Euclidean geometry, if we start with a point
A and a line
l, then we can only draw one line through
A that is parallel to
l. \nIn hyperbolic geometry, by contrast, there are
infinitely many lines through
A parallel to
l, and in elliptic geometry, parallel lines do not exist. \n(See the entries on
hyperbolic geometry and
elliptic geometry for more information.)
Another way to describe the differences between these geometries is as follows:\nconsider two lines in a plane that are both
perpendicular to a third line.\nIn Euclidean and hyperbolic geometry, the two lines are then parallel.\nIn Euclidean geometry, however, the lines remain at a constant
distance, while in hyperbolic geometry they "curve away" from each other, increasing their distance as one moves farther from the point of intersection with the common perpendicular.\nIn elliptic geometry, the lines "curve toward" each other, and eventually intersect; therefore no parallel lines exist in elliptic geometry.
Behavior of lines with a common perpendicular in each of the three types of geometry
History
While Euclidean geometry (named for the Greek mathematician Euclid) includes some of the oldest known mathematics, non-Euclidean geometries were not widely accepted as legitimate until the 19th century. \nThe debate that eventually led to the discovery of non-Euclidean geometries began almost as soon as Euclid's work Elements was written. \nIn the Elements, Euclid attempted to establish a fully logical basis for the mathematics known up to his era. \nIn so doing, he began with a limited number of assumptions (called axioms and postulates) and sought to prove all the other results (propositions) in the work. \nThe most notorious of the postulates is often referred to as "Euclid's Fifth Postulate," or simply the "parallel postulate", which in Euclid's original formulation is:
- "If a straight line falls on two straight lines in such a manner that the interior angles on the same side are together less than two right angles, then the straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles."
Other mathematicians have devised simpler forms of this property (see the article on the
parallel postulate for some examples of
equivalent statements). Regardless of the form of the postulate, however, it consistently appears to be more complicated than Euclid's other postulates (which include, for example, "Between any two points a straight line may be drawn").
For several hundred years,
geometers were troubled by the disparate complexity of the fifth postulate, and believed it could be proved as a theorem from the other four. \nMany attempted find a
proof by contradiction, most notably the
Italian Giovanni Gerolamo Saccheri. \nIn a work titled
Euclides ab Omni Naevo Vindicatus (
Euclid Freed from All Flaws), published in
1733, he quickly discarded elliptic geometry as a possibility (some others of Euclid's axioms must be modified for elliptic geometry to work) and set to work proving a great number of results in hyperbolic geometry. \nHe finally reached a point where he believed that his results demonstrated a contradiction in the system, thus showing that hyperbolic geometry is logically inconsistent. \nHis claim of inconsistency seems to have been based on Euclidean presuppositions, because no such contradiction was present.
\nA hundred years later, in
1829, the
Russian Nikolai Ivanovich Lobachevsky published a treatise of hyperbolic geometry. \nFor this reason, hyperbolic geometry is sometimes called Lobachevskian geometry.\nAbout the same time, the
Hungarian Janos Bolyai also wrote a treatise on hyperbolic geometry, which was published in
1832 as an appendix to a work of his father's. \nThe great mathematician
Karl Friedrich Gauss read the appendix and revealed to Bolyai that he had worked out the same results some time earlier. \nEach of these men thus discovered hyperbolic geometry independently, and none of their work should be disparaged in this light. \nLobachevsky's name is attached by right of earliest publication. The fundamental difference between these and earlier works, such as Saccheri's, is that they were the first to unabashedly claim that Euclidean geometry was not the only geometry, nor the only conceivable geometric structure for the universe. However, the possibility still remained that the axioms for hyperbolic geometry were logically inconsistent.
As had been mentioned, more work on Euclid's axioms needed to be done to establish elliptic geometry. \n
Bernhard Riemann, in a famous lecture in
1854, founded the field of
Riemannian geometry, discussing in particular the ideas now called manifolds,
Riemannian metric, and
curvature. \nHe constructed an infinite family of non-Euclidean geometries by giving a formula for a familiy of Riemannian metrics on the unit ball in Euclidean space. \nSometimes he is unjustly credited with only discovering
elliptic geometry, but in fact, this construction shows that his work was far-reaching, with his theorems holding for all geometries.
Euclidean geometry is modelled by our notion of a "flat
plane." \nThe simplest model for elliptic geometry is a sphere, where lines are "
great circles" (such as the
equator or the meridians on a
globe), and points opposite each other are identified (considered to be the same). \nEven after the work of Lobachevski, Gauss, and Bolyai, the question remained: does such a model exist for hyperbolic geometry? \nThis question was answered by
Eugenio Beltrami, in
1868, who proved that a surface called the
pseudosphere has the appropriate
curvature to model hyperbolic geometry. \nHis work was directly based on that of
Riemann. The significance of Beltrami's work lies in showing that hyperbolic geometry was logically consistent if Euclidean geometry was.
The development of non-Euclidean geometries proved very important to physics in the
20th century.\n
Einstein's
Theory of Relativity describes space as generally flat (i.e., Euclidean), but elliptically curved (i.e., non-Euclidean) in regions near where matter is present. Because the universe expands (see the
hubble constant), the space where no matter exists could be described by using a hyperbolic model.\nThis kind of geometry, where the curvature changes from point to point, is called pseudo-Euclidean geometry.
Reference
\n*Ian Stewart;
Flatterland; Perseus Publishing; ISBN 0-7382-0675-X (softcover, 2001)
External link
\n*MacTutor Archive article on non-Euclidean geometry
See also
\n*Projective geometry\n*
Spherical geometry\n*
Taxicab geometry