Non-Euclidean geometry

The term non-Euclidean geometry (also spelled: non-euclidian geometry) describes hyperbolic, elliptic and absolute geometry, which are contrasted with Euclidean geometry. The essential difference between Euclidean and non-Euclidean geometry is the nature of parallel lines. In Euclidean geometry, if we start with a line l and a point A, which is not on l, then we can only draw one line through A that is parallel to l. In hyperbolic geometry, by contrast, there are infinitely many lines through A parallel to l, and in elliptic geometry, parallel lines do not exist. (See the entries on hyperbolic geometry and elliptic geometry for more information.)

Another way to describe the differences between these geometries is as follows: Consider two lines in a two-dimensional plane that are both perpendicular to a third line. In Euclidean and hyperbolic geometry the two lines are then parallel. In Euclidean geometry the lines remain at a constant distance, intersecting only in the infinite; while in hyperbolic geometry they "curve away" from each other, increasing their distance as one moves further from the point of intersection with the common perpendicular. In elliptic geometry the lines "curve toward" each other and eventually intersect. Therefore no parallel lines exist in elliptic 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. The debate that eventually led to the discovery of non-Euclidean geometries began almost as soon as Euclid's work Elements was written. In the Elements, Euclid began with a limited number of assumptions (five axioms and five postulates) and sought to prove all the other results (propositions) in the work. The 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 parallel postulate for 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. Many attempted to find a proof by contradiction, most notably the Italian Giovanni Gerolamo Saccheri. In 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. He finally reached a point where he believed that his results demonstrated the impossibility of hyperbolic geometry. His claim seems to have been based on Euclidean presuppositions, because no logical contradiction was present. In this attempt to prove Euclidean geometry he instead unintentionally invented a new viable geometry. At this time it was widely believed that the universe worked according to the principles of Euclidean geometry.

A hundred years later, in 1829, the Russian Nikolai Ivanovich Lobachevsky published a treatise of hyperbolic geometry. For this reason, hyperbolic geometry is sometimes called Lobachevskian geometry. About the same time, the Hungarian János Bolyai also wrote a treatise on hyperbolic geometry, which was published in 1832 as an appendix to a work of his father's. The great mathematician Carl Friedrich Gauss read the appendix and revealed to Bolyai that he had worked out the same results some time earlier.

Lobachevsky'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. Lobachevsky termed Euclidean geometry, "ordinary geometry," and this new hyperbolic geometry, "imaginary geometry." 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. 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. He constructed an infinite family of non-Euclidean geometries by giving a formula for a family of Riemannian metrics on the unit ball in Euclidean space. Sometimes 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." The 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). Even after the work of Lobachevsky, Gauss, and Bolyai, the question remained: does such a model exist for hyperbolic geometry? This question 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 equiconsistent, so that hyperbolic geometry was logically consistent if Euclidean geometry was. (The reverse implication follows from the horosphere model of Euclidean geometry.)

The development of non-Euclidean geometries proved very important to physics in the 20th century. Given the limitation of the speed of light, velocity additions necessitate the use of hyperbolic geometry. 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. This kind of geometry, where the curvature changes from point to point, is called riemannian geometry.

There are other mathematical models of the plane in which the parallel postulate fails, for example the Dehn plane consisting of all points (x,y), where x and y are finite surreal numbers.

Fiction

Non-Euclidean geometry often makes appearances in works of Science Fiction and Fantasy. Its usage is most clearly tied with the influence of the 20th century Horror fiction writer H.P. Lovecraft. In his works, many unnatural things follow their own unique laws of geometry. This is said to be a profoundly unsettling sight, often to the point of driving those who look upon it insane.

Modern usage is similar, portraying Non-Euclidean geometry as a stark, mentally disturbing intrusion on the natural order. It is associated most commonly with beings from universes distinct from our own. This is inconsistent with the fact that our own universe is not, according to modern physical theories, itself Euclidean; however, like many other things in Science Fiction and Fantasy, the truthfulness of this characterization is simply accepted as a literary convention.

References

James W. Anderson, Hyperbolic Geometry, second edition, Springer, 2005

Eugenio Beltrami, Theoria fondamentale delgi spazil di curvatura constanta, Annali. di Mat., ser II 2 (1868), 232-255

Ian Stewart, Flatterland. New York: Perseus Publishing, 2001. ISBN 0-7382-0675-X (softcover)

Marvin Jay Greenberg, Euclidean and pseudo-Euclidean geometries: Development and history New York: W. H. Freeman, 1993. ISBN 0716724464

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