Non-Euclidean geometry Totally Explained

Everything about Non-euclidean Geometry totally explained

In mathematics, non-Euclidean geometry describes hyperbolic and elliptic geometry, which are contrasted with Euclidean geometry. The essential difference between Euclidean and non-Euclidean geometry is the nature of parallel lines. Euclid's 5th postulate is equivalent to Playfair's Postulate, which states that, within a two-dimensional plane, for any given line l and a point A, which isn't on l, there's exactly one line through A that doesn't intersect l. In hyperbolic geometry, by contrast, there are infinitely many lines through A not intersecting l, while in elliptic geometry, any pair of lines intersect. (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 straight lines indefinitely extended in a two-dimensional plane that are both perpendicular to a third line. In Euclidean geometry the lines remain at a constant distance from each other, and are known as parallels. In hyperbolic geometry they "curve away" from each other, increasing in distance as one moves further from the points of intersection with the common perpendicular; these lines are often called ultraparallels. In elliptic geometry the lines "curve toward" each other and eventually intersect.

Concepts of non-Euclidean geometry

Non-Euclidean geometry systems differ from Euclidean geometry in that they modify Euclid's fifth postulate, which is also known as the parallel postulate.
   In general, there are two forms of non-Euclidean geometry, hyperbolic geometry and elliptic geometry. In hyperbolic geometry there are many more than one distinct line through a particular point that won't intersect with another given line. In elliptic geometry there are no lines that won't intersect, as all that start separate will converge. In addition, elliptic geometry modifies Euclid's first postulate so that two points determine at least one line.
   Basing new systems on these assumptions, each is constructed with its own rules and postulates. Non-Euclidean geometries and in particular elliptic geometry play an important role in relativity theory and the geometry of spacetime.
   The concepts applied to certain non-Euclidean planes can only be shown in three dimensions. The Mobius strip and Klein bottle are both complete one-sided objects, impossible in a Euclidean plane.

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 (23 definitions, five common notions, 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, including the Arabian mathematician Ibn al-Haytham (Alhacen, 11th century), the Persian mathematicians Omar Khayyám (12th century) and Nasīr al-Dīn al-Tūsī (13th century), and the Italian mathematician Giovanni Girolamo Saccheri (18th century).
   The theorems of Alhacen, Khayyam and al-Tusi on quadrilaterals, including the Lambert quadrilateral and Saccheri quadrilateral, were "the first few theorems of the hyperbolic and the elliptic geometries." These theorems along with their alternative postulates, such as Playfair's axiom, played an important role in the later development of non-Euclidean geometry. These early attempts at challenging the fifth postulate had a considerable influence on its development among later European geometers, including Witelo, Levi ben Gerson, Alfonso, John Wallis and Saccheri. All of these early attempts made at trying to formulate non-Euclidean geometry however provided flawed proofs of the parallel postulate, containing assumptions that were essentially equivalent to the parallel postulate. These early attempts did, however, provide some early properties of the hyperbolic and elliptic geometries.
   In a work titled Euclides ab Omni Naevo Vindicatus (Euclid Freed from All Flaws), published in 1733, Saccheri 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 discovered a new viable geometry. At this time it was widely believed that the universe worked according to the principles of Euclidean geometry.
   The beginning of the 19th century would finally witness decisive steps in the creation of non-Euclidean geometry. Around 1830, the Hungarian mathematician János Bolyai and the Russian mathematician Nikolai Ivanovich Lobachevsky separately published treatises on hyperbolic geometry. Consequently, hyperbolic geometry is called Bolyai-Lobachevskian geometry, as both mathematicians, independent of each other, are the basic authors of non-Euclidean geometry. While Lobachevsky created a non-Euclidean geometry by negating the parallel postulate, Bolyai worked out a geometry where both the Euclidean and the hyperbolic geometry are possible depending on a parameter k. Bolyai ends his work by mentioning that it isn't possible to decide through mathematical reasoning alone if the geometry of the physical universe is Euclidean or non-Euclidean; this is a task for the physical sciences.
   Bolyai wrote his work between 1820 and 1823 and had finalized it in 1826 (written in German but lost). Lobachevsky published his first paper on the non-Euclidean geometry in 1829, in Russian, in a journal of the Kazan university. The starting date for printing the Appendix is 1829, but it actually came out in 1831. As of this date, none of these works were considered correct by any other mathematician, with the exception of Bolyais' work, with which Gauss agreed. In 1840 Lobachevsky published his work in German, and through this work the ideas of the non-Euclidean geometry came step by step to the mathematical community. Gauss had decided not to mention to other mathematicians the existence of Bolyai's Appendix; as a result, only Lobachevsky's name was associated with the non-Euclidean geometry. It took another 30 years until the mathematical community rediscovered the work of Bolyai and corrected the authorship.
   When the mathematician Carl Friedrich Gauss read the work of János Bolyai's (Appendix), he wrote to Bolyai that he'd worked out the same results some time earlier; however Gauss hadn't written these thoughts down. In all his correspondence and manuscripts only the very starting points of the non-Euclidean geometry can be found. There is no written evidence that Gauss had worked out the non-Euclidean geometry to an extent comparable to the works of Bolyai and Lobachevsky, so Gauss can't be considered as one of the basic authors of the non-Euclidean 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's 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.

Models of non-Euclidean geometry

Euclidean geometry is modelled by our notion of a "flat plane."

Elliptic geometry

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).
In the elliptic model, for any given line l and a point A, which isn't on l, all lines through A will intersect l.

Hyperbolic geometry

Even after the work of Lobachevsky, Gauss, and Bolyai, the question remained: does such a model exist for hyperbolic geometry? The model for hyperbolic geometry 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.) Another practical model of hyperbolic space was developed by Dr. Diana Taimina in 1997 using crochet.
In the hyperbolic model, for any given line l and a point A, which isn't on l, there are infinitely many lines through A that don't intersect l.

Other models

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.

Importance

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 (for example, Euclidean), but elliptically curved (for example, 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.

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. Although the theories of modern physics suggest that our universe isn't in fact Euclidean at all, most of these science-fiction stories refer to phenomena as non-Euclidean to minds using Euclidean geometry as an approximating schema (much as time dilation is non-intuitive, despite the fact that humans live in a relativistic universe).

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