Non-Euclidean geometry
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Behavior of lines with a common perpendicular in each of the three types of geometry

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 fifth postulate, the parallel 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 is not on l, there is exactly one line through A that does not intersect l. In hyperbolic geometry, by contrast, there are infinitely many lines through A not intersecting l, while in elliptic geometry, any line through A intersects l. (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.

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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 will not intersect with another given line. In elliptic geometry there are no lines that will not intersect, as all that start to 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 after 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 Arabic mathematician Ibn al-Haytham (Alhazen, 11th century),1 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 Ibn al-Haytham, 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.2 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.

Khayyam, however, may be somewhat of an exception. Unlike many commentators on Euclid before and after him (including Saccheri), Khayyam was not trying to prove the parallel postulate as such but to derive it from an equivalent postulate he formulated from "the principles of the Philosopher" (Aristotle): "Two convergent straight lines intersect and it is impossible for two convergent straight lines to diverge in the direction in which they converge."3 Khayyam then considered the three cases right, obtuse, and acute that the summit angles of a Saccheri quadrilateral can take and after proving a number of theorems about them, he correctly refuted the obtuse and acute cases based on his postulate and hence derived the classic postulate of Euclid. Another exception may be al-Tusi's son, Sadr al-Din (sometimes known as "Pseudo-Tusi"), who wrote a book on the subject in 1298, based on al-Tusi's later thoughts, which presented one of the earliest arguments for a non-Euclidean hypothesis equivalent to the parallel postulate. "He essentially revised both the Euclidean system of axioms and postulates and the proofs of many propositions from the Elements."45 His work was published in Rome in 1594 and was studied by European geometers, including Saccheri.4

Giordano Vitale, in his book Euclide restituo (1680, 1686), used the Saccheri quadrilateral to prove that if three points are equidistant on the base AB and the summit CD, then AB and CD are everywhere equidistant. 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 is not 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 had worked out the same results some time earlier; however Gauss had not 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 cannot 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 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.

On a sphere, the sum of the angles of a triangle is not equal to 180°. The surface of a sphere is not a Euclidean space, but locally the laws of the Euclidean geometry are good approximations. In a small triangle on the face of the earth, the sum of the angles is very nearly 180°.

Models of non-Euclidean geometry

For more details on this topic, see 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 is not 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 is not on l, there are infinitely many lines through A that do not 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. Furthermore, the use of synthetic geometry serves to elucidate the non-Euclidean nature of spacetime (see external reference "Synthetic Spacetime"). Einstein's general 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.

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 is not 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).

See also

Notes

  1. ^ Eder, Michelle (2000), Views of Euclid's Parallel Postulate in Ancient Greece and in Medieval Islam, Rutgers University, http://www.math.rutgers.edu/~cherlin/History/Papers2000/eder.html, retrieved on 23 January 2008 
  2. ^ Boris A. Rosenfeld & Adolf P. Youschkevitch, "Geometry", p. 470, in Roshdi Rashed & Régis Morelon (1996), Encyclopedia of the History of Arabic Science, Vol. 2, pp. 447-494, Routledge, London and New York:

    "Three scientists, Ibn al-Haytham, Khayyam and al-Tusi, had made the most considerable contribution to this branch of geometry whose importance came to be completely recognized only in the nineteenth century. In essence their propositions concerning the properties of quadrangles which they considered assuming that some of the angles of these figures were acute of obtuse, embodied the first few theorems of the hyperbolic and the elliptic geometries. Their other proposals showed that various geometric statements were equivalent to the Euclidean postulate V. It is extremely important that these scholars established the mutual connection between tthis postulate and the sum of the angles of a triangle and a quadrangle. By their works on the theory of parallel lines Arab mathematicians directly influenced the relevant investiagtions of their European couterparts. The first European attempt to prove the postulate on parallel lines - made by Witelo, the Polish scientists of the thirteenth century, while revising Ibn al-Haytham's Book of Optics (Kitab al-Manazir) - was undoubtedly prompted by Arabic sources. The proofs put forward in the fourteenth century by the Jewish scholar Levi ben Gerson, who lived in southern France, and by the above-mentioned Alfonso from Spain directly border on Ibn al-Haytham's demonstration. Above, we have demonstrated that Pseudo-Tusi's Exposition of Euclid had stimulated borth J. Wallis's and G. Saccheri's studies of the theory of parallel lines."

  3. ^ Boris A. Rosenfeld & Adolf P. Youschkevitch (1996), "Geometry", p. 467, in Roshdi Rashed & Régis Morelon (1996), Encyclopedia of the History of Arabic Science, Vol. 2, pp. 447-494, Routledge, ISBN 0415124115
  4. ^ a b Victor J. Katz (1998), History of Mathematics: An Introduction, p. 270-271, Addison-Wesley, ISBN 0321016181:

    "But in a manuscript probably written by his son Sadr al-Din in 1298, based on Nasir al-Din's later thoughts on the subject, there is a new argument based on another hypothesis, also equivalent to Euclid's, [...] The importance of this latter work is that it was published in Rome in 1594 and was studied by European geometers. In particular, it became the starting point for the work of Saccheri and ultimately for the discovery of non-Euclidean geometry."

  5. ^ Boris A. Rosenfeld and Adolf P. Youschkevitch (1996), "Geometry", in Roshdi Rashed, ed., Encyclopedia of the History of Arabic Science, Vol. 2, p. 447-494 [469], Routledge, London and New York:

    "In Pseudo-Tusi's Exposition of Euclid, [...] another statement is used instead of a postulate. It was independent of the Euclidean postulate V and easy to prove. [...] He essentially revised both the Euclidean system of axioms and postulates and the proofs of many propositions from the Elements."

References

  • Anderson, James W. Hyperbolic Geometry, second edition, Springer, 2005
  • Beltrami, Eugenio Teoria fondamentale degli spazî di curvatura costante, Annali. di Mat., ser II 2 (1868), 232-255
  • Greenberg, Marvin Jay Euclidean and Non-Euclidean Geometries: Development and History, 4th ed., New York: W. H. Freeman, 2007. ISBN 0716799480
  • Milnor, John W. (1982) Hyperbolic geometry: The first 150 years, Bull. Amer. Math. Soc. (N.S.) Volume 6, Number 1, pp. 9-24.
  • Stewart, Ian Flatterland. New York: Perseus Publishing, 2001. ISBN 0-7382-0675-X (softcover)

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