Inmathematics,non-Euclidean geometry consists of two geometries based onaxioms closely related to those that specifyEuclidean geometry. As Euclidean geometry lies at the intersection ofmetric geometry andaffine geometry, non-Euclidean geometry arises by either replacing theparallel postulate with an alternative, or relaxing the metric requirement. In the former case, one obtainshyperbolic geometry andelliptic geometry, the traditional non-Euclidean geometries. When the metric requirement is relaxed, then there are affine planes associated with theplanar algebras, which give rise tokinematic geometries that have also been called non-Euclidean geometry.
The essential difference between the metric geometries is the nature ofparallel lines.Euclid's fifth postulate, theparallel postulate, is equivalent toPlayfair's postulate, which states that, within a two-dimensional plane, for any given linel and a pointA, which is not onl, there is exactly one line throughA that does not intersectl. In hyperbolic geometry, by contrast, there areinfinitely many lines throughA not intersectingl, while in elliptic geometry, any line throughA intersectsl.
Another way to describe the differences between these geometries is to consider two straight lines indefinitely extended in a two-dimensional plane that are bothperpendicular to a third line (in the same plane):
In Euclidean geometry, the lines remain at a constantdistance from each other (meaning that a line drawn perpendicular to one line at any point will intersect the other line and the length of the line segment joining the points of intersection remains constant) and are known as parallels.
In hyperbolic geometry, they diverge from each other, increasing in distance as one moves further from the points of intersection with the common perpendicular; these lines are often calledultraparallels.
In elliptic geometry, the lines converge toward each other and intersect.
Euclidean geometry, named after theGreek mathematicianEuclid, includes some of the oldest known mathematics, and geometries that deviated from this were not widely accepted as legitimate until the 19th century.
The debate that eventually led to the discovery of the non-Euclidean geometries began almost as soon as Euclid wroteElements. In theElements, Euclid begins with a limited number of assumptions (23 definitions, five common notions, and five postulates) and seeks 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 theparallel 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. Regardless of the form of the postulate, however, it consistently appears more complicated thanEuclid's other postulates:
To draw a straight line from any point to any point.
To produce [extend] a finite straight line continuously in a straight line.
To describe a circle with any centre and distance [radius].
The theorems of Ibn al-Haytham, Khayyam and al-Tusi onquadrilaterals, including theLambert quadrilateral andSaccheri quadrilateral, were "the first few theorems of thehyperbolic and theelliptic geometries". These theorems along with their alternative postulates, such asPlayfair'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, includingWitelo,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, depending on assumptions that are now recognized as essentially equivalent to the parallel postulate. These early attempts did, however, provide some early properties of the hyperbolic and elliptic geometries.
Khayyam, for example, tried 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, which he didn't realize was equivalent to his own postulate. Another example is 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 another hypothesis equivalent to the parallel postulate. "He essentially revised both the Euclidean system of axioms and postulates and the proofs of many propositions from theElements."[4][5] His work was published inRome in 1594 and was studied by European geometers, including Saccheri[4] who criticised this work as well as that of Wallis.[6]
Giordano Vitale, in his bookEuclide 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 titledEuclides 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 nological contradiction was present. In this attempt to prove Euclidean geometry he instead unintentionally discovered a new viable geometry, but did not realize it.
In 1766Johann Lambert wrote, but did not publish,Theorie der Parallellinien in which he attempted, as Saccheri did, to prove the fifth postulate. He worked with a figure now known as aLambert quadrilateral, a quadrilateral with three right angles (can be considered half of a Saccheri quadrilateral). He quickly eliminated the possibility that the fourth angle is obtuse, as had Saccheri and Khayyam, and then proceeded to prove many theorems under the assumption of an acute angle. Unlike Saccheri, he never felt that he had reached a contradiction with this assumption. He had proved the non-Euclidean result that the sum of the angles in a triangle increases as the area of the triangle decreases, and this led him to speculate on the possibility of a model of the acute case on a sphere of imaginary radius. He did not carry this idea any further.[7]
At this time it was widely believed that the universe worked according to the principles of Euclidean geometry.[8]
The beginning of the 19th century would finally witness decisive steps in the creation of non-Euclidean geometry.Circa 1813,Carl Friedrich Gauss and independently around 1818, the German professor of lawFerdinand Karl Schweikart[9] had the germinal ideas of non-Euclidean geometry worked out, but neither published any results. Schweikart's nephewFranz Taurinus did publish important results of hyperbolic trigonometry in two papers in 1825 and 1826, yet while admitting the internal consistency of hyperbolic geometry, he still believed in the special role of Euclidean geometry.[10]
Then, in 1829–1830 theRussian mathematicianNikolai Ivanovich Lobachevsky and in 1832 theHungarian mathematicianJános Bolyai separately and independently published treatises on hyperbolic geometry. Consequently, hyperbolic geometry is called Lobachevskian or Bolyai-Lobachevskian geometry, as both mathematicians, independent of each other, are the basic authors of non-Euclidean geometry.Gauss mentioned to Bolyai's father, when shown the younger Bolyai's work, that he had developed such a geometry several years before,[11] though he did not publish. 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.
Bernhard Riemann, in a famous lecture in 1854, founded the field ofRiemannian geometry, discussing in particular the ideas now calledmanifolds,Riemannian metric, andcurvature.He constructed an infinite family of non-Euclidean geometries by giving a formula for a family of Riemannian metrics on the unit ball inEuclidean space. The simplest of these is calledelliptic geometry and it is considered a non-Euclidean geometry due to its lack of parallel lines.[12]
By formulating the geometry in terms of a curvaturetensor, Riemann allowed non-Euclidean geometry to apply to higher dimensions. Beltrami (1868) was the first to apply Riemann's geometry to spaces of negative curvature.
It was Gauss who coined the term "non-Euclidean geometry".[13] He was referring to his own work, which today we callhyperbolic geometry orLobachevskian geometry. Several modern authors still use the generic termnon-Euclidean geometry to meanhyperbolic geometry.[14]
Arthur Cayley noted that distance between points inside a conic could be defined in terms oflogarithm and the projectivecross-ratio function. The method has become called theCayley–Klein metric becauseFelix Klein exploited it to describe the non-Euclidean geometries in articles[15] in 1871 and 1873 and later in book form. The Cayley–Klein metrics provided working models of hyperbolic and elliptic metric geometries, as well as Euclidean geometry.
Klein is responsible for the terms "hyperbolic" and "elliptic" (in his system he called Euclidean geometryparabolic, a term that generally fell out of use[16]). His influence has led to the current usage of the term "non-Euclidean geometry" to mean either "hyperbolic" or "elliptic" geometry.
There are some mathematicians who would extend the list of geometries that should be called "non-Euclidean" in various ways.[17]
There are many kinds of geometry that are quite different from Euclidean geometry but are also not necessarily included in the conventional meaning of "non-Euclidean geometry", such as more general instances ofRiemannian geometry.
Euclidean geometry can be axiomatically described in several ways. However, Euclid's original system of five postulates (axioms) is not one of these, as his proofs relied on several unstated assumptions that should also have been taken as axioms.Hilbert's system consisting of 20 axioms[18] most closely follows the approach of Euclid and provides the justification for all of Euclid's proofs. Other systems, using different sets ofundefined terms obtain the same geometry by different paths. All approaches, however, have an axiom that is logically equivalent to Euclid's fifth postulate, the parallel postulate.Hilbert uses the Playfair axiom form, whileBirkhoff, for instance, uses the axiom that says that, "There exists a pair of similar but not congruent triangles." In any of these systems, removal of the one axiom equivalent to the parallel postulate, in whatever form it takes, and leaving all the other axioms intact, producesabsolute geometry. As the first 28 propositions of Euclid (inThe Elements) do not require the use of the parallel postulate or anything equivalent to it, they are all true statements in absolute geometry.[19]
To obtain a non-Euclidean geometry, the parallel postulate (or its equivalent)must be replaced by itsnegation. Negating thePlayfair's axiom form, since it is a compound statement (... there exists one and only one ...), can be done in two ways:
Either there will exist more than one line through the point parallel to the given line or there will exist no lines through the point parallel to the given line. In the first case, replacing the parallel postulate (or its equivalent) with the statement "In a plane, given a point P and a linel not passing through P, there exist two lines through P, which do not meetl" and keeping all the other axioms, yieldshyperbolic geometry.[20]
The second case is not dealt with as easily. Simply replacing the parallel postulate with the statement, "In a plane, given a point P and a linel not passing through P, all the lines through P meetl", does not give a consistent set of axioms. This follows since parallel lines exist in absolute geometry,[21] but this statement says that there are no parallel lines. This problem was known (in a different guise) to Khayyam, Saccheri and Lambert and was the basis for their rejecting what was known as the "obtuse angle case". To obtain a consistent set of axioms that includes this axiom about having no parallel lines, some other axioms must be tweaked. These adjustments depend upon the axiom system used. Among others, these tweaks have the effect of modifying Euclid's second postulate from the statement that line segments can be extended indefinitely to the statement that lines are unbounded.Riemann'selliptic geometry emerges as the most natural geometry satisfying this axiom.
Comparison of elliptic, Euclidean and hyperbolic geometries in two dimensionsOn 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 near 180°.
Models of non-Euclidean geometry aremathematical models of geometries which are non-Euclidean in the sense that it is not the case that exactly one line can be drawnparallel to a given linel through a point that is not onl. In hyperbolic geometric models, by contrast, there areinfinitely many lines throughA parallel tol, and in elliptic geometric models, parallel lines do not exist. (See the entries onhyperbolic geometry andelliptic geometry for more information.)
Euclidean geometry is modelled by our notion of a "flatplane." The simplest model for elliptic geometry is a sphere, where lines are "great circles" (such as theequator or themeridians on aglobe), and points opposite each other are identified (considered to be the same). Thepseudosphere has the appropriatecurvature to model hyperbolic geometry.
The simplest model forelliptic geometry is a sphere, where lines are "great circles" (such as theequator or themeridians on aglobe), and points opposite each other (calledantipodal points) are identified (considered the same). This is also one of the standard models of thereal projective plane. The difference is that as a model of elliptic geometry a metric is introduced permitting the measurement of lengths and angles, while as a model of the projective plane there is no such metric.
In the elliptic model, for any given linel and a pointA, which is not onl, all lines throughA will intersectl.
Even after the work of Lobachevsky, Gauss, and Bolyai, the question remained: "Does such a model exist forhyperbolic geometry?". The model forhyperbolic geometry was answered byEugenio Beltrami, in 1868, who first showed that a surface called thepseudosphere has the appropriatecurvature to model a portion ofhyperbolic space and in a second paper in the same year, defined theKlein model, which models the entirety of hyperbolic space, and used this to show that Euclidean geometry and hyperbolic geometry wereequiconsistent so that hyperbolic geometry waslogically consistent if and only if Euclidean geometry was. (The reverse implication follows from thehorosphere model of Euclidean geometry.)
In the hyperbolic model, within a two-dimensional plane, for any given linel and a pointA, which is not onl, there areinfinitely many lines throughA that do not intersectl.
In these models, the concepts of non-Euclidean geometries are represented by Euclidean objects in a Euclidean setting. This introduces a perceptual distortion wherein the straight lines of the non-Euclidean geometry are represented by Euclidean curves that visually bend. This "bending" is not a property of the non-Euclidean lines, only an artifice of the way they are represented.
In three dimensions, there are eight models of geometries.[22] There are Euclidean, elliptic, and hyperbolic geometries, as in the two-dimensional case; mixed geometries that are partially Euclidean and partially hyperbolic or spherical; twisted versions of the mixed geometries; and one unusual geometry that is completelyanisotropic (i.e. every direction behaves differently).
Euclidean and non-Euclidean geometries naturally have many similar properties, namely those that do not depend upon the nature of parallelism. This commonality is the subject ofabsolute geometry (also calledneutral geometry). However, the properties that distinguish one geometry from others have historically received the most attention.
Besides the behavior of lines with respect to a common perpendicular, mentioned in the introduction, we also have the following:
ALambert quadrilateral is a quadrilateral with three right angles. The fourth angle of a Lambert quadrilateral isacute if the geometry is hyperbolic, aright angle if the geometry is Euclidean orobtuse if the geometry is elliptic. Consequently,rectangles exist (a statement equivalent to the parallel postulate) only in Euclidean geometry.
ASaccheri quadrilateral is a quadrilateral with two sides of equal length, both perpendicular to a side called thebase. The other two angles of a Saccheri quadrilateral are called thesummit angles and they have equal measure. The summit angles of a Saccheri quadrilateral are acute if the geometry is hyperbolic, right angles if the geometry is Euclidean and obtuse angles if the geometry is elliptic.
The sum of the measures of the angles of any triangle is less than 180° if the geometry is hyperbolic, equal to 180° if the geometry is Euclidean, and greater than 180° if the geometry is elliptic. Thedefect of a triangle is the numerical value (180° − sum of the measures of the angles of the triangle). This result may also be stated as: the defect of triangles in hyperbolic geometry is positive, the defect of triangles in Euclidean geometry is zero, and the defect of triangles in elliptic geometry is negative.
Before the models of a non-Euclidean plane were presented by Beltrami, Klein, and Poincaré, Euclidean geometry stood unchallenged as themathematical model ofspace. Furthermore, since the substance of the subject insynthetic geometry was a chief exhibit of rationality, the Euclidean point of view represented absolute authority.
The discovery of the non-Euclidean geometries had a ripple effect which went far beyond the boundaries of mathematics and science. The philosopherImmanuel Kant's treatment of human knowledge had a special role for geometry. It was his prime example of synthetic a priori knowledge; not derived from the senses nor deduced through logic — our knowledge of space was a truth that we were born with. Unfortunately for Kant, his concept of this unalterably true geometry was Euclidean. Theology was also affected by the change from absolute truth to relative truth in the way that mathematics is related to the world around it, that was a result of this paradigm shift.[23]
Non-Euclidean geometry is an example of ascientific revolution in thehistory of science, in which mathematicians and scientists changed the way they viewed their subjects.[24] Some geometers calledLobachevsky the "Copernicus of Geometry" due to the revolutionary character of his work.[25][26]
The existence of non-Euclidean geometries impacted the intellectual life ofVictorian England in many ways[27] and in particular was one of the leading factors that caused a re-examination of the teaching of geometry based onEuclid's Elements. This curriculum issue was hotly debated at the time and was even the subject of a book,Euclid and his Modern Rivals, written by Charles Lutwidge Dodgson (1832–1898) better known asLewis Carroll, the author ofAlice in Wonderland.
For planar algebra, non-Euclidean geometry arises in the other cases.Whenε2 = +1, ahyperbolic unit. Thenz is asplit-complex number and conventionallyj replaces epsilon. Then
This approach to non-Euclidean geometry explains the non-Euclidean angles: the parameters ofslope in the dual number plane andhyperbolic angle in the split-complex plane correspond toangle in Euclidean geometry. Indeed, they each arise inpolar decomposition of a complex numberz.[29]
The non-Euclidean planar algebras support kinematic geometries in the plane. For instance, thesplit-complex numberz = eaj can represent a spacetime event one moment into the future of aframe of reference ofrapiditya. Furthermore, multiplication byz amounts to aLorentz boost mapping the frame with rapidity zero to that with rapiditya.
In 1895,H. G. Wells published the short story "The Remarkable Case of Davidson's Eyes". To appreciate this story one should know howantipodal points on a sphere are identified in a model of the elliptic plane. In the story, in the midst of a thunderstorm, Sidney Davidson sees "Waves and a remarkably neat schooner" while working in an electrical laboratory at Harlow Technical College. At the story's close, Davidson proves to have witnessed H.M.S.Fulmar offAntipodes Island.
Non-Euclidean geometry is sometimes connected with the influence of the 20th-centuryhorror fiction writerH. P. Lovecraft. In his works, many unnatural things follow their own unique laws of geometry: in Lovecraft'sCthulhu Mythos, the sunken city ofR'lyeh is characterized by its non-Euclidean geometry. It is heavily implied this is achieved as a side effect of not following the natural laws of this universe rather than simply using an alternate geometric model, as the sheer innate wrongness of it is said to be capable of driving those who look upon it insane.[35]
InThe Brothers Karamazov, Dostoevsky discusses non-Euclidean geometry through his character Ivan.
Christopher Priest's novelInverted World describes the struggle of living on a planet with the form of a rotatingpseudosphere.
Robert Heinlein'sThe Number of the Beast utilizes non-Euclidean geometry to explain instantaneous transport through space and time and between parallel and fictional universes.
Zeno Rogue'sHyperRogue is aroguelike game set on thehyperbolic plane, allowing the player to experience many properties of this geometry. Many mechanics, quests, and locations are strongly dependent on the features of hyperbolic geometry.[36]
"Three scientists, Ibn al-Haytham, Khayyam, and al-Tusi, had made the most considerable contribution to this branch of geometry, whose importance was completely recognized only in the nineteenth century. In essence, their propositions concerning the properties of quadrangle—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 this 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 investigations of their European counterparts. The first European attempt to prove the postulate on parallel lines – made byWitelo, the Polish scientists of the thirteenth century, while revisingIbn al-Haytham'sBook of Optics (Kitab al-Manazir) – was undoubtedly prompted by Arabic sources. The proofs put forward in the fourteenth century by the Jewish scholarLevi 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 thatPseudo-Tusi's Exposition of Euclid had stimulated both J. Wallis's and G.Saccheri's studies of the theory of parallel lines."
"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."
"InPseudo-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 theElements."
^A notable exception is David Hume, who as early as 1739 seriously entertained the possibility that our universe was non-Euclidean; see David Hume (1739/1978)A Treatise of Human Nature, L.A. Selby-Bigge, ed. (Oxford: Oxford University Press), pp. 51–52.
^In a letter of December 1818, Ferdinand Karl Schweikart (1780–1859) sketched a few insights into non-Euclidean geometry. The letter was forwarded to Gauss in 1819 by Gauss's former student Gerling. In his reply to Gerling, Gauss praised Schweikart and mentioned his own, earlier research into non-Euclidean geometry. See:
Carl Friedrich Gauss,Werke (Leipzig, Germany: B. G. Teubner, 1900), vol. 8,pp. 180–182.
Letters by Schweikart and the writings of his nephewFranz Adolph Taurinus, who also was interested in non-Euclidean geometry and who in 1825 published a brief book on the parallel axiom, appear in: Paul Stäckel and Friedrich Engel,Die theorie der Parallellinien von Euklid bis auf Gauss, eine Urkundensammlung der nichteuklidischen Geometrie (The theory of parallel lines from Euclid to Gauss, an archive of non-Euclidean geometry), (Leipzig, Germany: B. G. Teubner, 1895),pages 243 ff.
^In the letter to Wolfgang (Farkas) Bolyai of March 6, 1832 Gauss claims to have worked on the problem for thirty or thirty-five years (Faber 1983, p. 162). In his 1824 letter to Taurinus (Faber 1983, p. 158) he claimed that he had been working on the problem for over 30 years and provided enough detail to show that he actually had worked out the details. According toFaber (1983, p. 156) it wasn't until around 1813 that Gauss had come to accept the existence of a new geometry.
^However, other axioms besides the parallel postulate must be changed to make this a feasible geometry.
^Felix Klein,Elementary Mathematics from an Advanced Standpoint: Geometry, Dover, 1948 (Reprint of English translation of 3rd Edition, 1940. First edition in German, 1908.) p. 176.
^*William Thurston.Three-dimensional geometry and topology. Vol. 1. Edited by Silvio Levy. Princeton Mathematical Series, 35. Princeton University Press, Princeton, NJ, 1997. x+311 pp.ISBN0-691-08304-5 (in depth explanation of the eight geometries and the proof that there are only eight)
^Imre Toth, "Gott und Geometrie: Eine viktorianische Kontroverse,"Evolutionstheorie und ihre Evolution, Dieter Henrich, ed. (Schriftenreihe der Universität Regensburg, band 7, 1982) pp. 141–204.
^This is a quote from G. B. Halsted's translator's preface to his 1914 translation ofThe Theory of Parallels: "WhatVesalius was toGalen, whatCopernicus was toPtolemy that was Lobachevsky toEuclid." —W. K. Clifford
^Isaak Yaglom (1968)Complex Numbers in Geometry, translated by E. Primrose from 1963 Russian original, appendix "Non-Euclidean geometries in the plane and complex numbers", pp 195–219,Academic Press, N.Y.
^Richard C. Tolman (2004)Theory of Relativity of Motion, page 194, §180 Non-Euclidean angle, §181 Kinematical interpretation of angle in terms of velocity
^Isaak Yaglom (1979) A simple non-Euclidean geometry and its physical basis : an elementary account of Galilean geometry and the Galilean principle of relativity, SpringerISBN0-387-90332-1
A'Campo, Norbert; Papadopoulos, Athanase (2012), "Notes on non-Euclidean geometry",Strasbourg Master Class on Geometry, Zürich, Switzerland: European Mathematical Society, pp. 1–182,doi:10.4171/105-1/1,ISBN978-3-03719-105-7,OCLC776505866
Lambert, Johann Heinrich (2014), Papadopoulos, Athanase; Théret, Guillaume (eds.),La théorie des lignes parallèles de Johann Heinrich Lambert, Paris: Librairie Albert Blanchard,ISBN978-2-85367-266-5
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