Non-Euclidean geometry

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

Non-Euclidean geometry is the term used to refer to two specific geometries which are, loosely speaking, obtained by negating the Euclidean parallel postulate, namely hyperbolic and elliptic geometry. This is one term which, for historical reasons, has a meaning in mathematics which is much narrower than it appears to have in the general English language. There are a great many geometries which are not Euclidean geometry, but only these two are referred to as the non-Euclidean geometries.

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 and a point A, which is not on , there is exactly one line through A that does not intersect . In hyperbolic geometry, by contrast, there are infinitely many lines through A not intersecting , while in elliptic geometry, any line through A intersects (see the entries on hyperbolic geometry, elliptic geometry, and absolute geometry for more information).

Another way to describe the differences between these geometries is to 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 even if extended to infinity, 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.



Early 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 at least a thousand 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, 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 the Elements."[4][5] His work was published in Rome 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 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, but did not realize it.

In 1766 Johann 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 that today we call a Lambert 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.

Creation of non-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. Gauss mentioned to Bolyai's father, when shown the younger Bolyai's work, that he had developed such a geometry about 20 years before, 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 of Riemannian geometry, discussing in particular the ideas now called manifolds, Riemannian metric, and curvature. He constructed an infinite family of geometries which are not Euclidean by giving a formula for a family of Riemannian metrics on the unit ball in Euclidean space. The simplest of these is called elliptic geometry and it is considered to be a non-Euclidean geometry due to its lack of parallel lines.[8]

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°.


It was Gauss who coined the term "non-euclidean geometry".[9] He was referring to his own work which today we call hyperbolic geometry. Several modern authors still consider "non-euclidean geometry" and "hyperbolic geometry" to be synonyms. In 1871, Felix Klein, by adapting a metric discussed by Arthur Cayley in 1852, was able to bring metric properties into a projective setting and was therefore able to unify the treatments of hyperbolic, euclidean and elliptic geometry under the umbrella of projective geometry.[10] Klein is responsible for the terms "hyperbolic" and "elliptic" (in his system he called Euclidean geometry "parabolic", a term which has not survived the test of time). 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.[11] In other disciplines, most notably mathematical physics, the term "non-euclidean" is often taken to mean not Euclidean.

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 and a point A, which is not on , all lines through A will intersect .

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 and only if Euclidean geometry was. (The reverse implication follows from the horosphere model of Euclidean geometry.)

In the hyperbolic model, within a two-dimensional plane, for any given line and a point A, which is not on , there are infinitely many lines through A that do not intersect .


Non-Euclidean geometry is an example of a paradigm shift in the history of science.[12] Before the models of a non-Euclidean plane were presented by Beltrami, Cayley, and Klein, Euclidean geometry stood unchallenged as the mathematical model of space. Furthermore, since the substance of the subject in synthetic geometry was a chief exhibit of rationality, the Euclidean point of view represented absolute authority. Non-Euclidean geometry, though assimilated by learned investigators, continues to be suspect for those not having exposure to hyperbolic and elliptical concepts.

The discovery of the non-Euclidean geometries had a ripple effect which went far beyond the boundaries of mathematics and science. The philosopher Immanuel 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 mathematics that was a result of this paradigm shift.[13]

The existence of non-Euclidean geometries impacted the "intellectual life" of Victorian England in many ways[14] and in particular was one of the leading factors that caused a re-examination of the teaching of geometry based on Euclid's Elements. This curriculum issue was hotly debated at the time and even produced a play, Euclid and his Modern Rivals, written by the author of Alice in Wonderland.[15]

Planar algebras

In analytic geometry a plane is described with Cartesian coordinates : C = {(x,y) : x, y in R}. The points are sometimes identified with hypercomplex numbers z = x + y ε where the square of ε is in {−1, 0, +1}. The Euclidean plane corresponds to the case ε2 = −1 since the modulus of z is given by

z z^\ast = (x + y \epsilon) (x - y \epsilon) = x^2 + y^2

and this quantity is the square of the Euclidean distance between z and the origin. For instance, {z : z z* = 1} is the unit circle.

For planar algebra, non-Euclidean geometry arises in the other cases. When \epsilon ^2 = +1, then z is a split-complex number and conventionally j replaces epsilon. Then

z z^\ast = (x + yj) (x - yj) = x^2 - y^2 \!

and {z : z z* = 1} is the unit hyperbola.

When \epsilon ^2 = 0, then z is a dual number.[16]

This approach to non-Euclidean geometry explains the non-Euclidean angles: the parameters of slope in the dual number plane and hyperbolic angle in the split-complex plane correspond to angle in Euclidean geometry. Indeed, they each arise in polar decomposition of a complex number z.[17]

Kinematic geometries

Hyperbolic geometry found an application in kinematics with the cosmology introduced by Herman Minkowski in 1908. Minkowski introduced terms like worldline and proper time into mathematical physics. He realized that the submanifold, of events one moment of proper time into the future, could be considered a hyperbolic space of three dimensions.[18] [19] Already in the 1890s Alexander Macfarlane was charting this submanifold through his Algebra of Physics and hyperbolic quaternions, though Macfarlane didn’t use cosmological language as Minkowski did in 1908. The relevant structure is now called the hyperboloid model of hyperbolic geometry.

The non-Euclidean planar algebras support kinematic geometries in the plane. For instance, the split-complex number z = eaj can represent a spacetime event one moment into the future of a frame of reference of rapidity a. Furthermore, multiplication by z amounts to a Lorentz boost mapping the frame with rapidity zero to that with rapidity a.

Kinematic study makes use of the dual numbers z = x + y \epsilon, \quad \epsilon^2 = 0, to represent the classical description of motion in absolute time and space: The equations x^\prime = x + vt,\quad t^\prime = t are equivalent to a shear mapping in linear algebra:

\begin{pmatrix}x' \\ t' \end{pmatrix} = \begin{pmatrix}1 & v \\ 0 & 1 \end{pmatrix}\begin{pmatrix}x \\ t \end{pmatrix}.

With dual numbers the mapping is t^\prime + x^\prime \epsilon = (1 + v \epsilon)(t + x \epsilon) = t + (x + vt)\epsilon.[20]

Another view of special relativity as a non-Euclidean geometry was advanced by E. B. Wilson and Gilbert Lewis in Proceedings of the American Academy of Arts and Sciences in 1912. They revamped the analytic geometry implicit in the split-complex number algebra into synthetic geometry of premises and deductions.[21][22]


Non-Euclidean geometry often makes appearances in works of science fiction and fantasy.

In 1895 H. G. Wells published the short story The Remarkable Case of Davidson’s Eyes. To appreciate this story one should know how antipodal 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 off Antipodes Island.

Non-Euclidean geometry is sometimes connected 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: In Lovecraft's Cthulhu Mythos, the sunken city of R'lyeh is characterized by its non-Euclidean geometry. This is said to be a profoundly unsettling sight, often to the point of driving those who look upon it insane. The main character in Robert Pirsig's Zen and the Art of Motorcycle Maintenance mentioned Riemannian Geometry on multiple occasions.

In The Brothers Karamazov, Dostoevsky discusses non-Euclidean geometry through his main character Ivan.

Christopher Priest's The Inverted World describes the struggle of living on a planet with the form of a rotating pseudosphere.

Robert Heinlein's The Number of the Beast utilizes non-Euclidean geometry to explain instantaneous transport through space and time and between parallel and fictional universes.

Alexander Bruce's Antichamber uses non-Euclidean geometry to create a brilliant, minimal, Escher-like world, where geometry and space follow unfamiliar rules.

In the Renegade Legion science fiction setting for FASA's wargame, role-playing-game and fiction, faster-than-light travel and communications is possible through the use of Hsieh Ho's Polydimensional Non-Euclidean Geometry, published sometime in the middle of the twenty-second century.

See also


  1. ^ Eder, Michelle (2000), Views of Euclid's Parallel Postulate in Ancient Greece and in Medieval Islam, Rutgers University,, retrieved 2008-01-23 
  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 0-415-12411-5
  4. ^ a b Victor J. Katz (1998), History of Mathematics: An Introduction, p. 270–271, Addison–Wesley, ISBN 0-321-01618-1:

    "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."
  6. ^ MacTutor's Giovanni Girolamo Saccheri
  7. ^ O'Connor, J.J.; Robertson, E.F. "Johann Heinrich Lambert". Retrieved 16 September 2011. 
  8. ^ However, other axioms besides the parallel postulate must be changed in order to make this a feasible geometry.
  9. ^ Felix Klein, Elementary Mathematics from an Advanced Standpoint: Geometry, Dover, 1948 (reprint of English translation of 3rd Edition, 1940. First edition in German, 1908) pg. 176
  10. ^ F. Klein, Über die sogenannte nichteuklidische Geometrie, Mathematische Annalen, 4(1871).
  11. ^ for instance, Manning 1963 and Yaglom 1968
  12. ^ see (Trudeau 1987)
  13. ^ 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.
  14. ^ (Richards 1988)
  15. ^ Lewis Carroll, see reference below.
  16. ^ Yaglom 1968
  17. ^ Richard C. Tolman (2004) Theory of Relativity of Motion, page 194, §180 Non-Euclidean angle, §181 Kinematical interpretation of angle in terms of velocity
  18. ^ Herman Minkowski (1908–9). "Space and Time" (Wikisource).
  19. ^ Scott Walter (1999) Non-Euclidean Style of Special Relativity
  20. ^ Isaak Yaglom (1979) A simple non-Euclidean geometry and its physical basis : an elementary account of Galilean geometry and the Galilean principle of relativity, Springer ISBN 0387903321
  21. ^ Edwin B. Wilson & Gilbert N. Lewis (1912) "The Space-time Manifold of Relativity. The Non-Euclidean Geometry of Mechanics and Electromagnetics" Proceedings of the American Academy of Arts and Sciences 48:387–507
  22. ^ Synthetic Spacetime, a digest of the axioms used, and theorems proved, by Wilson and Lewis. Archived by WebCite


  • 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
  • Blumenthal, Leonard M. (1980), A Modern View of Geometry, New York: Dover, ISBN 0-486-63962-2 
  • Carroll, Lewis Euclid and His Modern Rivals, New York: Barnes and Noble, 2009 (reprint) ISBN 978-1-4351-2348-9
  • H. S. M. Coxeter (1942) Non-Euclidean Geometry, University of Toronto Press, reissued 1998 by Mathematical Association of America, ISBN 0-88385-522-4 .
  • Jeremy Gray (1989) Ideas of Space: Euclidean, Non-Euclidean, and Relativistic, 2nd edition, Clarendon Press.
  • Greenberg, Marvin Jay Euclidean and Non-Euclidean Geometries: Development and History, 4th ed., New York: W. H. Freeman, 2007. ISBN 0-7167-9948-0
  • Nikolai Lobachevsky (2010) Pangeometry, Translator and Editor: A. Papadopoulos, Heritage of European Mathematics Series, Vol. 4, European Mathematical Society.
  • Manning, Henry Parker (1963), Introductory Non-Euclidean Geometry, New York: Dover 
  • Meschkowski, Herbert (1964). Noneuclidean Geometry. New York: Academic Press. 
  • Milnor, John W. (1982) Hyperbolic geometry: The first 150 years, Bull. Amer. Math. Soc. (N.S.) Volume 6, Number 1, pp. 9–24.
  • Richards, Joan L. (1988), Mathematical Visions: The Pursuit of Geometry in Victorian England, Boston: Academic Press, ISBN 0-12-587445-6 
  • Stewart, Ian Flatterland. New York: Perseus Publishing, 2001. ISBN 0-7382-0675-X (softcover)
  • John Stillwell (1996) Sources of Hyperbolic Geometry, American Mathematical Society ISBN 0-8218-0529-0 .
  • Trudeau, Richard J. (1987), The Non-Euclidean Revolution, Boston: Birkhauser, ISBN 0-8176-3311-1 
  • 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.

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