Non-Euclidean geometry
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Geometers |
In
Principles
The essential difference between the metric geometries is the nature of
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 the same plane):
- In Euclidean geometry, the lines remain at a constant distance 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 "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 intersect.
History
Background
Euclidean geometry, named after the Greek mathematician Euclid, 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 wrote Elements. In the Elements, 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 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. Regardless of the form of the postulate, however, it consistently appears more complicated than Euclid's other postulates:
1. To draw a straight line from any point to any point.
2. To produce [extend] a finite straight line continuously in a straight line.
3. To describe a circle with any centre and distance [radius].
4. That all right angles are equal to one another.
For at least a thousand years,
The theorems of Ibn al-Haytham, Khayyam and al-Tusi on
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 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 now known as 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.[8]
Discovery of non-Euclidean geometry
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 law Ferdinand Karl Schweikart[9] had the germinal ideas of non-Euclidean geometry worked out, but neither published any results. Schweikart's nephew Franz 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 the
By formulating the geometry in terms of a curvature tensor, 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.
Terminology
It was Gauss who coined the term "non-Euclidean geometry".[13] He was referring to his own work, which today we call hyperbolic geometry or Lobachevskian geometry. Several modern authors still use the generic term non-Euclidean geometry to mean hyperbolic geometry.[14]
Arthur Cayley noted that distance between points inside a conic could be defined in terms of logarithm and the projective cross-ratio function. The method has become called the Cayley–Klein metric because Felix 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 geometry parabolic, 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 of Riemannian geometry.
Axiomatic basis of non-Euclidean 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 of undefined 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, while Birkhoff, 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, produces absolute geometry. As the first 28 propositions of Euclid (in The 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 its negation. Negating the Playfair'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 line l not passing through P, there exist two lines through P, which do not meet l" and keeping all the other axioms, yields hyperbolic 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 line l not passing through P, all the lines through P meet l", does not give a consistent set of axioms. This follows since parallel lines exist in absolute geometry,Riemann's elliptic geometryemerges as the most natural geometry satisfying this axiom.
Models
Models of non-Euclidean geometry are
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). The pseudosphere has the appropriate curvature to model hyperbolic geometry.
Elliptic geometry
The simplest model for
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
In the hyperbolic model, within a two-dimensional plane, 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.
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.
Three-dimensional non-Euclidean geometry
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 completely anisotropic (i.e. every direction behaves differently).
Uncommon properties
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 of absolute geometry (also called neutral 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:
- A obtuse if the geometry is elliptic. Consequently, rectanglesexist (a statement equivalent to the parallel postulate) only in Euclidean geometry.
- A Saccheri quadrilateral is a quadrilateral with two sides of equal length, both perpendicular to a side called the base. The other two angles of a Saccheri quadrilateral are called the summit 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. The defect 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.
Importance
Before the models of a non-Euclidean plane were presented by Beltrami, Klein, and Poincaré, 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.
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 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 a
The existence of non-Euclidean geometries impacted the intellectual life of
Planar algebras
In
The Euclidean plane corresponds to the case ε2 = −1 since the modulus of z is given by
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 ε2 = +1, then z is a split-complex number and conventionally j replaces epsilon. Then
and {z | z z* = 1} is the unit hyperbola.
When ε2 = 0, then z is a dual number.[28]
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.[29]
Kinematic geometries
Hyperbolic geometry found an application in
The non-Euclidean planar algebras support kinematic geometries in the plane. For instance, the
Kinematic study makes use of the dual numbers to represent the classical description of motion in
With dual numbers the mapping is [32]
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.[33][34]
Fiction
Non-Euclidean geometry often makes appearances in works of science fiction and fantasy.
- In 1895, 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. 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]
- The main character in Robert Pirsig's Zen and the Art of Motorcycle Maintenance mentioned Riemannian geometryon multiple occasions.
- In The Brothers Karamazov, Dostoevsky discusses non-Euclidean geometry through his character Ivan.
- Christopher Priest's novel 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.
- Zeno Rogue's hyperbolic 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]
- In the faster-than-light traveland communications is possible through the use of Hsieh Ho's Polydimensional Non-Euclidean Geometry, published sometime in the middle of the 22nd century.
- In Ian Stewart's Flatterland the protagonist Victoria Line visits all kinds of non-Euclidean worlds.
See also
- Hyperbolic space
- Lénárt sphere
- Projective geometry
- Non-Euclidean surface growth
- Parallel (geometry)#In non-Euclidean geometry
- Spherical geometry#Relation to similar geometries
Notes
- ^ Eder, Michelle (2000), Views of Euclid's Parallel Postulate in Ancient Greece and in Medieval Islam, Rutgers University, retrieved 2008-01-23
- Saccheri's studies of the theory of parallel lines."
- ISBN 0-415-12411-5
- ^ 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."
- ^ Boris A. Rosenfeld and Adolf P. Youschkevitch (1996), "Geometry", in Roshdi Rashed, ed., Encyclopedia of the History of Arabic Science, vol. 2, pp. 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."
- ^ MacTutor's Giovanni Girolamo Saccheri
- ^ O'Connor, J.J.; Robertson, E.F. "Johann Heinrich Lambert". Retrieved 16 September 2011.
- ^ 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.
- English translations of Schweikart's letter and Gauss's reply to Gerling appear in:Course notes: "Gauss and non-Euclidean geometry", University of Waterloo, Ontario, Canada; see especially pages 10 and 11.
- Letters by Schweikart and the writings of his nephew Franz 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.
- ^ Bonola, R. (1912). Non-Euclidean geometry: A critical and historical study of its development. Chicago: Open Court.
- ^ 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 to Faber (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.
- ^ For example:
Kulczycki, Stefan (1961). Non-Euclidean Geometry. Pergamon. p. 53.
Iwasawa, Kenkichi (1993). Algebraic Functions. American Mathematical Society. p. 140.ISBN 9780821845950. - ^ F. Klein, Über die sogenannte nichteuklidische Geometrie, Mathematische Annalen, 4(1871).
- ^ The Euclidean plane is still referred to as parabolic in the context of conformal geometry: see Uniformization theorem.
- ^ for instance, Manning 1963 and Yaglom 1968
- ^ a 21st axiom appeared in the French translation of Hilbert's Grundlagen der Geometrie according to Smart 1997, p. 416
- ^ (Smart 1997, p. 366)
- ^ while only two lines are postulated, it is easily shown that there must be an infinite number of such lines.
- ^ Book I Proposition 27 of Euclid's Elements
- ISBN 0-691-08304-5(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.
- ^ see Trudeau 1987, pp. vii–viii
- ISBN 978-0-671-62818-5. Author attributes this quote to another mathematician, William Kingdon Clifford.
- ^ (Richards 1988)
- ^ 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
- ^ Hermann Minkowski (1908–9). "Space and Time" (Wikisource).
- ^ Scott Walter (1999) Non-Euclidean Style of Special Relativity
- ISBN 0-387-90332-1
- 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 Sciences48:387–507
- ^ Synthetic Spacetime, a digest of the axioms used, and theorems proved, by Wilson and Lewis. Archived by WebCite
- ^ "The Call of Cthulhu".
- ^ "HyperRogue website".
References
- A'Campo, Norbert and Papadopoulos, Athanase, (2012) Notes on hyperbolic geometry, in: Strasbourg Master class on Geometry, pp. 1–182, IRMA Lectures in Mathematics and Theoretical Physics, vol. 18, Zürich: European Mathematical Society (EMS), 461 pages, .
- 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
- ISBN 978-1-4351-2348-9
- ISBN 0-88385-522-4.
- Faber, Richard L. (1983), Foundations of Euclidean and Non-Euclidean Geometry, New York: Marcel Dekker, ISBN 0-8247-1748-1
- Jeremy Gray (1989) Ideas of Space: Euclidean, Non-Euclidean, and Relativistic, 2nd edition, Clarendon Press.
- ISBN 0-7167-9948-0
- Morris Kline (1972) Mathematical Thought from Ancient to Modern Times, Chapter 36 Non-Euclidean Geometry, pp 861–81, Oxford University Press.
- ISBN 9789814340489.
- 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.
- ISBN 0-12-587445-6
- Smart, James R. (1997), Modern Geometries (5th Ed.), Pacific Grove: Brooks/Cole, ISBN 0-534-35188-3
- ISBN 0-7382-0675-X(softcover)
- ISBN 0-8218-0529-0.
- Trudeau, Richard J. (1987), The Non-Euclidean Revolution, Boston: Birkhauser, ISBN 0-8176-3311-1
- ISBN 978-2-85367-266-5
External links
- Media related to Non-Euclidean geometry at Wikimedia Commons
- Roberto Bonola (1912) Non-Euclidean Geometry, Open Court, Chicago.
- MacTutor Archive article on non-Euclidean geometry
- Non-euclidean geometry at PlanetMath.
- Non-Euclidean geometries from Encyclopedia of Math of European Mathematical Society and Springer Science+Business Media
- Synthetic Spacetime, a digest of the axioms used, and theorems proved, by Wilson and Lewis. Archived by WebCite.