In mathematics, non-
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. Contents 1 History 1.1 Background 1.2 Discovery of non-Euclidean geometry 1.3 Terminology 2 Axiomatic basis of non-Euclidean geometry 3 Models of non-Euclidean geometry 3.1 Elliptic geometry 3.2 Hyperbolic geometry 3.3 Three-dimensional non-Euclidean geometry 4 Uncommon properties 5 Importance 6 Planar algebras 7 Kinematic geometries 8 Fiction 9 See also 10 Notes 11 References 12 External links History[edit]
See also:
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 to be 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, 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
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.[18]
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,[19] but this statement says
that there are no parallel lines. This problem was known (in a
different guise) to Khayyam,
Models of non-Euclidean geometry[edit] Further information: Models of non-Euclidean geometry 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
Two dimensional
Euclidean and non-Euclidean geometries naturally have many similar properties, namely those which do not depend upon the nature of parallelism. This commonality is the subject of absolute geometry (also called neutral geometry). However, the properties which distinguish one geometry from the others are the ones which 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
Importance[edit]
Before the models of a non-Euclidean plane were presented by Beltrami,
Klein, and Poincaré,
z z ∗ = ( x + y ϵ ) ( x − y ϵ ) = x 2 + y 2 displaystyle zz^ ast =(x+yepsilon )(x-yepsilon )=x^ 2 +y^ 2 and this quantity is the square of the
z z ∗ = ( x + y j ) ( x − y j ) = x 2 − y 2 displaystyle zz^ ast =(x+ymathbf j )(x-ymathbf j )=x^ 2 -y^ 2 ! and z z z* = 1 is the unit hyperbola.
When ε2 = 0, then z is a dual number.[26]
This approach to non-
z = x + y ϵ , ϵ 2 = 0 , displaystyle z=x+yepsilon ,quad epsilon ^ 2 =0, to represent the classical description of motion in absolute time and space: The equations x ′ = x + v t , t ′ = t displaystyle x^ prime =x+vt,quad t^ prime =t are equivalent to a shear mapping in linear algebra: ( x ′ t ′ ) = ( 1 v 0 1 ) ( x t ) . displaystyle begin pmatrix x'\t'end pmatrix = begin pmatrix 1&v\0&1end pmatrix begin pmatrix x\tend pmatrix . With dual numbers the mapping is t ′ + x ′ ϵ = ( 1 + v ϵ ) ( t + x ϵ ) = t + ( x + v t ) ϵ . displaystyle t^ prime +x^ prime epsilon =(1+vepsilon )(t+xepsilon )=t+(x+vt)epsilon . [30]
Another view of special relativity as a non-
In 1895
See also[edit] Hyperbolic space Lénárt sphere Projective geometry Notes[edit] ^ Eder, Michelle (2000), Views of Euclid's Parallel Postulate in Ancient Greece and in Medieval Islam, Rutgers University, retrieved 2008-01-23 ^ 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 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 by Witelo, the Polish scientists of the thirteenth century, while
revising Ibn al-Haytham's
^ 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 ^ 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
^ 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." ^ 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), volume 8, pages 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 (1794-1874), 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, pg. 162). In his 1824 letter to Taurinus (Faber 1983, pg.
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, pg. 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
in order to make this a feasible geometry.
^ Felix Klein, Elementary
References[edit] A'Campo, Norbert and Papadopoulos, Athanase, (2012) Notes on
hyperbolic geometry, in: Strasbourg Master class on Geometry,
pp. 1–182, IRMA Lectures in
External links[edit] Wikiquote has quotations related to: Non-Euclidean geometry Roberto Bonola (1912) Non-Euclidean Geometry, Open Court, Chicago. MacTutor Archive article on non-Euclidean geometry Non-euclidean geometry at PlanetMath.org. 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. Authority control LCCN: sh85054155 GND: 4042073-5 BNF: cb119798569 (d |