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In mathematicsnon-Euclidean geometry consists of two geometries based on axioms closely related to those specifying Euclidean geometry. As Euclidean geometry lies at the intersection of metric geometry and affine geometrynon-Euclidean geometry arises when either the metric requirement is relaxed, or the parallel postulate is replaced with an alternative one. In the latter case one obtains hyperbolic geometry and elliptic geometrythe traditional non-Euclidean geometries.
When the metric requirement is relaxed, then there are affine planes associated with the planar algebras which give rise to kinematic geometries that have also been called non-Euclidean geometry. The essential difference between the metric geometries is the nature of parallel lines. 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:.
Euclidean geometrynamed after the Greek mathematician Euclidincludes 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’s work Elements was written. In the ElementsEuclid 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. Regardless of the form of the postulate, however, it consistently appears to be more complicated than Euclid’s other postulates:.
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. The theorems of Ibn al-Haytham, Khayyam and al-Tusi on quadrilateralsincluding the Lambert quadrilateral and Saccheri quadrilateralwere “the first few theorems of the hyperbolic and the elliptic geometries. These early attempts at challenging the fifth postulate had a considerable influence on its development among later European geometers, including WiteloLevi ben GersonAlfonsoJohn Wallis and Saccheri.
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: Another example is al-Tusi’s son, Sadr al-Din sometimes known as “Pseudo-Tusi”who wrote a book on the subject inbased on al-Tusi’s later thoughts, which presented another hypothesis equivalent to the parallel postulate.
Giordano Vitalein his book Euclide restituo, 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 Flawspublished inSaccheri 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 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 quadrilaterala 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. At this time it was heometras 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. CircaCarl Friedrich Gauss and independently geomettasthe German professor of law Ferdinand Karl Schweikart  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 andyet while admitting the internal consistency of hyperbolic geometry, he still believed in the special role of Euclidean 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 several years before,  though he did not publish.
Bolyai ends his work eucildianas 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 Riemannin a famous lecture infounded the field of Riemannian geometrydiscussing in particular the ideas now called manifoldsRiemannian metricand 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. By formulating the geometry euclidiznas terms of a curvature tensorRiemann allowed non-Euclidean geometry to be applied to higher dimensions.
It was Gauss who coined the term “non-Euclidean geometry”. Several modern authors still consider “non-Euclidean geometry” and “hyperbolic geometry” to be synonyms. 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  in and 73 and later in book form. The Cayley-Klein metrics provided working models of bo 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 which generally fell out of use .
His influence eucliidianas 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. Euclidean geometry can be axiomatically described in several ways. Unfortunately, Euclid’s original system of five postulates axioms is not one of these as his proofs relied on several unstated assumptions which should also have been taken as axioms.
Hilbert’s system consisting of 20 axioms  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.
In all approaches, however, there is an axiom which is logically equivalent to Euclid’s fifth postulate, the parallel postulate.
Geometrías no euclidianas by carlos rodriguez on Prezi
Hilbert uses the Playfair axiom form, while Birkhofffor instance, uses the axiom which says that “there exists a pair of similar but not congruent triangles. 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.
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 Two dimensional 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 globeand points opposite each other called antipodal points are identified considered to be the same. This is also one of the standard models of the real 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.
Even after the work of Lobachevsky, Gauss, and Bolyai, the question remained: The model for hyperbolic geometry was answered by Eugenio Beltramiinwho 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 which models 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 these models the concepts of non-Euclidean geometries are being represented by Euclidean objects in a Euclidean setting.
This introduces a perceptual distortion wherein the straight lines of the non-Euclidean geometry are being represented by Euclidean curves which visually bend. This “bending” is not a property of the non-Euclidean lines, only an artifice of the way they are being represented. In three dimensions, there are eight models of geometries. 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:. 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. 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. Non-Euclidean geometry is an example of a scientific revolution in the history of sciencein which mathematicians and scientists changed the way they viewed their subjects.
The existence of non-Euclidean geometries impacted the intellectual life of Victorian England in many ways  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 was even the subject of a book, Euclid and his Modern Rivalswritten by Charles Lutwidge Dodgson — better known as Lewis Carrollthe author of Alice in Wonderland. In analytic geometry a plane is described with Cartesian coordinates: For planar algebra, non-Euclidean geometry arises in the other cases. This approach to non-Euclidean geometry explains the non-Euclidean angles: Indeed, they each arise in polar decomposition of a complex number z.
Non-Euclidean geometry – Wikipedia
Hyperbolic geometry found an application in kinematics with the physical cosmology introduced by Hermann Minkowski in Minkowski introduced terms like worldline and proper time into mathematical physics. He realized that the submanifoldof events one moment of proper time into the future, could be considered a hyperbolic space of three dimensions. The relevant structure is now called the hyperboloid model of hyperbolic geometry.
The non-Euclidean planar algebras support kinematic geometries in the plane. Furthermore, multiplication by z amounts to a Geomettras boost mapping the frame with rapidity zero to that with rapidity a. Another view of special relativity as a non-Euclidean geometry was advanced by E.
They revamped the analytic geometry implicit in the split-complex number algebra into synthetic geometry geomwtras premises and deductions.
Non-Euclidean geometry often makes appearances in works of science fiction and fantasy.
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 beometras 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 proofs put forward in the fourteenth century by the Jewish scholar Levi ben Gersonwho lived in southern France, and by the above-mentioned Alfonso from Spain directly border on Ibn al-Haytham’s demonstration.