18 Facts About Non-Euclidean geometry

Euclidean Non-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.

All of these early attempts made at trying to formulate non-Euclidean geometry provided flawed proofs of the parallel postulate, containing assumptions that were essentially equivalent to the parallel postulate.

Non-Euclidean geometry finally reached a point where he believed that his results demonstrated the impossibility of hyperbolic geometry.

Non-Euclidean geometry's claim seems to have been based on Euclidean presuppositions, because no logical contradiction was present.

Non-Euclidean geometry worked with a figure now known as a Lambert quadrilateral, a quadrilateral with three right angles .

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Non-Euclidean geometry 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.

Non-Euclidean geometry 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.

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.

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 Non-Euclidean geometry, discussing in particular the ideas now called manifolds, Riemannian metric, and curvature.

Non-Euclidean geometry constructed an infinite family of non-Euclidean geometries by giving a formula for a family of Riemannian metrics on the unit ball in Euclidean space.

Non-Euclidean geometry was referring to his own work, which today we call hyperbolic geometry.

Simplest model for elliptic Non-Euclidean geometry is a sphere, where lines are "great circles", and points opposite each other are identified .

Furthermore, since the substance of the subject in synthetic Non-Euclidean geometry was a chief exhibit of rationality, the Euclidean point of view represented absolute authority.

Non-Euclidean geometry is an example of a scientific revolution in the history of science, in which mathematicians and scientists changed the way they viewed their subjects.

Hyperbolic Non-Euclidean geometry found an application in kinematics with the physical cosmology introduced by Hermann Minkowski in 1908.

Non-Euclidean geometry realized that the submanifold, of events one moment of proper time into the future, could be considered a hyperbolic space of three dimensions.

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