In mathematics, non-Euclidean geometry is a small set of geometries based on axioms closely related to those specifying Euclidean geometry. As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean geometry arises when either the metric requirement is relaxed, or the parallel postulate is set aside. In the latter case one obtains hyperbolic geometry and elliptic geometry, the 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. 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 ℓ and a point A, which is not on ℓ, there is exactly one line through A that does not intersect ℓ. In hyperbolic geometry, by contrast, there are infinitely many lines through A not intersecting ℓ, while in elliptic geometry, any line through A intersects ℓ (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 intersect.
Other articles related to "geometry":
... of parallels and his proof of properties of figures in non-Euclidean geometries contributed to the eventual development of non-Euclidean geometry ... to prove the Parallel Postulate, they set out to develop a self-consistent geometry in which that postulate was false ... In this they were successful, thus creating the first non-Euclidean geometry ...
... In non-Euclidean geometry, squares are more generally polygons with 4 equal sides and equal angles ... In spherical geometry, a square is a polygon whose edges are great circle arcs of equal distance, which meet at equal angles ... Unlike the square of plane geometry, the angles of such a square are larger than a right angle ...
... Elements contained five postulates that form the basis for Euclidean geometry ... Ivanovich Lobachevsky separately published treatises on a type of geometry that does not include the parallel postulate, called hyperbolic geometry ... In this geometry, an infinite number of parallel lines pass through the point P ...
... Non-Euclidean geometry often makes appearances in works of science fiction and fantasy ... Non-Euclidean geometry is sometimes connected with the influence of the 20th century horror fiction writer H ... 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 ...
Famous quotes containing the word geometry:
“The geometry of landscape and situation seems to create its own systems of time, the sense of a dynamic element which is cinematising the events of the canvas, translating a posture or ceremony into dynamic terms. The greatest movie of the 20th century is the Mona Lisa, just as the greatest novel is Grays Anatomy.”
—J.G. (James Graham)