# Non-Euclidean geometry

 "Gaston Bachelard showed that new theories integrate old theories in new paradigms, changing the sense of concepts (for instance, the concept of mass, used by Newton and Einstein in two different senses). Thus, non-Euclidean geometry did not contradict Euclidean geometry, but integrated it into a larger framework." --Sholem Stein "He said that the geometry of the dream-place he saw was abnormal, non-Euclidean, and loathsomely redolent of spheres and dimensions apart from ours." --"The Call of Cthulhu", H. P. Lovecraft
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In mathematics, non-Euclidean geometry describes hyperbolic and elliptic geometry, which are contrasted with Euclidean geometry. The essential difference between Euclidean and non-Euclidean geometry is the nature of parallel lines. Euclid's 5th 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 pair of lines intersect. (See the entries on hyperbolic geometry and elliptic geometry for more information.)

Another way to describe the differences between these geometries is as follows: 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, 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 eventually intersect.

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