Geometry

The foregoing is a summary of Euclidean geometry, based on Euclid's postulates. Euclid's fifth postulate is equivalent to assuming that through a given point not on a given line, there is exactly one line parallel to the given line. When one assumes that there is no such line, elliptical geometry emerges. When one assumes that there is more than one such line, the result is hyperbolic geometry. These geometries are called "non-Euclidean." Non-Euclidean geometries are as correct and consistent as Euclidean, but describe special spaces. Geometry can also be extended to more than three dimensions. Other special geometries include projective geometry, affine geometry, and topology.