Non-Euclidean geometry is a form of geometry that rejects Euclid's fifth postulate that parallel lines extend to infinity and never intersect. There are two main forms of non-Euclidean geometry, hyperbolic and elliptic. The former allows parallel lines to intersect, the latter does not have parallel lines. In Euclidean geometry because parallel lines go on to infinity and never intersect the sum of angles in triangles is always exactly 180 degrees, in hyperbolic geometry because parallel lines do intersect triangles are always less than 180 degrees. In elliptic geometry triangles are always greater than 180 degrees because there are no parallel lines. The roots of Non-Euclidean geometry were in works by Gauss, and his pupil Riemann. Later nineteenth century mathematicians such as Janos Bolyai and Nikolai I. Lobachevsky developed this considerably.
A geometry is defined as the pure mathematics of points and lines and curves and surfaces. It works by defining a set of axioms which clarify how points and lines are constituted, for example: two unique points define a straight line, and deriving from these results. Some geometries such as the Fano Plane many have seven points or fewer. The simplest geometry is the empty set.