On Axioms and Postulates (optional material)
Euclid's "fifth postulate" (or "parallel postulate") asserts that through any point not on a line, there exists one and only one line parallel to the given line (in the plane determined by the given point and line). Lobachevsky (1793-1856) developed an alternative "hyperbolic" geometry, based on the assumption that there exists not just one, but multiple parallel lines through the given point. "Elliptic" geometry, on the other hand, assumes that there exist no such parallels. Riemann (1826-1866), whose geometric concepts were incorporated into Einstein's general relativity theory, showed how a more general geometry could combine hyperbolic and elliptic aspects.

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